Hindawi Publishing CorporationISRNMathematical PhysicsVolume 2013 Article ID 184325 7 pageshttpdxdoiorg1011552013184325

Research ArticleOn the Problem of Electromagnetic Waves Propagating alonga Nonlinear Inhomogeneous Cylindrical Waveguide

Yury G Smirnov and Dmitry V Valovik

Department of Mathematics and Supercomputing Penza State University Krasnaya Street 40 Penza 440026 Russia

Correspondence should be addressed to Dmitry V Valovik dvalovikmailru

Received 23 April 2013 Accepted 6 June 2013

Academic Editors G Cleaver J Garecki F Sugino and G F Torres del Castillo

Copyright copy 2013 Y G Smirnov and D V Valovik This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Electromagnetic TEwave propagation in an inhomogeneous nonlinear cylindrical waveguide is consideredThe permittivity insidethe waveguide is described by the Kerr law Inhomogeneity of the waveguide is modeled by a nonconstant term in the Kerr lawPhysical problem is reduced to a nonlinear eigenvalue problem for ordinary differential equations Existence of propagating wavesis proved with the help of fixed point theorem and contracting mapping method For numerical solution an iteration methodis suggested and its convergence is proved Existence of eigenvalues of the problem (propagation constants) is proved and theirlocalization is found Conditions of k waves existence are found

1 Introduction

Electromagnetic wave propagation in linear (homogeneousand inhomogeneous) waveguide plane layers and cylindricalwaveguides with circular cross section is of particular interestin linear optics (see eg [1 2]) In nonlinear optics waveg-uides (plane and cylindrical) filled with nonlinear mediumhave been the focus of a number of studies [3ndash11] Howevermany of researches are devoted to study homogeneous nonlin-ear waveguides [6ndash11]

Problems of electromagnetic wave propagation in non-linear waveguides (plane and cylindrical) lead to nonlinearboundary and transmission eigenvalue problems for ordinarydifferential equations Eigenvalues in these problems corre-spond to propagation constants of the waveguides In theseproblems differential equations depend nonlinearly either onsought-for functions and the spectral parameter Boundaryandor transmission conditions depend nonlinearly on thespectral parameter The main goal is to prove existence ofeigenvalues and determine their localization Existence andlocalization can be derived from the dispersion equation(DE) DE is an equation with respect to spectral parameterThere are two ways to obtain the DE The first one isto integrate the differential equations and obtain usingboundary andor transmission conditions the DE This way

is of very limited applicability as it is very rarely possibleto find explicit solutions of nonlinear differential equationsHowever there are some problems in which this way works(see eg [10 12 13]) The second one is a very generalapproach based on reduction of the differential equations tointegral equations using the Green function This approachwe call integral equation approachHerewe consider this verymethod Inspite of the fact that by this method the DE isfound in an implicit form it is possible to prove existence ofeigenvalues and find their localization

Electromagnetic guided waves in a cylindrical waveguidewith Kerr nonlinearity are considered in [6] It is one of thefirst studies which we know about where electromagneticwave propagation in nonlinear medium is considered ina rigorous electromagnetic statement Then there were alot of researches devoted to study Kerr nonlinearity inhomogeneous plane and cylindrical waveguides For moredetails about Kerr nonlinearity and homogeneous planeand cylindrical waveguides see the following references TEguidedwaves in a plane layer were investigated in [12 14] andadditional results were obtained in [13] TM guided waves ina plane layer were investigated in [15ndash20] TE guided wavesin a cylindrical waveguide were investigated in [21ndash23] TMguided waves in a cylindrical waveguide were investigated in[24]

2 ISRNMathematical Physics

Inmost cases it is very difficult (if at all possible) to obtainexact solutions of the equations in nonlinear waveguidingproblems However integral equation approach can help inthis case [21ndash25] In this approach a problem is reduced toan integral equation whose kernel depends on the Greenfunction of the linear part of the differential equations of theproblem Two circumstances are important for the followinganalysis First in the case of a homogeneous waveguidethis Green function can be found explicitly Second thedispersion equation of the nonlinear homogeneous case canbe written as DElin + 119879nonlin = 0 where DElin is a linearproblem term and 119879nonlin is an extra nonlinear term Here thelinear problem term is written in an explicit form Moreoverthe equation DElin = 0 is well known and examined DE forthe linear problem Its roots are also known All this allowsto prove existence of the nonlinear problem solutions at leastnear to the linear problem solutions

Here we investigate guided waves in a nonlinear inhomo-geneous cylindrical waveguide filled with Kerr medium Thewaveguide is placed in cylindrical coordinate system 119874120588120593119911where axis 119911 coincides with axis of the waveguide Inhomo-geneity is modeled by a function that depends on radius ofthe waveguide The permittivity inside the waveguide is 120576 =1205762(120588)+119886|E|2 where 120576

2(120588) is the inhomogeneity 119886 is a constant

in the Kerr law and E is complex amplitude If 1205762(120588) equiv

const we have a nonlinear homogeneous waveguide Thenonconstant term 120576

2(120588) dramatically changes the situation

In this case we cannot find explicitly the necessary Greenfunction so we investigate it in an implicit form Thedispersion equation of the nonlinear inhomogeneous casecan be also written as DElin + 119879nonlin = 0 However in thiscase the term DElin is written in an implicit form as opposedto the case of a homogeneous waveguide and its roots areunknown So at first we prove that the equation for thelinear inhomogeneous problem DElin = 0 has roots anddefine localization of the rootsThen we prove that nonlinearproblem has solutions

Integral equation approach has been already used for anonlinear inhomogeneous waveguiding problem [26] How-ever in study [26] authors apply integral equation approachin the way as they would solve the problem for a homoge-neous waveguide To be precise the authors use the Greenfunction for constant 120576

2that helps them to determine the

Green function in explicit form We pay heed that thereare no theoretical results (existence of eigenvalues and theirlocalization) in [26] We emphasize that for inhomogeneouswaveguides important and general results can be obtainedwith the method we use in this paper in which the Greenfunction has implicit form

In spite of the fact that the method here looks similar tothe method in [21ndash24] we solve radically different problemas we consider inhomogeneous nonlinear waveguide

2 Statement of the Problem

Let us consider three-dimensional space R3 with cylindricalcoordinate system 119874120588120593119911 The space is filled by isotropicmedium with constant permittivity 120576

1ge 1205760 where 120576

0is

the permittivity of free space In this medium a cylindrical

waveguide is placed The waveguide is filled by isotropicnonmagnetic medium and has cross section 119882 = (120588 120593) 1205882lt 1198772 0 le 120593 lt 2120587 and its generating line (the

waveguide axis) is parallel to the axis 119874119911 We will considerelectromagnetic waves propagating along the waveguide axisEverywhere below 120583 = 120583

0is the permeability of free space

We use Maxwellrsquos equations in the following form [27]

rot H = 120597119905D

rot E = minus120597119905B

(1)

where D = 120576E B = 120583H and 120597119905= 120597120597119905 Field (E H) is the total

fieldFrom formulae (1) we obtain

rot E = minus120597119905(120583H)

rot H = 120597119905(120576E)

(2)

Real monochromatic field (E H) in the medium can bewritten in the following form

E (120588 120593 119911 119905) = E+ (120588 120593 119911) cos120596119905 + Eminus (120588 120593 119911) sin120596119905

H (120588 120593 119911 119905) = H+ (120588 120593 119911) cos120596119905 +Hminus (120588 120593 119911) sin120596119905(3)

where 120596 is circular frequency E+ Eminus H+ and Hminus are realrequired vectors

Let us form complex amplitudes EH

E = E+ + 119894Eminus H = H+ + 119894Hminus (4)

It is clear that

E = Re E119890minus119894120596119905 H = Re H119890minus119894120596119905 (5)

where

E = (119864120588 119864120593 119864119911)

119879

H = (119867120588 119867120593 119867119911)

119879

(6)

and components in (6) depend on three spatial variablesIt is known (see eg [3 6 28]) that Kerr law in isotropic

medium for a monochromatic wave E119890minus119894120596119905 has the form 120576 =1205762+119886|E|2 where E is complex amplitude 120576

2is a constant part

of the permittivity 120576 119886 is the coefficient of nonlinearityWe obtain that in this case dependence of Maxwellrsquos

equations on 119905 is the same as in the case of constant 120576 insidethe waveguideThis allows us to writeMaxwellrsquos equations (2)in the form

rot (E119890minus119894120596119905) = 119894120596120583H119890minus119894120596119905

rot (H119890minus119894120596119905) = minus119894120596120576E119890minus119894120596119905(7)

Complex amplitudes (4) satisfy the Maxwell equations

rotE = 119894120596120583H

rotH = minus119894120596120576E(8)

the continuity condition for the tangential components onthe media interfaces (on the boundary of the waveguide) and

ISRNMathematical Physics 3

0

z

120588

120593

R

120576

1205830

1205761

1205830

Figure 1 Geometry of the problem

the radiation condition at infinity the electromagnetic fieldexponentially decays as 120588 rarr infin

The permittivity in the entire space has the form

120576 = 1205760

1205761 120588 gt 119877

1205762(120588) + 119886|E|2 120588 lt 119877

(9)

where 119886 is a real positive value 1205762(120588) gt 120576

1 Here 120576

2(120588) is a

linear part of the permittivityThe solutions to the Maxwell equations are sought in the

entire spaceThereby passing from time-dependent equations (1) to

time-independent equations (8) is grounded on previousconsideration

Geometry of the problem is shown in Figure 1 Thewaveguide is infinite along axis 119874119911

Let us consider TE waves with harmonical dependenceon time

E119890minus119894120596119905 = 119890minus119894120596119905(0 119864120593 0)

119879

H119890minus119894120596119905 = 119890minus119894120596119905(119867120588 0119867119911)

119879

(10)

where EH are the complex amplitudesSubstituting the complex amplitudes into Maxwell equa-

tions (8) we obtain

1

120588

120597119867119911

120597120593

= 0

120597119867120588

120597119911

minus

120597119867119911

120597120588

= minus119894120596120576119864120593

1

120588

120597119867120588

120597120593

= 0

120597119864120593

120597119911

= minus119894120596120583119867120588

1

120588

120597 (120588119864120593)

120597120588

= 119894120596120583119867119911

(11)

It is obvious from the first and the third equations of thissystem that119867

119911and119867

120588do not depend on 120593 This implies that

119864120593does not depend on 120593Independence of the components on 120593 can be explained

if we chose dependence on 120593 in the form 119890119894119899120593 with 119899 = 0

Waves propagating along waveguide axis 119874119911 dependharmonically on 119911 This means that the fields componentshave the form

119864120593= 119864120593(120588) 119890119894120574119911 119867

120588= 119867120588(120588) 119890119894120574119911

119867119911= 119867119911(120588) 119890119894120574119911

(12)

where 120574 is the unknown spectral parameter of the problem(propagation constant)

So we obtain from system (11) that

119894120574119867120588(120588) minus 119867

1015840

119911(120588) = minus119894120596120576119864

120593(120588)

119894120574119864120593(120588) = minus119894120596120583119867

120588(120588)

1

120588

(120588119864120593(120588))

1015840

= 119894120596120583119867119911(120588)

(13)

where ( sdot )1015840 equiv 119889119889120588Then 119867

119911(120588) = (1119894120596120583)(1120588)(120588119864

120593(120588))1015840 and 119867

120588(120588) =

minus(120574120596120583)119864120593(120588) From the first equation of the latter system

we obtain

(

1

120588

(120588119864120593(120588))

1015840

)

1015840

+ (1205962120583120576 minus 120574

2) 119864120593(120588) = 0 (14)

Denoting by 119906(120588) = 119864120593(120588) we obtain

11990610158401015840+

1

120588

1199061015840minus

1

1205882119906 + (119896

2

0120576 minus 1205742) 119906 = 0 (15)

and 120576 = 1205761205760 where

120576 =

1205761 120588 gt 119877

1205762(120588) + 119886119906

2 120588 lt 119877

(16)

and 11989620= 12059621205831205760

Also we assume that function 119906 is sufficiently smooth

119906 (120588) isin 119862 [0 +infin) cap 1198621[0 +infin) cap 119862

2(0 119877) cap 119862

2(119877 +infin)

(17)

Physical nature of the problem implies these conditionsWe will seek 120574 under conditions 1198962

01205761lt 120574

2lt

1198962

0min120588isin[0119877]

1205762(120588)

In the domain 120588 gt 119877 we have 120576 = 1205761 From (15) we obtain

the equation

11990610158401015840+

1

120588

1199061015840minus

1

1205882119906 + 1198962

1119906 = 0 (18)

where 11989621= 1198962

01205761minus 1205742 It is the Bessel equation

In the domain 120588 lt 119877 we have 120576 = 1205762(120588) + 119886119906

2 From (15)we obtain the equation

11990610158401015840+

1

120588

1199061015840minus

1

1205882119906 + 1198962(120588) 119906 + 120572119906

3= 0 (19)

where 1198962(120588) = 11989622(120588) minus 120574

2 11989622(120588) = 119896

2

01205762(120588) and 120572 = 1198861198962

0

4 ISRNMathematical Physics

Tangential components of electromagnetic field areknown to be continuous atmedia interfaces Hence we obtain

119864120593(119877 + 0) = 119864

120593(119877 minus 0) 119867

119911(119877 + 0) = 119867

119911(119877 minus 0)

(20)

Further we have 119867119911(120588) = (1119894120596120583)(1120588)119864

120593(120588) + 119864

1015840

120593(120588))

Since 119864120593(120588) and 119867

119911(120588) are continuous at the point 120588 = 119877

therefore 1198641015840120593(120588) is continuous at 120588 = 119877 These conditions

imply the transmission conditions for functions 119906(120588) and1199061015840(120588)

[119906]120588=119877 = 0 [1199061015840]120588=119877= 0 (21)

where [119891]119909=1199090= lim119909rarr1199090minus0

119891(119909) minus lim119909rarr1199090+0

119891(119909)Let us formulate the transmission eigenvalue problem

(problem P) It is necessary to find eigenvalues 120574 andcorrespond to them nonzero eigenfunctions 119906(120588) such that119906(120588) satisfy (18) (19) transmission conditions (21) and theradiation condition at infinity eigenfunctions exponentiallydecay as 120588 rarr infin

The general solution of (18) is taken in the followingform 119906(120588) = 119887119867(1)

1(1198961120588) + 119887

1119867(2)

1(1198961120588) where 119867(1)

1and 119867(2)

1

are the Hankel functions of the first and the second kindsrespectively In accordance with the radiation condition weobtain that 119887

1= 0 then the solution has the form 119906(120588) =

119887119867(1)

1(1198961120588) 120588 gt 119877 where 119887 is a constant If Re 119896

1= 0 then

119906 (120588) =1198871198701(10038161003816100381610038161198961

1003816100381610038161003816120588) 120588 gt 119877 (22)

as 119867(1)1(119894119911) = minus(2120587)119870

1(119911) and 119870

1(119911) is the Macdonald

functionThe radiation condition is fulfilled since 119870

1(|1198961|120588) rarr 0

as 120588 rarr infin

3 Nonlinear Integral Equation andDispersion Equation

Consider nonlinear equation (19) written in the form

(1205881199061015840)

1015840

+ (1198962(120588) 120588 minus

1

120588

) 119906 + 1205721205881199063= 0 (23)

and the linear equation

(1205881199061015840)

1015840

+ (1198962(120588) 120588 minus

1

120588

) 119906 = 0 (24)

The latter equation can be written in the operator form as

119871119896119906 = 0 119871

119896=

119889

119889120588

(120588

119889

119889120588

) + (1198962(120588) 120588 minus

1

120588

) (25)

(here we place index 119896 in order to stress that the operator andthe Green function depend on 119896(120588))

Suppose that the Green function119866119896(120588 1205880 120582) exists for the

following boundary value problem

119871119896119866119896= minus120575 (120588 minus 120588

0) 119866|120588=0 = 119866

101584010038161003816100381610038161003816120588=119877= 0 (0 lt 120588

0lt 119877)

(26)

In this case the Green function has the representation(see eg [29 30])

119866119896(120588 1205880 120582) = minus

V119894(120588) V119894(1205880)

120582 minus 120582119894

+ 1198661(120588 1205880 120582) (27)

in the vicinity of eigenvalue 120582119894 Here 120582 = 1205742 and 119866

1(120588 1205880 120582)

is regular with respect to 120582 in the vicinity of 120582119894 120582119899 V119899(120588) are

complete orthonormal (real) eigenvalues and eigenfunctionssystems of boundary eigenvalue problem

(120588V1015840

119899)

1015840

+ (1198962

2(120588) 120588 minus

1

120588

) V119899= 120582119899120588V119899

V119899

1003816100381610038161003816120588=0= V1015840

119899

10038161003816100381610038161003816 120588=119877

= 0

(28)

The Green function exists if 120582 = 120582119894

For 1205762equiv const explicit form of theGreen function is given

in [21]Let us write (19) in the operator form

119871119896119906 + 120572119861 (119906) = 0 119861 (119906) = 120588119906

3(120588) (29)

Using the second Green formula [31]

int

119877

0

(V119871119896119906 minus 119906119871

119896V) 119889120588 = int

119877

0

(V(1205881199061015840)

1015840

minus 119906(120588V1015840)

1015840

) 119889120588

= 119877 (1199061015840(119877) V (119877) minus V

1015840(119877) 119906 (119877))

(30)

and assuming that V = 119866 we obtain that

int

119877

0

(119866119896119871119896119906 minus 119906119871

119896119866119896) 119889119901

= 119877 (1199061015840(119877 minus 0) 119866

119896(119877 1205880) minus 1198661015840

119896(119877 1205880) 119906 (119877 minus 0))

= 1198771199061015840(119877 minus 0) 119866

119896(119877 1205880)

(31)

From the previous formulae we obtain

int

119877

0

119906119871119896119866119896119889120588 = minusint

119877

0

119906 (120588) 120575 (120588 minus 1205880) 119889120588 = minus119906 (120588

0)

int

119877

0

119866119896119871119896119906119889120588 = minus120572int

119877

0

119866119896(120588 1205880) 1205881199063(120588) 119889120588

(32)

Taking into account these results and using (29) weobtain the nonlinear integral representation of solution 119906(120588

0)

of (19) on the segment [0 119877]

119906 (1205880) = 120572int

119877

0

119866119896(120588 1205880) 1205881199063(120588) 119889120588

+ 1198771199061015840(119877 minus 0) 119866

119896(119877 1205880) 0 le 120588

0le 119877

(33)

Using transmission conditions 1199061015840(119877 minus 0) = 1199061015840(119877 + 0) wecan rewrite (33)

119906 (1205880) = 120572int

119877

0

119866119896(120588 1205880) 1205881199063(120588) 119889120588 + 119891 (120588

0) 0 le 120588

0le 119877

(34)

where 119891(1205880) = 119877119906

1015840(119877 + 0)119866

119896(119877 1205880)

ISRNMathematical Physics 5

Using (34) and transmission condition119906(119877minus0) = 119906(119877+0)we obtain the dispersion equation (DE) with respect to thepropagation constant

119906 (119877 + 0) = 120572int

119877

0

119866119896(120588 119877) 120588119906

3(120588) 119889120588 + 119877119906

1015840(119877 + 0) 119866 (119877 119877)

(35)

Let us denote by 119873(120588 1205880 120582) = 120572119866

119896(120588 1205880 120582)120588 and

consider integral equation (34)

119906 (1205880) = int

119877

0

119873(120588 1205880) 1199063(120588) 119889120588 + 119891 (120588

0) (36)

in 119862[0 119877] [32] It is assumed that 119891 isin 119862[0 119877] and 120582 = 120582119894

The kernel 119873(120588 1205880) is continuous in the square 0 le

120588 1205880le 119877

Let us consider linear integral operator 119873119908 =

int

119877

0119873(120588 120588

0)119908(120588)119889120588 in 119862[0 119877] It is bounded completely

continuous and 119873 = max1205880isin[0119877]

int

119877

0|119873(120588 120588

0)|119889120588

Since nonlinear operator 1198610(119906) = 119906

3(120588) is bounded and

continuous in 119862[0 119877] therefore nonlinear operator 119865(119906) =int

119877

0119873(120588 120588

0)1199063(120588)119889120588 + 119891(120588

0) is completely continuous in any

bounded set in 119862[0 119877]The following theorems (about existence of a unique

solution and continuous dependence of the solution on theparameter) can be proved in the same way as for the case of ahomogeneous nonlinear cylindrical waveguide (for details ofproofs see [22 33])

Proposition 1 If 120572 le 1198602 where

119860 =

2

3

1

10038171003817100381710038171198911003817100381710038171003817radic310038171003817100381710038171198731

1003817100381710038171003817

10038171003817100381710038171198731

1003817100381710038171003817= max1205880isin[0119877]

int

119877

0

1003816100381610038161003816120588119866119896(120588 1205880)1003816100381610038161003816119889120588

(37)

then (36) has a unique continuous solution 119906 isin 119862[0 119877] suchthat 119906 le 119903

lowast where

119903lowast= minus

2

radic3 119873

cos(13

arccos(3radic3

2

10038171003817100381710038171198911003817100381710038171003817radic119873) minus

2120587

3

)

(38)

is a root of the equation 1198731199033 + 119891 = 119903

Note that 119860 gt 0 does not depend on 120572

Proposition 2 Let the kernel119873 and the right-hand side 119891 ofequation (36) depend continuously on the parameter 120582 isin Λ

0

119873(120588 1205880 120582) sub 119862([0 119877]times[0 119877]timesΛ

0)119891(1205880 120582) sub 119862([0 119877]timesΛ

0)

on some segment Λ0of the real number axis Let also

0 lt1003817100381710038171003817119891 (120582)

1003817100381710038171003817lt

2

3radic3 119873 (120582)

(39)

Then for 120582 isin Λ0 a unique solution 119906(120588 120582) of (36) exists

and depends continuously on 120582 119906(120588 120582) sub 119862([0 119877] times Λ0)

4 Iteration Method

Approximate solutions 119906119899of integral equation (36) repre-

sented in the form 119906 = 119882(119906) can be found by means of theiteration process 119906

119899+1= 119882(119906

119899) 119899 = 0 1

1199060= 0 119906

119899+1= 120572int

119877

0

119866119896(120588 1205880) 1205881199063

119899119889120588 + 119891

119899 = 0 1

(40)

The sequence119906119899converges uniformly to solution119906 of (36)

by virtue of the fact that 119865(119906) is a contracting operator Theestimate of the convergence rate of iteration process (40) isalso known Let us formulate these results as the following(for proof see [22])

Proposition 3 The sequence of approximate solutions 119906119899of

(36) obtained by means of iteration process (40) converges inthe norm of space 119862[0 119877] to (unique) exact solution 119906 of thisequationThe following estimate of the convergence rate is valid119906119899minus 119906 le (119902

119899(1 minus 119902))119891(119906

0) 119899 rarr infin where 119902 = 31198731199032

lowastlt 1

is the coefficient of contraction of mapping 119865

5 Theorem of Existence

Taking into account formula (22) DE (35) can be representedin the form

1198701(10038161003816100381610038161198961

1003816100381610038161003816119877) minus

10038161003816100381610038161198961

10038161003816100381610038161198771198701015840

1(10038161003816100381610038161198961

1003816100381610038161003816119877) 119866119896(119877 119877 120582)

=

120572

119887

int

119877

0

119866119896(120588 119877 120582) 120588119906

3(120588) 119889120588

(41)

As it can be seen DE (41) depends on 119887 Here 119887 is an initialcondition This is the peculiarity of this (and not only this)nonlinear problem For the linear problem (if 120572 = 0) weobtain as it is expected the DE that does not depend on theinitial condition

From the properties of Bessel functions it follows that

minus10038161003816100381610038161198961

10038161003816100381610038161198771198701015840

1(10038161003816100381610038161198961

1003816100381610038161003816119877) =

10038161003816100381610038161198961

10038161003816100381610038161198771198700(10038161003816100381610038161198961

1003816100381610038161003816119877) + 119870

1(10038161003816100381610038161198961

1003816100381610038161003816119877) (42)

Now we can rewrite DE (41) in the following form

119892 (120582) = 120572119865 (120582) (43)

where

119892 (120582) = 1198701(10038161003816100381610038161198961

1003816100381610038161003816119877)

+ (10038161003816100381610038161198961

10038161003816100381610038161198771198700(10038161003816100381610038161198961

1003816100381610038161003816119877) + 119870

1(10038161003816100381610038161198961

1003816100381610038161003816119877))119866

119896

(119877 119877 120582)

119865 (120582) = int

119877

0

119866119896(120588 119877 120582) 120588119906

3(120588) 119889120588

(44)We should note that DE (41) depends on frequency 120596

implicitly If one obtains 120582lowast for chosen119877lowast (radius of the innercore) such that 119892(120582lowast) = 120572119865(120582lowast) is satisfied then one cancalculate 120596lowast which satisfies the propagation constants 120574lowast =radic120582lowast using formulae in the beginning of this section

6 ISRNMathematical Physics

The zeros of the function Φ(120574) equiv 119892(120582) minus 120572119865(120582) are thosevalues of 120582 for which a nonzero solution of the problem Pexists The following assertion gives us sufficient conditionsfor the existence of zeros of the function Φ

Let us consider the question about existence of solutionsof the linear problem 119892(120582) = 0

This equation can be rewritten in the form

119866119896(119877 119877 120582) = minus

1198701(10038161003816100381610038161198961

1003816100381610038161003816119877)

10038161003816100381610038161198961

10038161003816100381610038161198771198700(10038161003816100381610038161198961

1003816100381610038161003816119877) + 119870

1(10038161003816100381610038161198961

1003816100381610038161003816119877)

(45)

From expression 119866119896(119877 119877 120582) = minusV2

119894(119877)(120582 minus 120582

119894) +

1198661(119877 119877 120582) it follows that 119866

119896(119877 119877 120582) continuously varies

from minusinfin to +infin when 120582 varies from 120582119894to 120582119894+1

As value 119870

1(|1198961|119877)(|119896

1|1198771198700(|1198961|119877) + 119870

1(|1198961|119877)) is

bounded then there is at least one root of equation 119892(120582) = 0and this root lies between 120582

119894and 120582

119894+1

Finally it is necessary to prove that term V119894(119877) does

not vanish in expression 119866119896(119877 119877 120582) We prove this fact by

contradiction Let V119894(119877) = 0 Consider a Cauchy problem

for equation 120588V10158401015840119894+ V1015840119894+ (1198962

2(120588)120588 minus 1120588)V

119894= 120582119894120588V119894with

initial conditions V119894|120588=119877= V1015840119894|120588=119877= 0 as 120588 isin [120575 119877] where

120575 gt 0 From the general theory of ordinary differentialequations (see eg [34]) it is known that solution V

119894(120588) of

consideredCauchy problem exists and is unique as 120588 isin [120575 119877]In this case this solution coincides with function V

119894(119877) as

120588 isin [120575 119877] Function V119894(119877) is the function which is contained

in Greenrsquos function representation (27) On the other handa solution of the Cauchy problem for a linear equation withzero initial condition is the trivial solution This contradictswith representation (27) ofGreenrsquos function119866

119896(120588 1205880 120582) in the

vicinity of 120582 = 120582119894

Consider nonlinear problem Let inequalities

1205761lt 1205820lt 1205821lt sdot sdot sdot lt 120582

119896minus1lt 120582119896lt 1205762 (46)

hold where 119896 ge 1 and 1205762= min

120588isin[0119877]1205762(120588)

We can choose sufficiently small 120575119894gt 0 such that the

Green function 119866119896(120588 1205880 120582) exists and is continuous on Γ =

⋃119896

119894=1Γ119894 where

Γ119894= [radic120582

119894minus1+ 120575119894minus1 radic120582119894minus 120575119894] 119894 = 1 119896 (47)

and the following inequality 119892(radic120582119894minus1+ 120575119894minus1)119892(radic120582

119894minus 120575119894) lt 0

is satisfiedIt follows from the choice of 120575

119894that 119865(120582) is bounded

Moreover product 120572119865(120582) can be made sufficiently small bychoosing appropriate 120572 (the estimation is given at the end ofthis section) Let us consider DE Φ(120582) = 0 As it is shownbefore function 119892(120582) is continuous and reverse sign when 120582varies from 120582

119894minus1+120575119894minus1

to 120582119894minus120575119894 As function 119865(120582) is bounded

then it is clear that equation Φ(120582) = 0 has at least 119896 roots 120582119894

119894 = 1 119896 if we choose appropriate 120572 Here 120582119894isin (120582119894minus1+120575119894minus1 120582119894minus

120575119894) 119894 = 1 119896On the basis of previous consideration we can formulate

the main result of this paper

Theorem 4 Let the values 1205761 1205762= min

120588isin[0119877]1205762(120588) 120572 satisfy

condition 1205762gt 1205761gt 0 and let the following inequalities

1205761lt 1205820lt 1205821lt sdot sdot sdot lt 120582

119896minus1lt 120582119896lt 1205762hold where 119896 ge 1 is

an integerThen there is a value 1205720gt 0 such that for any 120572 le 120572

0

at least 119896 values 120574119894 119894 = 1 119896 exist such that the problem P has a

nonzero solution and 120574119894isin (radic120582

119894minus1+ 120575119894minus1 radic120582119894minus 120575119894)

Proof The Green function exists for all 120574 isin Γ It is also clearthat function 119860(120574) = 23119891(120574)radic3119873

1(120574) is continuous as

120574 isin Γ Let 1198601= min

120574isinΓ119860(120574) and 120572 lt 1198602

1 In accordance

with Proposition 1 there is a unique solution 119906 = 119906(120574) of(36) for any 120574 isin Γ This solution is continuous and 119906 le119903lowast= 119903lowast(120574) Let 119903

00= max

120574isinΓ119903lowast(120574) The following estimation

|119865(120582)| le 1198621199033

00is valid where 119862 is a constant

Function 119892(120574) is continuous and equation 119892(120574) = 0 has atleast one root 120574

119894inside segment Γ

119894 that isradic120582

119894minus1+ 120575119894minus1lt 120574119894lt

radic120582119894minus 120575119894 Let us denote119872

1= min

0leile119896minus1|119892(radic120582119894 + 120575119894)|1198722 =min1le119894le119896|119892(radic120582

119894minus 120575119894)| Value = min119872

11198722 is positive

and does not depend on 120572If 120572 le 1198621199033

00 then

(119892(radic120582119894minus1+ 120575119894minus1) minus 120572119865(radic120582

119894minus1+ 120575119894minus1))

times (119892(radic120582119894minus 120575119894) minus 120572119865(radic120582

119894minus 120575119894)) lt 0

(48)

As 119892(120582) minus 120572119865(120582) is continuous it follows that equation119892(120582)minus120572119865(120582) = 0 has a root 120574

119894inside Γ

119894 that isradic120582

119894+120575119894lt 120574119894lt

radic120582119894+1minus 120575119894+1

We can choose 1205720= min1198602

1 119862119903

3

00

From Theorem 4 it follows that under the previousassumptions there exist axially symmetrical propagating TEwaves in cylindrical dielectric waveguides of circular cross-section filled with a nonmagnetic isotropic inhomogeneousmedium with Kerr nonlinearity This result generalizes thewell-known similar statement for dielectric waveguides ofcircular cross-section filled with a linearmedium (ie 120572 = 0)

It should be noticed that the value 1205720can be effectively

estimated

6 Conclusion

In this study we suggest and develop a method to inves-tigate the problem of existence of electromagnetic wavesthat propagate along axis of an inhomogeneous nonlinearcylindrical waveguideThe nonlinearity inside the waveguideis described by the Kerr law the inhomogeneity is describedby a function that depends on radius of the waveguide

Here we show that the integral equation approach allowsus to investigate quite general problem for nonlinear inhomo-geneous waveguides

We should say that this method can be used to proveexistence of guided waves in a nonlinear inhomogeneouswaveguide for TM waves

Numerical results can be obtained with the help ofiteration procedure from Section 4

A separate paper will be devoted to development of acouple of numerical methods for this problem

ISRNMathematical Physics 7

Acknowledgments

This work is partially supported by the RFBR (Grants nos 11-07-00330-A 12-07-97010-RA) theMinistry of Education andScience of the Russian Federation (Grant no 14B37211950)

References

[1] M J AdamsAn Introduction To OpticalWaveguide JohnWileyamp Sons Chichester UK 1981

[2] A Snyder and J LoveOpticalWaveguideTheory Chapman andHall London UK 1983

[3] N N Akhmediev and A Ankevich Solitons Nonlinear Pulsesand Beams Chapman and Hall London UK 1997

[4] Y R Shen The Principles of Nonlinear Optics John Wiley ampSons New York NY USA 1984

[5] H M Gibbs Optical Bistability Controlling Light with LightAcademic Press New York NY USA 1985

[6] P N Eleonskii L G Oganersquosyants and V P Silin ldquoCylindricalnonlinear waveguidesrdquo Journal of Experimental andTheoreticalPhysics vol 35 no 1 pp 44ndash47 1972

[7] KM Leung ldquoP-polarized nonlinear surface polaritons inmate-rials with intensity-dependent dielectric functionsrdquo PhysicalReview B vol 32 no 8 pp 5093ndash5101 1985

[8] R I Joseph and D N Christodoulides ldquoExact field decompo-sition for tm waves in nonlinear mediardquo Optics Letters vol 12no 10 pp 826ndash828 1987

[9] D V Valovik ldquoPropagation of tmwaves in a layer with arbitrarynonlinearityrdquo Computational Mathematics and MathematicalPhysics vol 51 no 9 pp 1622ndash1632 2011

[10] D V Valovik ldquoPropagation of electromagnetic TE waves in anonlinear mediumwith saturationrdquo Journal of CommunicationsTechnology and Electronics vol 56 no 11 pp 1311ndash1316 2011

[11] G Yu Smirnov and D V Valovik Electromagnetic WavePropagation inNon-Linear LayeredWaveguide Structures PenzaState University Press Penza Russia 2011

[12] H W Schurmann V S Serov and Y V Shestopalov ldquoTE-polarized waves guided by a lossless nonlinear three-layerstructurerdquo Physical Review E vol 58 no 1 pp 1040ndash1050 1998

[13] D V Valovik ldquoPropagation of electromagnetic waves in anonlinearmetamaterial layerrdquo Journal of Communications Tech-nology and Electronics vol 56 no 5 pp 544ndash556 2011

[14] HW SchurmannV S Serov andYuV Shestopalov ldquoSolutionsto the Helmholtz equation for TE-guided waves in a three-layerstructure with Kerr-type nonlinearityrdquo Journal of Physics A vol35 no 50 pp 10789ndash10801 2002

[15] D V Valovik and Yu G Smirnov ldquoPropagation of tm wavesin a kerr nonlinear layerrdquo Computational Mathematics andMathematical Physics vol 48 no 12 pp 2217ndash2225 2008

[16] DVValovik andYG Smirnov ldquoCalculation of the propagationconstants of TM electromagnetic waves in a nonlinear layerrdquoJournal of Communications Technology and Electronics vol 53no 8 pp 883ndash889 2008

[17] DVValovik andYG Smirnov ldquoCalculation of the propagationconstants and fields of polarized electromagnetic TM wavesin a nonlinear anisotropic layerrdquo Journal of CommunicationsTechnology and Electronics vol 54 no 4 pp 391ndash398 2009

[18] G Yu Smirnov and D V Valovik ldquoBoundary eigenvalueproblem for maxwell equations in a nonlinear dielectric layerrdquoApplied Mathematics no 1 pp 29ndash36 2010

[19] D V Valovik and Y G Smirnov ldquoNonlinear effects in theproblem of propagation of TM electromagnetic waves in a Kerrnonlinear layerrdquo Journal of Communications Technology andElectronics vol 56 no 3 pp 283ndash288 2011

[20] K A Yuskaeva V S Serov and H W Schurmann ldquoTm-electromagnetic guided waves in a (kerr-) nonlinear three-layerstructurerdquo in Proceedings of the Progress in ElectromagneticsResearch Symposium (PIERS rsquo09) pp 364ndash369 Moscow RussiaAugust 2009

[21] H W Schurmann G Yu Smirnov and V Yu ShestopalovldquoPropagation of te-waves in cylindrical nonlinear dielectricwaveguidesrdquo Physical Review E vol 71 no 1 Article ID 0166142005

[22] Y Smirnov H W Schurmann and Y Shestopalov ldquoIntegralequation approach for the propagation of TE-waves in a non-linear dielectric cylindrical waveguiderdquo Journal of NonlinearMathematical Physics vol 11 no 2 pp 256ndash268 2004

[23] Yu G Smirnov and S N Kupriyanova ldquoPropagation of electro-magnetic waves in cylindrical dielectric waveguides filled witha nonlinear mediumrdquo Computational Mathematics and Mathe-matical Physics vol 44 no 10 pp 1850ndash1860 2004

[24] G Yu Smirnov and E A Horosheva ldquoOn the solvabilityof the nonlinear boundary eigenvalue problem for tm wavespropagation in a circle cylyindrical nonlinear waveguiderdquoIzvestiya Vysshikh Uchebnykh Zavedenij Povolzh Region Fiziki-Matematicheskie Nauki no 3 pp 55ndash70 2010 (Russian)

[25] G Yu Smirnov andD V Valovik ldquoCoupled electromagnetic te-tm wave propagation in a layer with kerr nonlinearityrdquo Journalof Mathematical Physics vol 53 no 12 Article ID 123530 pp1ndash24 2012

[26] V S Serov K A Yuskaeva and H W Schurmann ldquoIntegralequations approach to tm-electromagnetic waves guided bya (linearnonlinear) dielectric film with a spatially varyingpermittivityrdquo in Proceedings of the Progress in ElectromagneticsResearch Symposium (PIERS rsquo09) pp 1915ndash1919 Moscow Rus-sia August 2009

[27] J A Stretton Electromagnetic Theory McGraw Hill New YorkNY USA 1941

[28] L D Landau E M Lifshitz and L P Pitaevskii Course ofTheoretical Physics vol 8 of Electrodynamics of ContinuousMedia Butterworth-Heinemann Oxford UK 1993

[29] M A Naimark Linear Differential Operators Part I ElementaryTheory of Linear Differential Operators Frederick Ungar NewYork NY USA 1967

[30] M A Naimark Linear Differential Operators Part II LinearDifferential Operators in Hilbertspace Frederick Ungar NewYork NY USA 1968

[31] I Stakgold Greenrsquos Functions and Boundary Value ProblemsJohn Wiley amp Sons New York NY USA 1979

[32] V A Trenogin Functional Analysis Nauka Moscow Russia1993

[33] Y G Smirnov ldquoPropagation of electromagnetic waves in cylin-drical dielectric waveguides filled with a nonlinear mediumrdquoJournal of Communications Technology and Electronics vol 50no 2 pp 179ndash185 2005

[34] P I Lizorkin Course of Differential and Integral Equations withSupplementary Chapters of Calculus Nauka Moscow Russia1981

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 ISRNMathematical Physics

Inmost cases it is very difficult (if at all possible) to obtainexact solutions of the equations in nonlinear waveguidingproblems However integral equation approach can help inthis case [21ndash25] In this approach a problem is reduced toan integral equation whose kernel depends on the Greenfunction of the linear part of the differential equations of theproblem Two circumstances are important for the followinganalysis First in the case of a homogeneous waveguidethis Green function can be found explicitly Second thedispersion equation of the nonlinear homogeneous case canbe written as DElin + 119879nonlin = 0 where DElin is a linearproblem term and 119879nonlin is an extra nonlinear term Here thelinear problem term is written in an explicit form Moreoverthe equation DElin = 0 is well known and examined DE forthe linear problem Its roots are also known All this allowsto prove existence of the nonlinear problem solutions at leastnear to the linear problem solutions

Here we investigate guided waves in a nonlinear inhomo-geneous cylindrical waveguide filled with Kerr medium Thewaveguide is placed in cylindrical coordinate system 119874120588120593119911where axis 119911 coincides with axis of the waveguide Inhomo-geneity is modeled by a function that depends on radius ofthe waveguide The permittivity inside the waveguide is 120576 =1205762(120588)+119886|E|2 where 120576

2(120588) is the inhomogeneity 119886 is a constant

in the Kerr law and E is complex amplitude If 1205762(120588) equiv

const we have a nonlinear homogeneous waveguide Thenonconstant term 120576

2(120588) dramatically changes the situation

In this case we cannot find explicitly the necessary Greenfunction so we investigate it in an implicit form Thedispersion equation of the nonlinear inhomogeneous casecan be also written as DElin + 119879nonlin = 0 However in thiscase the term DElin is written in an implicit form as opposedto the case of a homogeneous waveguide and its roots areunknown So at first we prove that the equation for thelinear inhomogeneous problem DElin = 0 has roots anddefine localization of the rootsThen we prove that nonlinearproblem has solutions

Integral equation approach has been already used for anonlinear inhomogeneous waveguiding problem [26] How-ever in study [26] authors apply integral equation approachin the way as they would solve the problem for a homoge-neous waveguide To be precise the authors use the Greenfunction for constant 120576

2that helps them to determine the

Green function in explicit form We pay heed that thereare no theoretical results (existence of eigenvalues and theirlocalization) in [26] We emphasize that for inhomogeneouswaveguides important and general results can be obtainedwith the method we use in this paper in which the Greenfunction has implicit form

In spite of the fact that the method here looks similar tothe method in [21ndash24] we solve radically different problemas we consider inhomogeneous nonlinear waveguide

2 Statement of the Problem

Let us consider three-dimensional space R3 with cylindricalcoordinate system 119874120588120593119911 The space is filled by isotropicmedium with constant permittivity 120576

1ge 1205760 where 120576

0is

the permittivity of free space In this medium a cylindrical

waveguide is placed The waveguide is filled by isotropicnonmagnetic medium and has cross section 119882 = (120588 120593) 1205882lt 1198772 0 le 120593 lt 2120587 and its generating line (the

waveguide axis) is parallel to the axis 119874119911 We will considerelectromagnetic waves propagating along the waveguide axisEverywhere below 120583 = 120583

0is the permeability of free space

We use Maxwellrsquos equations in the following form [27]

rot H = 120597119905D

rot E = minus120597119905B

(1)

where D = 120576E B = 120583H and 120597119905= 120597120597119905 Field (E H) is the total

fieldFrom formulae (1) we obtain

rot E = minus120597119905(120583H)

rot H = 120597119905(120576E)

(2)

Real monochromatic field (E H) in the medium can bewritten in the following form

E (120588 120593 119911 119905) = E+ (120588 120593 119911) cos120596119905 + Eminus (120588 120593 119911) sin120596119905

H (120588 120593 119911 119905) = H+ (120588 120593 119911) cos120596119905 +Hminus (120588 120593 119911) sin120596119905(3)

where 120596 is circular frequency E+ Eminus H+ and Hminus are realrequired vectors

Let us form complex amplitudes EH

E = E+ + 119894Eminus H = H+ + 119894Hminus (4)

It is clear that

E = Re E119890minus119894120596119905 H = Re H119890minus119894120596119905 (5)

where

E = (119864120588 119864120593 119864119911)

119879

H = (119867120588 119867120593 119867119911)

119879

(6)

and components in (6) depend on three spatial variablesIt is known (see eg [3 6 28]) that Kerr law in isotropic

medium for a monochromatic wave E119890minus119894120596119905 has the form 120576 =1205762+119886|E|2 where E is complex amplitude 120576

2is a constant part

of the permittivity 120576 119886 is the coefficient of nonlinearityWe obtain that in this case dependence of Maxwellrsquos

equations on 119905 is the same as in the case of constant 120576 insidethe waveguideThis allows us to writeMaxwellrsquos equations (2)in the form

rot (E119890minus119894120596119905) = 119894120596120583H119890minus119894120596119905

rot (H119890minus119894120596119905) = minus119894120596120576E119890minus119894120596119905(7)

Complex amplitudes (4) satisfy the Maxwell equations

rotE = 119894120596120583H

rotH = minus119894120596120576E(8)

the continuity condition for the tangential components onthe media interfaces (on the boundary of the waveguide) and

ISRNMathematical Physics 3

0

z

120588

120593

R

120576

1205830

1205761

1205830

Figure 1 Geometry of the problem

the radiation condition at infinity the electromagnetic fieldexponentially decays as 120588 rarr infin

The permittivity in the entire space has the form

120576 = 1205760

1205761 120588 gt 119877

1205762(120588) + 119886|E|2 120588 lt 119877

(9)

where 119886 is a real positive value 1205762(120588) gt 120576

1 Here 120576

2(120588) is a

linear part of the permittivityThe solutions to the Maxwell equations are sought in the

entire spaceThereby passing from time-dependent equations (1) to

time-independent equations (8) is grounded on previousconsideration

Geometry of the problem is shown in Figure 1 Thewaveguide is infinite along axis 119874119911

Let us consider TE waves with harmonical dependenceon time

E119890minus119894120596119905 = 119890minus119894120596119905(0 119864120593 0)

119879

H119890minus119894120596119905 = 119890minus119894120596119905(119867120588 0119867119911)

119879

(10)

where EH are the complex amplitudesSubstituting the complex amplitudes into Maxwell equa-

tions (8) we obtain

1

120588

120597119867119911

120597120593

= 0

120597119867120588

120597119911

minus

120597119867119911

120597120588

= minus119894120596120576119864120593

1

120588

120597119867120588

120597120593

= 0

120597119864120593

120597119911

= minus119894120596120583119867120588

1

120588

120597 (120588119864120593)

120597120588

= 119894120596120583119867119911

(11)

It is obvious from the first and the third equations of thissystem that119867

119911and119867

120588do not depend on 120593 This implies that

119864120593does not depend on 120593Independence of the components on 120593 can be explained

if we chose dependence on 120593 in the form 119890119894119899120593 with 119899 = 0

Waves propagating along waveguide axis 119874119911 dependharmonically on 119911 This means that the fields componentshave the form

119864120593= 119864120593(120588) 119890119894120574119911 119867

120588= 119867120588(120588) 119890119894120574119911

119867119911= 119867119911(120588) 119890119894120574119911

(12)

where 120574 is the unknown spectral parameter of the problem(propagation constant)

So we obtain from system (11) that

119894120574119867120588(120588) minus 119867

1015840

119911(120588) = minus119894120596120576119864

120593(120588)

119894120574119864120593(120588) = minus119894120596120583119867

120588(120588)

1

120588

(120588119864120593(120588))

1015840

= 119894120596120583119867119911(120588)

(13)

where ( sdot )1015840 equiv 119889119889120588Then 119867

119911(120588) = (1119894120596120583)(1120588)(120588119864

120593(120588))1015840 and 119867

120588(120588) =

minus(120574120596120583)119864120593(120588) From the first equation of the latter system

we obtain

(

1

120588

(120588119864120593(120588))

1015840

)

1015840

+ (1205962120583120576 minus 120574

2) 119864120593(120588) = 0 (14)

Denoting by 119906(120588) = 119864120593(120588) we obtain

11990610158401015840+

1

120588

1199061015840minus

1

1205882119906 + (119896

2

0120576 minus 1205742) 119906 = 0 (15)

and 120576 = 1205761205760 where

120576 =

1205761 120588 gt 119877

1205762(120588) + 119886119906

2 120588 lt 119877

(16)

and 11989620= 12059621205831205760

Also we assume that function 119906 is sufficiently smooth

119906 (120588) isin 119862 [0 +infin) cap 1198621[0 +infin) cap 119862

2(0 119877) cap 119862

2(119877 +infin)

(17)

Physical nature of the problem implies these conditionsWe will seek 120574 under conditions 1198962

01205761lt 120574

2lt

1198962

0min120588isin[0119877]

1205762(120588)

In the domain 120588 gt 119877 we have 120576 = 1205761 From (15) we obtain

the equation

11990610158401015840+

1

120588

1199061015840minus

1

1205882119906 + 1198962

1119906 = 0 (18)

where 11989621= 1198962

01205761minus 1205742 It is the Bessel equation

In the domain 120588 lt 119877 we have 120576 = 1205762(120588) + 119886119906

2 From (15)we obtain the equation

11990610158401015840+

1

120588

1199061015840minus

1

1205882119906 + 1198962(120588) 119906 + 120572119906

3= 0 (19)

where 1198962(120588) = 11989622(120588) minus 120574

2 11989622(120588) = 119896

2

01205762(120588) and 120572 = 1198861198962

0

4 ISRNMathematical Physics

Tangential components of electromagnetic field areknown to be continuous atmedia interfaces Hence we obtain

119864120593(119877 + 0) = 119864

120593(119877 minus 0) 119867

119911(119877 + 0) = 119867

119911(119877 minus 0)

(20)

Further we have 119867119911(120588) = (1119894120596120583)(1120588)119864

120593(120588) + 119864

1015840

120593(120588))

Since 119864120593(120588) and 119867

119911(120588) are continuous at the point 120588 = 119877

therefore 1198641015840120593(120588) is continuous at 120588 = 119877 These conditions

imply the transmission conditions for functions 119906(120588) and1199061015840(120588)

[119906]120588=119877 = 0 [1199061015840]120588=119877= 0 (21)

where [119891]119909=1199090= lim119909rarr1199090minus0

119891(119909) minus lim119909rarr1199090+0

119891(119909)Let us formulate the transmission eigenvalue problem

(problem P) It is necessary to find eigenvalues 120574 andcorrespond to them nonzero eigenfunctions 119906(120588) such that119906(120588) satisfy (18) (19) transmission conditions (21) and theradiation condition at infinity eigenfunctions exponentiallydecay as 120588 rarr infin

The general solution of (18) is taken in the followingform 119906(120588) = 119887119867(1)

1(1198961120588) + 119887

1119867(2)

1(1198961120588) where 119867(1)

1and 119867(2)

1

are the Hankel functions of the first and the second kindsrespectively In accordance with the radiation condition weobtain that 119887

1= 0 then the solution has the form 119906(120588) =

119887119867(1)

1(1198961120588) 120588 gt 119877 where 119887 is a constant If Re 119896

1= 0 then

119906 (120588) =1198871198701(10038161003816100381610038161198961

1003816100381610038161003816120588) 120588 gt 119877 (22)

as 119867(1)1(119894119911) = minus(2120587)119870

1(119911) and 119870

1(119911) is the Macdonald

functionThe radiation condition is fulfilled since 119870

1(|1198961|120588) rarr 0

as 120588 rarr infin

3 Nonlinear Integral Equation andDispersion Equation

Consider nonlinear equation (19) written in the form

(1205881199061015840)

1015840

+ (1198962(120588) 120588 minus

1

120588

) 119906 + 1205721205881199063= 0 (23)

and the linear equation

(1205881199061015840)

1015840

+ (1198962(120588) 120588 minus

1

120588

) 119906 = 0 (24)

The latter equation can be written in the operator form as

119871119896119906 = 0 119871

119896=

119889

119889120588

(120588

119889

119889120588

) + (1198962(120588) 120588 minus

1

120588

) (25)

(here we place index 119896 in order to stress that the operator andthe Green function depend on 119896(120588))

Suppose that the Green function119866119896(120588 1205880 120582) exists for the

following boundary value problem

119871119896119866119896= minus120575 (120588 minus 120588

0) 119866|120588=0 = 119866

101584010038161003816100381610038161003816120588=119877= 0 (0 lt 120588

0lt 119877)

(26)

In this case the Green function has the representation(see eg [29 30])

119866119896(120588 1205880 120582) = minus

V119894(120588) V119894(1205880)

120582 minus 120582119894

+ 1198661(120588 1205880 120582) (27)

in the vicinity of eigenvalue 120582119894 Here 120582 = 1205742 and 119866

1(120588 1205880 120582)

is regular with respect to 120582 in the vicinity of 120582119894 120582119899 V119899(120588) are

complete orthonormal (real) eigenvalues and eigenfunctionssystems of boundary eigenvalue problem

(120588V1015840

119899)

1015840

+ (1198962

2(120588) 120588 minus

1

120588

) V119899= 120582119899120588V119899

V119899

1003816100381610038161003816120588=0= V1015840

119899

10038161003816100381610038161003816 120588=119877

= 0

(28)

The Green function exists if 120582 = 120582119894

For 1205762equiv const explicit form of theGreen function is given

in [21]Let us write (19) in the operator form

119871119896119906 + 120572119861 (119906) = 0 119861 (119906) = 120588119906

3(120588) (29)

Using the second Green formula [31]

int

119877

0

(V119871119896119906 minus 119906119871

119896V) 119889120588 = int

119877

0

(V(1205881199061015840)

1015840

minus 119906(120588V1015840)

1015840

) 119889120588

= 119877 (1199061015840(119877) V (119877) minus V

1015840(119877) 119906 (119877))

(30)

and assuming that V = 119866 we obtain that

int

119877

0

(119866119896119871119896119906 minus 119906119871

119896119866119896) 119889119901

= 119877 (1199061015840(119877 minus 0) 119866

119896(119877 1205880) minus 1198661015840

119896(119877 1205880) 119906 (119877 minus 0))

= 1198771199061015840(119877 minus 0) 119866

119896(119877 1205880)

(31)

From the previous formulae we obtain

int

119877

0

119906119871119896119866119896119889120588 = minusint

119877

0

119906 (120588) 120575 (120588 minus 1205880) 119889120588 = minus119906 (120588

0)

int

119877

0

119866119896119871119896119906119889120588 = minus120572int

119877

0

119866119896(120588 1205880) 1205881199063(120588) 119889120588

(32)

Taking into account these results and using (29) weobtain the nonlinear integral representation of solution 119906(120588

0)

of (19) on the segment [0 119877]

119906 (1205880) = 120572int

119877

0

119866119896(120588 1205880) 1205881199063(120588) 119889120588

+ 1198771199061015840(119877 minus 0) 119866

119896(119877 1205880) 0 le 120588

0le 119877

(33)

Using transmission conditions 1199061015840(119877 minus 0) = 1199061015840(119877 + 0) wecan rewrite (33)

119906 (1205880) = 120572int

119877

0

119866119896(120588 1205880) 1205881199063(120588) 119889120588 + 119891 (120588

0) 0 le 120588

0le 119877

(34)

where 119891(1205880) = 119877119906

1015840(119877 + 0)119866

119896(119877 1205880)

ISRNMathematical Physics 5

Using (34) and transmission condition119906(119877minus0) = 119906(119877+0)we obtain the dispersion equation (DE) with respect to thepropagation constant

119906 (119877 + 0) = 120572int

119877

0

119866119896(120588 119877) 120588119906

3(120588) 119889120588 + 119877119906

1015840(119877 + 0) 119866 (119877 119877)

(35)

Let us denote by 119873(120588 1205880 120582) = 120572119866

119896(120588 1205880 120582)120588 and

consider integral equation (34)

119906 (1205880) = int

119877

0

119873(120588 1205880) 1199063(120588) 119889120588 + 119891 (120588

0) (36)

in 119862[0 119877] [32] It is assumed that 119891 isin 119862[0 119877] and 120582 = 120582119894

The kernel 119873(120588 1205880) is continuous in the square 0 le

120588 1205880le 119877

Let us consider linear integral operator 119873119908 =

int

119877

0119873(120588 120588

0)119908(120588)119889120588 in 119862[0 119877] It is bounded completely

continuous and 119873 = max1205880isin[0119877]

int

119877

0|119873(120588 120588

0)|119889120588

Since nonlinear operator 1198610(119906) = 119906

3(120588) is bounded and

continuous in 119862[0 119877] therefore nonlinear operator 119865(119906) =int

119877

0119873(120588 120588

0)1199063(120588)119889120588 + 119891(120588

0) is completely continuous in any

bounded set in 119862[0 119877]The following theorems (about existence of a unique

solution and continuous dependence of the solution on theparameter) can be proved in the same way as for the case of ahomogeneous nonlinear cylindrical waveguide (for details ofproofs see [22 33])

Proposition 1 If 120572 le 1198602 where

119860 =

2

3

1

10038171003817100381710038171198911003817100381710038171003817radic310038171003817100381710038171198731

1003817100381710038171003817

10038171003817100381710038171198731

1003817100381710038171003817= max1205880isin[0119877]

int

119877

0

1003816100381610038161003816120588119866119896(120588 1205880)1003816100381610038161003816119889120588

(37)

then (36) has a unique continuous solution 119906 isin 119862[0 119877] suchthat 119906 le 119903

lowast where

119903lowast= minus

2

radic3 119873

cos(13

arccos(3radic3

2

10038171003817100381710038171198911003817100381710038171003817radic119873) minus

2120587

3

)

(38)

is a root of the equation 1198731199033 + 119891 = 119903

Note that 119860 gt 0 does not depend on 120572

Proposition 2 Let the kernel119873 and the right-hand side 119891 ofequation (36) depend continuously on the parameter 120582 isin Λ

0

119873(120588 1205880 120582) sub 119862([0 119877]times[0 119877]timesΛ

0)119891(1205880 120582) sub 119862([0 119877]timesΛ

0)

on some segment Λ0of the real number axis Let also

0 lt1003817100381710038171003817119891 (120582)

1003817100381710038171003817lt

2

3radic3 119873 (120582)

(39)

Then for 120582 isin Λ0 a unique solution 119906(120588 120582) of (36) exists

and depends continuously on 120582 119906(120588 120582) sub 119862([0 119877] times Λ0)

4 Iteration Method

Approximate solutions 119906119899of integral equation (36) repre-

sented in the form 119906 = 119882(119906) can be found by means of theiteration process 119906

119899+1= 119882(119906

119899) 119899 = 0 1

1199060= 0 119906

119899+1= 120572int

119877

0

119866119896(120588 1205880) 1205881199063

119899119889120588 + 119891

119899 = 0 1

(40)

The sequence119906119899converges uniformly to solution119906 of (36)

by virtue of the fact that 119865(119906) is a contracting operator Theestimate of the convergence rate of iteration process (40) isalso known Let us formulate these results as the following(for proof see [22])

Proposition 3 The sequence of approximate solutions 119906119899of

(36) obtained by means of iteration process (40) converges inthe norm of space 119862[0 119877] to (unique) exact solution 119906 of thisequationThe following estimate of the convergence rate is valid119906119899minus 119906 le (119902

119899(1 minus 119902))119891(119906

0) 119899 rarr infin where 119902 = 31198731199032

lowastlt 1

is the coefficient of contraction of mapping 119865

5 Theorem of Existence

Taking into account formula (22) DE (35) can be representedin the form

1198701(10038161003816100381610038161198961

1003816100381610038161003816119877) minus

10038161003816100381610038161198961

10038161003816100381610038161198771198701015840

1(10038161003816100381610038161198961

1003816100381610038161003816119877) 119866119896(119877 119877 120582)

=

120572

119887

int

119877

0

119866119896(120588 119877 120582) 120588119906

3(120588) 119889120588

(41)

As it can be seen DE (41) depends on 119887 Here 119887 is an initialcondition This is the peculiarity of this (and not only this)nonlinear problem For the linear problem (if 120572 = 0) weobtain as it is expected the DE that does not depend on theinitial condition

From the properties of Bessel functions it follows that

minus10038161003816100381610038161198961

10038161003816100381610038161198771198701015840

1(10038161003816100381610038161198961

1003816100381610038161003816119877) =

10038161003816100381610038161198961

10038161003816100381610038161198771198700(10038161003816100381610038161198961

1003816100381610038161003816119877) + 119870

1(10038161003816100381610038161198961

1003816100381610038161003816119877) (42)

Now we can rewrite DE (41) in the following form

119892 (120582) = 120572119865 (120582) (43)

where

119892 (120582) = 1198701(10038161003816100381610038161198961

1003816100381610038161003816119877)

+ (10038161003816100381610038161198961

10038161003816100381610038161198771198700(10038161003816100381610038161198961

1003816100381610038161003816119877) + 119870

1(10038161003816100381610038161198961

1003816100381610038161003816119877))119866

119896

(119877 119877 120582)

119865 (120582) = int

119877

0

119866119896(120588 119877 120582) 120588119906

3(120588) 119889120588

(44)We should note that DE (41) depends on frequency 120596

implicitly If one obtains 120582lowast for chosen119877lowast (radius of the innercore) such that 119892(120582lowast) = 120572119865(120582lowast) is satisfied then one cancalculate 120596lowast which satisfies the propagation constants 120574lowast =radic120582lowast using formulae in the beginning of this section

6 ISRNMathematical Physics

The zeros of the function Φ(120574) equiv 119892(120582) minus 120572119865(120582) are thosevalues of 120582 for which a nonzero solution of the problem Pexists The following assertion gives us sufficient conditionsfor the existence of zeros of the function Φ

Let us consider the question about existence of solutionsof the linear problem 119892(120582) = 0

This equation can be rewritten in the form

119866119896(119877 119877 120582) = minus

1198701(10038161003816100381610038161198961

1003816100381610038161003816119877)

10038161003816100381610038161198961

10038161003816100381610038161198771198700(10038161003816100381610038161198961

1003816100381610038161003816119877) + 119870

1(10038161003816100381610038161198961

1003816100381610038161003816119877)

(45)

From expression 119866119896(119877 119877 120582) = minusV2

119894(119877)(120582 minus 120582

119894) +

1198661(119877 119877 120582) it follows that 119866

119896(119877 119877 120582) continuously varies

from minusinfin to +infin when 120582 varies from 120582119894to 120582119894+1

As value 119870

1(|1198961|119877)(|119896

1|1198771198700(|1198961|119877) + 119870

1(|1198961|119877)) is

bounded then there is at least one root of equation 119892(120582) = 0and this root lies between 120582

119894and 120582

119894+1

Finally it is necessary to prove that term V119894(119877) does

not vanish in expression 119866119896(119877 119877 120582) We prove this fact by

contradiction Let V119894(119877) = 0 Consider a Cauchy problem

for equation 120588V10158401015840119894+ V1015840119894+ (1198962

2(120588)120588 minus 1120588)V

119894= 120582119894120588V119894with

initial conditions V119894|120588=119877= V1015840119894|120588=119877= 0 as 120588 isin [120575 119877] where

120575 gt 0 From the general theory of ordinary differentialequations (see eg [34]) it is known that solution V

119894(120588) of

consideredCauchy problem exists and is unique as 120588 isin [120575 119877]In this case this solution coincides with function V

119894(119877) as

120588 isin [120575 119877] Function V119894(119877) is the function which is contained

in Greenrsquos function representation (27) On the other handa solution of the Cauchy problem for a linear equation withzero initial condition is the trivial solution This contradictswith representation (27) ofGreenrsquos function119866

119896(120588 1205880 120582) in the

vicinity of 120582 = 120582119894

Consider nonlinear problem Let inequalities

1205761lt 1205820lt 1205821lt sdot sdot sdot lt 120582

119896minus1lt 120582119896lt 1205762 (46)

hold where 119896 ge 1 and 1205762= min

120588isin[0119877]1205762(120588)

We can choose sufficiently small 120575119894gt 0 such that the

Green function 119866119896(120588 1205880 120582) exists and is continuous on Γ =

⋃119896

119894=1Γ119894 where

Γ119894= [radic120582

119894minus1+ 120575119894minus1 radic120582119894minus 120575119894] 119894 = 1 119896 (47)

and the following inequality 119892(radic120582119894minus1+ 120575119894minus1)119892(radic120582

119894minus 120575119894) lt 0

is satisfiedIt follows from the choice of 120575

119894that 119865(120582) is bounded

Moreover product 120572119865(120582) can be made sufficiently small bychoosing appropriate 120572 (the estimation is given at the end ofthis section) Let us consider DE Φ(120582) = 0 As it is shownbefore function 119892(120582) is continuous and reverse sign when 120582varies from 120582

119894minus1+120575119894minus1

to 120582119894minus120575119894 As function 119865(120582) is bounded

then it is clear that equation Φ(120582) = 0 has at least 119896 roots 120582119894

119894 = 1 119896 if we choose appropriate 120572 Here 120582119894isin (120582119894minus1+120575119894minus1 120582119894minus

120575119894) 119894 = 1 119896On the basis of previous consideration we can formulate

the main result of this paper

Theorem 4 Let the values 1205761 1205762= min

120588isin[0119877]1205762(120588) 120572 satisfy

condition 1205762gt 1205761gt 0 and let the following inequalities

1205761lt 1205820lt 1205821lt sdot sdot sdot lt 120582

119896minus1lt 120582119896lt 1205762hold where 119896 ge 1 is

an integerThen there is a value 1205720gt 0 such that for any 120572 le 120572

0

at least 119896 values 120574119894 119894 = 1 119896 exist such that the problem P has a

nonzero solution and 120574119894isin (radic120582

119894minus1+ 120575119894minus1 radic120582119894minus 120575119894)

Proof The Green function exists for all 120574 isin Γ It is also clearthat function 119860(120574) = 23119891(120574)radic3119873

1(120574) is continuous as

120574 isin Γ Let 1198601= min

120574isinΓ119860(120574) and 120572 lt 1198602

1 In accordance

with Proposition 1 there is a unique solution 119906 = 119906(120574) of(36) for any 120574 isin Γ This solution is continuous and 119906 le119903lowast= 119903lowast(120574) Let 119903

00= max

120574isinΓ119903lowast(120574) The following estimation

|119865(120582)| le 1198621199033

00is valid where 119862 is a constant

Function 119892(120574) is continuous and equation 119892(120574) = 0 has atleast one root 120574

119894inside segment Γ

119894 that isradic120582

119894minus1+ 120575119894minus1lt 120574119894lt

radic120582119894minus 120575119894 Let us denote119872

1= min

0leile119896minus1|119892(radic120582119894 + 120575119894)|1198722 =min1le119894le119896|119892(radic120582

119894minus 120575119894)| Value = min119872

11198722 is positive

and does not depend on 120572If 120572 le 1198621199033

00 then

(119892(radic120582119894minus1+ 120575119894minus1) minus 120572119865(radic120582

119894minus1+ 120575119894minus1))

times (119892(radic120582119894minus 120575119894) minus 120572119865(radic120582

119894minus 120575119894)) lt 0

(48)

As 119892(120582) minus 120572119865(120582) is continuous it follows that equation119892(120582)minus120572119865(120582) = 0 has a root 120574

119894inside Γ

119894 that isradic120582

119894+120575119894lt 120574119894lt

radic120582119894+1minus 120575119894+1

We can choose 1205720= min1198602

1 119862119903

3

00

From Theorem 4 it follows that under the previousassumptions there exist axially symmetrical propagating TEwaves in cylindrical dielectric waveguides of circular cross-section filled with a nonmagnetic isotropic inhomogeneousmedium with Kerr nonlinearity This result generalizes thewell-known similar statement for dielectric waveguides ofcircular cross-section filled with a linearmedium (ie 120572 = 0)

It should be noticed that the value 1205720can be effectively

estimated

6 Conclusion

In this study we suggest and develop a method to inves-tigate the problem of existence of electromagnetic wavesthat propagate along axis of an inhomogeneous nonlinearcylindrical waveguideThe nonlinearity inside the waveguideis described by the Kerr law the inhomogeneity is describedby a function that depends on radius of the waveguide

Here we show that the integral equation approach allowsus to investigate quite general problem for nonlinear inhomo-geneous waveguides

We should say that this method can be used to proveexistence of guided waves in a nonlinear inhomogeneouswaveguide for TM waves

Numerical results can be obtained with the help ofiteration procedure from Section 4

A separate paper will be devoted to development of acouple of numerical methods for this problem

ISRNMathematical Physics 7

Acknowledgments

This work is partially supported by the RFBR (Grants nos 11-07-00330-A 12-07-97010-RA) theMinistry of Education andScience of the Russian Federation (Grant no 14B37211950)

References

[1] M J AdamsAn Introduction To OpticalWaveguide JohnWileyamp Sons Chichester UK 1981

[2] A Snyder and J LoveOpticalWaveguideTheory Chapman andHall London UK 1983

[3] N N Akhmediev and A Ankevich Solitons Nonlinear Pulsesand Beams Chapman and Hall London UK 1997

[4] Y R Shen The Principles of Nonlinear Optics John Wiley ampSons New York NY USA 1984

[5] H M Gibbs Optical Bistability Controlling Light with LightAcademic Press New York NY USA 1985

[6] P N Eleonskii L G Oganersquosyants and V P Silin ldquoCylindricalnonlinear waveguidesrdquo Journal of Experimental andTheoreticalPhysics vol 35 no 1 pp 44ndash47 1972

[7] KM Leung ldquoP-polarized nonlinear surface polaritons inmate-rials with intensity-dependent dielectric functionsrdquo PhysicalReview B vol 32 no 8 pp 5093ndash5101 1985

[8] R I Joseph and D N Christodoulides ldquoExact field decompo-sition for tm waves in nonlinear mediardquo Optics Letters vol 12no 10 pp 826ndash828 1987

[9] D V Valovik ldquoPropagation of tmwaves in a layer with arbitrarynonlinearityrdquo Computational Mathematics and MathematicalPhysics vol 51 no 9 pp 1622ndash1632 2011

[10] D V Valovik ldquoPropagation of electromagnetic TE waves in anonlinear mediumwith saturationrdquo Journal of CommunicationsTechnology and Electronics vol 56 no 11 pp 1311ndash1316 2011

[11] G Yu Smirnov and D V Valovik Electromagnetic WavePropagation inNon-Linear LayeredWaveguide Structures PenzaState University Press Penza Russia 2011

[12] H W Schurmann V S Serov and Y V Shestopalov ldquoTE-polarized waves guided by a lossless nonlinear three-layerstructurerdquo Physical Review E vol 58 no 1 pp 1040ndash1050 1998

[13] D V Valovik ldquoPropagation of electromagnetic waves in anonlinearmetamaterial layerrdquo Journal of Communications Tech-nology and Electronics vol 56 no 5 pp 544ndash556 2011

[14] HW SchurmannV S Serov andYuV Shestopalov ldquoSolutionsto the Helmholtz equation for TE-guided waves in a three-layerstructure with Kerr-type nonlinearityrdquo Journal of Physics A vol35 no 50 pp 10789ndash10801 2002

[15] D V Valovik and Yu G Smirnov ldquoPropagation of tm wavesin a kerr nonlinear layerrdquo Computational Mathematics andMathematical Physics vol 48 no 12 pp 2217ndash2225 2008

[16] DVValovik andYG Smirnov ldquoCalculation of the propagationconstants of TM electromagnetic waves in a nonlinear layerrdquoJournal of Communications Technology and Electronics vol 53no 8 pp 883ndash889 2008

[17] DVValovik andYG Smirnov ldquoCalculation of the propagationconstants and fields of polarized electromagnetic TM wavesin a nonlinear anisotropic layerrdquo Journal of CommunicationsTechnology and Electronics vol 54 no 4 pp 391ndash398 2009

[18] G Yu Smirnov and D V Valovik ldquoBoundary eigenvalueproblem for maxwell equations in a nonlinear dielectric layerrdquoApplied Mathematics no 1 pp 29ndash36 2010

[19] D V Valovik and Y G Smirnov ldquoNonlinear effects in theproblem of propagation of TM electromagnetic waves in a Kerrnonlinear layerrdquo Journal of Communications Technology andElectronics vol 56 no 3 pp 283ndash288 2011

[20] K A Yuskaeva V S Serov and H W Schurmann ldquoTm-electromagnetic guided waves in a (kerr-) nonlinear three-layerstructurerdquo in Proceedings of the Progress in ElectromagneticsResearch Symposium (PIERS rsquo09) pp 364ndash369 Moscow RussiaAugust 2009

[21] H W Schurmann G Yu Smirnov and V Yu ShestopalovldquoPropagation of te-waves in cylindrical nonlinear dielectricwaveguidesrdquo Physical Review E vol 71 no 1 Article ID 0166142005

[22] Y Smirnov H W Schurmann and Y Shestopalov ldquoIntegralequation approach for the propagation of TE-waves in a non-linear dielectric cylindrical waveguiderdquo Journal of NonlinearMathematical Physics vol 11 no 2 pp 256ndash268 2004

[23] Yu G Smirnov and S N Kupriyanova ldquoPropagation of electro-magnetic waves in cylindrical dielectric waveguides filled witha nonlinear mediumrdquo Computational Mathematics and Mathe-matical Physics vol 44 no 10 pp 1850ndash1860 2004

[24] G Yu Smirnov and E A Horosheva ldquoOn the solvabilityof the nonlinear boundary eigenvalue problem for tm wavespropagation in a circle cylyindrical nonlinear waveguiderdquoIzvestiya Vysshikh Uchebnykh Zavedenij Povolzh Region Fiziki-Matematicheskie Nauki no 3 pp 55ndash70 2010 (Russian)

[25] G Yu Smirnov andD V Valovik ldquoCoupled electromagnetic te-tm wave propagation in a layer with kerr nonlinearityrdquo Journalof Mathematical Physics vol 53 no 12 Article ID 123530 pp1ndash24 2012

[26] V S Serov K A Yuskaeva and H W Schurmann ldquoIntegralequations approach to tm-electromagnetic waves guided bya (linearnonlinear) dielectric film with a spatially varyingpermittivityrdquo in Proceedings of the Progress in ElectromagneticsResearch Symposium (PIERS rsquo09) pp 1915ndash1919 Moscow Rus-sia August 2009

[27] J A Stretton Electromagnetic Theory McGraw Hill New YorkNY USA 1941

[28] L D Landau E M Lifshitz and L P Pitaevskii Course ofTheoretical Physics vol 8 of Electrodynamics of ContinuousMedia Butterworth-Heinemann Oxford UK 1993

[29] M A Naimark Linear Differential Operators Part I ElementaryTheory of Linear Differential Operators Frederick Ungar NewYork NY USA 1967

[30] M A Naimark Linear Differential Operators Part II LinearDifferential Operators in Hilbertspace Frederick Ungar NewYork NY USA 1968

[31] I Stakgold Greenrsquos Functions and Boundary Value ProblemsJohn Wiley amp Sons New York NY USA 1979

[32] V A Trenogin Functional Analysis Nauka Moscow Russia1993

[33] Y G Smirnov ldquoPropagation of electromagnetic waves in cylin-drical dielectric waveguides filled with a nonlinear mediumrdquoJournal of Communications Technology and Electronics vol 50no 2 pp 179ndash185 2005

[34] P I Lizorkin Course of Differential and Integral Equations withSupplementary Chapters of Calculus Nauka Moscow Russia1981

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRNMathematical Physics 3

0

z

120588

120593

R

120576

1205830

1205761

1205830

Figure 1 Geometry of the problem

the radiation condition at infinity the electromagnetic fieldexponentially decays as 120588 rarr infin

The permittivity in the entire space has the form

120576 = 1205760

1205761 120588 gt 119877

1205762(120588) + 119886|E|2 120588 lt 119877

(9)

where 119886 is a real positive value 1205762(120588) gt 120576

1 Here 120576

2(120588) is a

linear part of the permittivityThe solutions to the Maxwell equations are sought in the

entire spaceThereby passing from time-dependent equations (1) to

time-independent equations (8) is grounded on previousconsideration

Geometry of the problem is shown in Figure 1 Thewaveguide is infinite along axis 119874119911

Let us consider TE waves with harmonical dependenceon time

E119890minus119894120596119905 = 119890minus119894120596119905(0 119864120593 0)

119879

H119890minus119894120596119905 = 119890minus119894120596119905(119867120588 0119867119911)

119879

(10)

where EH are the complex amplitudesSubstituting the complex amplitudes into Maxwell equa-

tions (8) we obtain

1

120588

120597119867119911

120597120593

= 0

120597119867120588

120597119911

minus

120597119867119911

120597120588

= minus119894120596120576119864120593

1

120588

120597119867120588

120597120593

= 0

120597119864120593

120597119911

= minus119894120596120583119867120588

1

120588

120597 (120588119864120593)

120597120588

= 119894120596120583119867119911

(11)

It is obvious from the first and the third equations of thissystem that119867

119911and119867

120588do not depend on 120593 This implies that

119864120593does not depend on 120593Independence of the components on 120593 can be explained

if we chose dependence on 120593 in the form 119890119894119899120593 with 119899 = 0

Waves propagating along waveguide axis 119874119911 dependharmonically on 119911 This means that the fields componentshave the form

119864120593= 119864120593(120588) 119890119894120574119911 119867

120588= 119867120588(120588) 119890119894120574119911

119867119911= 119867119911(120588) 119890119894120574119911

(12)

where 120574 is the unknown spectral parameter of the problem(propagation constant)

So we obtain from system (11) that

119894120574119867120588(120588) minus 119867

1015840

119911(120588) = minus119894120596120576119864

120593(120588)

119894120574119864120593(120588) = minus119894120596120583119867

120588(120588)

1

120588

(120588119864120593(120588))

1015840

= 119894120596120583119867119911(120588)

(13)

where ( sdot )1015840 equiv 119889119889120588Then 119867

119911(120588) = (1119894120596120583)(1120588)(120588119864

120593(120588))1015840 and 119867

120588(120588) =

minus(120574120596120583)119864120593(120588) From the first equation of the latter system

we obtain

(

1

120588

(120588119864120593(120588))

1015840

)

1015840

+ (1205962120583120576 minus 120574

2) 119864120593(120588) = 0 (14)

Denoting by 119906(120588) = 119864120593(120588) we obtain

11990610158401015840+

1

120588

1199061015840minus

1

1205882119906 + (119896

2

0120576 minus 1205742) 119906 = 0 (15)

and 120576 = 1205761205760 where

120576 =

1205761 120588 gt 119877

1205762(120588) + 119886119906

2 120588 lt 119877

(16)

and 11989620= 12059621205831205760

Also we assume that function 119906 is sufficiently smooth

119906 (120588) isin 119862 [0 +infin) cap 1198621[0 +infin) cap 119862

2(0 119877) cap 119862

2(119877 +infin)

(17)

Physical nature of the problem implies these conditionsWe will seek 120574 under conditions 1198962

01205761lt 120574

2lt

1198962

0min120588isin[0119877]

1205762(120588)

In the domain 120588 gt 119877 we have 120576 = 1205761 From (15) we obtain

the equation

11990610158401015840+

1

120588

1199061015840minus

1

1205882119906 + 1198962

1119906 = 0 (18)

where 11989621= 1198962

01205761minus 1205742 It is the Bessel equation

In the domain 120588 lt 119877 we have 120576 = 1205762(120588) + 119886119906

2 From (15)we obtain the equation

11990610158401015840+

1

120588

1199061015840minus

1

1205882119906 + 1198962(120588) 119906 + 120572119906

3= 0 (19)

where 1198962(120588) = 11989622(120588) minus 120574

2 11989622(120588) = 119896

2

01205762(120588) and 120572 = 1198861198962

0

4 ISRNMathematical Physics

Tangential components of electromagnetic field areknown to be continuous atmedia interfaces Hence we obtain

119864120593(119877 + 0) = 119864

120593(119877 minus 0) 119867

119911(119877 + 0) = 119867

119911(119877 minus 0)

(20)

Further we have 119867119911(120588) = (1119894120596120583)(1120588)119864

120593(120588) + 119864

1015840

120593(120588))

Since 119864120593(120588) and 119867

119911(120588) are continuous at the point 120588 = 119877

therefore 1198641015840120593(120588) is continuous at 120588 = 119877 These conditions

imply the transmission conditions for functions 119906(120588) and1199061015840(120588)

[119906]120588=119877 = 0 [1199061015840]120588=119877= 0 (21)

where [119891]119909=1199090= lim119909rarr1199090minus0

119891(119909) minus lim119909rarr1199090+0

119891(119909)Let us formulate the transmission eigenvalue problem

(problem P) It is necessary to find eigenvalues 120574 andcorrespond to them nonzero eigenfunctions 119906(120588) such that119906(120588) satisfy (18) (19) transmission conditions (21) and theradiation condition at infinity eigenfunctions exponentiallydecay as 120588 rarr infin

The general solution of (18) is taken in the followingform 119906(120588) = 119887119867(1)

1(1198961120588) + 119887

1119867(2)

1(1198961120588) where 119867(1)

1and 119867(2)

1

are the Hankel functions of the first and the second kindsrespectively In accordance with the radiation condition weobtain that 119887

1= 0 then the solution has the form 119906(120588) =

119887119867(1)

1(1198961120588) 120588 gt 119877 where 119887 is a constant If Re 119896

1= 0 then

119906 (120588) =1198871198701(10038161003816100381610038161198961

1003816100381610038161003816120588) 120588 gt 119877 (22)

as 119867(1)1(119894119911) = minus(2120587)119870

1(119911) and 119870

1(119911) is the Macdonald

functionThe radiation condition is fulfilled since 119870

1(|1198961|120588) rarr 0

as 120588 rarr infin

3 Nonlinear Integral Equation andDispersion Equation

Consider nonlinear equation (19) written in the form

(1205881199061015840)

1015840

+ (1198962(120588) 120588 minus

1

120588

) 119906 + 1205721205881199063= 0 (23)

and the linear equation

(1205881199061015840)

1015840

+ (1198962(120588) 120588 minus

1

120588

) 119906 = 0 (24)

The latter equation can be written in the operator form as

119871119896119906 = 0 119871

119896=

119889

119889120588

(120588

119889

119889120588

) + (1198962(120588) 120588 minus

1

120588

) (25)

(here we place index 119896 in order to stress that the operator andthe Green function depend on 119896(120588))

Suppose that the Green function119866119896(120588 1205880 120582) exists for the

following boundary value problem

119871119896119866119896= minus120575 (120588 minus 120588

0) 119866|120588=0 = 119866

101584010038161003816100381610038161003816120588=119877= 0 (0 lt 120588

0lt 119877)

(26)

In this case the Green function has the representation(see eg [29 30])

119866119896(120588 1205880 120582) = minus

V119894(120588) V119894(1205880)

120582 minus 120582119894

+ 1198661(120588 1205880 120582) (27)

in the vicinity of eigenvalue 120582119894 Here 120582 = 1205742 and 119866

1(120588 1205880 120582)

is regular with respect to 120582 in the vicinity of 120582119894 120582119899 V119899(120588) are

complete orthonormal (real) eigenvalues and eigenfunctionssystems of boundary eigenvalue problem

(120588V1015840

119899)

1015840

+ (1198962

2(120588) 120588 minus

1

120588

) V119899= 120582119899120588V119899

V119899

1003816100381610038161003816120588=0= V1015840

119899

10038161003816100381610038161003816 120588=119877

= 0

(28)

The Green function exists if 120582 = 120582119894

For 1205762equiv const explicit form of theGreen function is given

in [21]Let us write (19) in the operator form

119871119896119906 + 120572119861 (119906) = 0 119861 (119906) = 120588119906

3(120588) (29)

Using the second Green formula [31]

int

119877

0

(V119871119896119906 minus 119906119871

119896V) 119889120588 = int

119877

0

(V(1205881199061015840)

1015840

minus 119906(120588V1015840)

1015840

) 119889120588

= 119877 (1199061015840(119877) V (119877) minus V

1015840(119877) 119906 (119877))

(30)

and assuming that V = 119866 we obtain that

int

119877

0

(119866119896119871119896119906 minus 119906119871

119896119866119896) 119889119901

= 119877 (1199061015840(119877 minus 0) 119866

119896(119877 1205880) minus 1198661015840

119896(119877 1205880) 119906 (119877 minus 0))

= 1198771199061015840(119877 minus 0) 119866

119896(119877 1205880)

(31)

From the previous formulae we obtain

int

119877

0

119906119871119896119866119896119889120588 = minusint

119877

0

119906 (120588) 120575 (120588 minus 1205880) 119889120588 = minus119906 (120588

0)

int

119877

0

119866119896119871119896119906119889120588 = minus120572int

119877

0

119866119896(120588 1205880) 1205881199063(120588) 119889120588

(32)

Taking into account these results and using (29) weobtain the nonlinear integral representation of solution 119906(120588

0)

of (19) on the segment [0 119877]

119906 (1205880) = 120572int

119877

0

119866119896(120588 1205880) 1205881199063(120588) 119889120588

+ 1198771199061015840(119877 minus 0) 119866

119896(119877 1205880) 0 le 120588

0le 119877

(33)

Using transmission conditions 1199061015840(119877 minus 0) = 1199061015840(119877 + 0) wecan rewrite (33)

119906 (1205880) = 120572int

119877

0

119866119896(120588 1205880) 1205881199063(120588) 119889120588 + 119891 (120588

0) 0 le 120588

0le 119877

(34)

where 119891(1205880) = 119877119906

1015840(119877 + 0)119866

119896(119877 1205880)

ISRNMathematical Physics 5

Using (34) and transmission condition119906(119877minus0) = 119906(119877+0)we obtain the dispersion equation (DE) with respect to thepropagation constant

119906 (119877 + 0) = 120572int

119877

0

119866119896(120588 119877) 120588119906

3(120588) 119889120588 + 119877119906

1015840(119877 + 0) 119866 (119877 119877)

(35)

Let us denote by 119873(120588 1205880 120582) = 120572119866

119896(120588 1205880 120582)120588 and

consider integral equation (34)

119906 (1205880) = int

119877

0

119873(120588 1205880) 1199063(120588) 119889120588 + 119891 (120588

0) (36)

in 119862[0 119877] [32] It is assumed that 119891 isin 119862[0 119877] and 120582 = 120582119894

The kernel 119873(120588 1205880) is continuous in the square 0 le

120588 1205880le 119877

Let us consider linear integral operator 119873119908 =

int

119877

0119873(120588 120588

0)119908(120588)119889120588 in 119862[0 119877] It is bounded completely

continuous and 119873 = max1205880isin[0119877]

int

119877

0|119873(120588 120588

0)|119889120588

Since nonlinear operator 1198610(119906) = 119906

3(120588) is bounded and

continuous in 119862[0 119877] therefore nonlinear operator 119865(119906) =int

119877

0119873(120588 120588

0)1199063(120588)119889120588 + 119891(120588

0) is completely continuous in any

bounded set in 119862[0 119877]The following theorems (about existence of a unique

solution and continuous dependence of the solution on theparameter) can be proved in the same way as for the case of ahomogeneous nonlinear cylindrical waveguide (for details ofproofs see [22 33])

Proposition 1 If 120572 le 1198602 where

119860 =

2

3

1

10038171003817100381710038171198911003817100381710038171003817radic310038171003817100381710038171198731

1003817100381710038171003817

10038171003817100381710038171198731

1003817100381710038171003817= max1205880isin[0119877]

int

119877

0

1003816100381610038161003816120588119866119896(120588 1205880)1003816100381610038161003816119889120588

(37)

then (36) has a unique continuous solution 119906 isin 119862[0 119877] suchthat 119906 le 119903

lowast where

119903lowast= minus

2

radic3 119873

cos(13

arccos(3radic3

2

10038171003817100381710038171198911003817100381710038171003817radic119873) minus

2120587

3

)

(38)

is a root of the equation 1198731199033 + 119891 = 119903

Note that 119860 gt 0 does not depend on 120572

Proposition 2 Let the kernel119873 and the right-hand side 119891 ofequation (36) depend continuously on the parameter 120582 isin Λ

0

119873(120588 1205880 120582) sub 119862([0 119877]times[0 119877]timesΛ

0)119891(1205880 120582) sub 119862([0 119877]timesΛ

0)

on some segment Λ0of the real number axis Let also

0 lt1003817100381710038171003817119891 (120582)

1003817100381710038171003817lt

2

3radic3 119873 (120582)

(39)

Then for 120582 isin Λ0 a unique solution 119906(120588 120582) of (36) exists

and depends continuously on 120582 119906(120588 120582) sub 119862([0 119877] times Λ0)

4 Iteration Method

Approximate solutions 119906119899of integral equation (36) repre-

sented in the form 119906 = 119882(119906) can be found by means of theiteration process 119906

119899+1= 119882(119906

119899) 119899 = 0 1

1199060= 0 119906

119899+1= 120572int

119877

0

119866119896(120588 1205880) 1205881199063

119899119889120588 + 119891

119899 = 0 1

(40)

The sequence119906119899converges uniformly to solution119906 of (36)

by virtue of the fact that 119865(119906) is a contracting operator Theestimate of the convergence rate of iteration process (40) isalso known Let us formulate these results as the following(for proof see [22])

Proposition 3 The sequence of approximate solutions 119906119899of

(36) obtained by means of iteration process (40) converges inthe norm of space 119862[0 119877] to (unique) exact solution 119906 of thisequationThe following estimate of the convergence rate is valid119906119899minus 119906 le (119902

119899(1 minus 119902))119891(119906

0) 119899 rarr infin where 119902 = 31198731199032

lowastlt 1

is the coefficient of contraction of mapping 119865

5 Theorem of Existence

Taking into account formula (22) DE (35) can be representedin the form

1198701(10038161003816100381610038161198961

1003816100381610038161003816119877) minus

10038161003816100381610038161198961

10038161003816100381610038161198771198701015840

1(10038161003816100381610038161198961

1003816100381610038161003816119877) 119866119896(119877 119877 120582)

=

120572

119887

int

119877

0

119866119896(120588 119877 120582) 120588119906

3(120588) 119889120588

(41)

As it can be seen DE (41) depends on 119887 Here 119887 is an initialcondition This is the peculiarity of this (and not only this)nonlinear problem For the linear problem (if 120572 = 0) weobtain as it is expected the DE that does not depend on theinitial condition

From the properties of Bessel functions it follows that

minus10038161003816100381610038161198961

10038161003816100381610038161198771198701015840

1(10038161003816100381610038161198961

1003816100381610038161003816119877) =

10038161003816100381610038161198961

10038161003816100381610038161198771198700(10038161003816100381610038161198961

1003816100381610038161003816119877) + 119870

1(10038161003816100381610038161198961

1003816100381610038161003816119877) (42)

Now we can rewrite DE (41) in the following form

119892 (120582) = 120572119865 (120582) (43)

where

119892 (120582) = 1198701(10038161003816100381610038161198961

1003816100381610038161003816119877)

+ (10038161003816100381610038161198961

10038161003816100381610038161198771198700(10038161003816100381610038161198961

1003816100381610038161003816119877) + 119870

1(10038161003816100381610038161198961

1003816100381610038161003816119877))119866

119896

(119877 119877 120582)

119865 (120582) = int

119877

0

119866119896(120588 119877 120582) 120588119906

3(120588) 119889120588

(44)We should note that DE (41) depends on frequency 120596

implicitly If one obtains 120582lowast for chosen119877lowast (radius of the innercore) such that 119892(120582lowast) = 120572119865(120582lowast) is satisfied then one cancalculate 120596lowast which satisfies the propagation constants 120574lowast =radic120582lowast using formulae in the beginning of this section

6 ISRNMathematical Physics

The zeros of the function Φ(120574) equiv 119892(120582) minus 120572119865(120582) are thosevalues of 120582 for which a nonzero solution of the problem Pexists The following assertion gives us sufficient conditionsfor the existence of zeros of the function Φ

Let us consider the question about existence of solutionsof the linear problem 119892(120582) = 0

This equation can be rewritten in the form

119866119896(119877 119877 120582) = minus

1198701(10038161003816100381610038161198961

1003816100381610038161003816119877)

10038161003816100381610038161198961

10038161003816100381610038161198771198700(10038161003816100381610038161198961

1003816100381610038161003816119877) + 119870

1(10038161003816100381610038161198961

1003816100381610038161003816119877)

(45)

From expression 119866119896(119877 119877 120582) = minusV2

119894(119877)(120582 minus 120582

119894) +

1198661(119877 119877 120582) it follows that 119866

119896(119877 119877 120582) continuously varies

from minusinfin to +infin when 120582 varies from 120582119894to 120582119894+1

As value 119870

1(|1198961|119877)(|119896

1|1198771198700(|1198961|119877) + 119870

1(|1198961|119877)) is

bounded then there is at least one root of equation 119892(120582) = 0and this root lies between 120582

119894and 120582

119894+1

Finally it is necessary to prove that term V119894(119877) does

not vanish in expression 119866119896(119877 119877 120582) We prove this fact by

contradiction Let V119894(119877) = 0 Consider a Cauchy problem

for equation 120588V10158401015840119894+ V1015840119894+ (1198962

2(120588)120588 minus 1120588)V

119894= 120582119894120588V119894with

initial conditions V119894|120588=119877= V1015840119894|120588=119877= 0 as 120588 isin [120575 119877] where

120575 gt 0 From the general theory of ordinary differentialequations (see eg [34]) it is known that solution V

119894(120588) of

consideredCauchy problem exists and is unique as 120588 isin [120575 119877]In this case this solution coincides with function V

119894(119877) as

120588 isin [120575 119877] Function V119894(119877) is the function which is contained

in Greenrsquos function representation (27) On the other handa solution of the Cauchy problem for a linear equation withzero initial condition is the trivial solution This contradictswith representation (27) ofGreenrsquos function119866

119896(120588 1205880 120582) in the

vicinity of 120582 = 120582119894

Consider nonlinear problem Let inequalities

1205761lt 1205820lt 1205821lt sdot sdot sdot lt 120582

119896minus1lt 120582119896lt 1205762 (46)

hold where 119896 ge 1 and 1205762= min

120588isin[0119877]1205762(120588)

We can choose sufficiently small 120575119894gt 0 such that the

Green function 119866119896(120588 1205880 120582) exists and is continuous on Γ =

⋃119896

119894=1Γ119894 where

Γ119894= [radic120582

119894minus1+ 120575119894minus1 radic120582119894minus 120575119894] 119894 = 1 119896 (47)

and the following inequality 119892(radic120582119894minus1+ 120575119894minus1)119892(radic120582

119894minus 120575119894) lt 0

is satisfiedIt follows from the choice of 120575

119894that 119865(120582) is bounded

Moreover product 120572119865(120582) can be made sufficiently small bychoosing appropriate 120572 (the estimation is given at the end ofthis section) Let us consider DE Φ(120582) = 0 As it is shownbefore function 119892(120582) is continuous and reverse sign when 120582varies from 120582

119894minus1+120575119894minus1

to 120582119894minus120575119894 As function 119865(120582) is bounded

then it is clear that equation Φ(120582) = 0 has at least 119896 roots 120582119894

119894 = 1 119896 if we choose appropriate 120572 Here 120582119894isin (120582119894minus1+120575119894minus1 120582119894minus

120575119894) 119894 = 1 119896On the basis of previous consideration we can formulate

the main result of this paper

Theorem 4 Let the values 1205761 1205762= min

120588isin[0119877]1205762(120588) 120572 satisfy

condition 1205762gt 1205761gt 0 and let the following inequalities

1205761lt 1205820lt 1205821lt sdot sdot sdot lt 120582

119896minus1lt 120582119896lt 1205762hold where 119896 ge 1 is

an integerThen there is a value 1205720gt 0 such that for any 120572 le 120572

0

at least 119896 values 120574119894 119894 = 1 119896 exist such that the problem P has a

nonzero solution and 120574119894isin (radic120582

119894minus1+ 120575119894minus1 radic120582119894minus 120575119894)

Proof The Green function exists for all 120574 isin Γ It is also clearthat function 119860(120574) = 23119891(120574)radic3119873

1(120574) is continuous as

120574 isin Γ Let 1198601= min

120574isinΓ119860(120574) and 120572 lt 1198602

1 In accordance

with Proposition 1 there is a unique solution 119906 = 119906(120574) of(36) for any 120574 isin Γ This solution is continuous and 119906 le119903lowast= 119903lowast(120574) Let 119903

00= max

120574isinΓ119903lowast(120574) The following estimation

|119865(120582)| le 1198621199033

00is valid where 119862 is a constant

Function 119892(120574) is continuous and equation 119892(120574) = 0 has atleast one root 120574

119894inside segment Γ

119894 that isradic120582

119894minus1+ 120575119894minus1lt 120574119894lt

radic120582119894minus 120575119894 Let us denote119872

1= min

0leile119896minus1|119892(radic120582119894 + 120575119894)|1198722 =min1le119894le119896|119892(radic120582

119894minus 120575119894)| Value = min119872

11198722 is positive

and does not depend on 120572If 120572 le 1198621199033

00 then

(119892(radic120582119894minus1+ 120575119894minus1) minus 120572119865(radic120582

119894minus1+ 120575119894minus1))

times (119892(radic120582119894minus 120575119894) minus 120572119865(radic120582

119894minus 120575119894)) lt 0

(48)

As 119892(120582) minus 120572119865(120582) is continuous it follows that equation119892(120582)minus120572119865(120582) = 0 has a root 120574

119894inside Γ

119894 that isradic120582

119894+120575119894lt 120574119894lt

radic120582119894+1minus 120575119894+1

We can choose 1205720= min1198602

1 119862119903

3

00

From Theorem 4 it follows that under the previousassumptions there exist axially symmetrical propagating TEwaves in cylindrical dielectric waveguides of circular cross-section filled with a nonmagnetic isotropic inhomogeneousmedium with Kerr nonlinearity This result generalizes thewell-known similar statement for dielectric waveguides ofcircular cross-section filled with a linearmedium (ie 120572 = 0)

It should be noticed that the value 1205720can be effectively

estimated

6 Conclusion

In this study we suggest and develop a method to inves-tigate the problem of existence of electromagnetic wavesthat propagate along axis of an inhomogeneous nonlinearcylindrical waveguideThe nonlinearity inside the waveguideis described by the Kerr law the inhomogeneity is describedby a function that depends on radius of the waveguide

Here we show that the integral equation approach allowsus to investigate quite general problem for nonlinear inhomo-geneous waveguides

We should say that this method can be used to proveexistence of guided waves in a nonlinear inhomogeneouswaveguide for TM waves

Numerical results can be obtained with the help ofiteration procedure from Section 4

A separate paper will be devoted to development of acouple of numerical methods for this problem

ISRNMathematical Physics 7

Acknowledgments

This work is partially supported by the RFBR (Grants nos 11-07-00330-A 12-07-97010-RA) theMinistry of Education andScience of the Russian Federation (Grant no 14B37211950)

References

[1] M J AdamsAn Introduction To OpticalWaveguide JohnWileyamp Sons Chichester UK 1981

[2] A Snyder and J LoveOpticalWaveguideTheory Chapman andHall London UK 1983

[3] N N Akhmediev and A Ankevich Solitons Nonlinear Pulsesand Beams Chapman and Hall London UK 1997

[4] Y R Shen The Principles of Nonlinear Optics John Wiley ampSons New York NY USA 1984

[5] H M Gibbs Optical Bistability Controlling Light with LightAcademic Press New York NY USA 1985

[6] P N Eleonskii L G Oganersquosyants and V P Silin ldquoCylindricalnonlinear waveguidesrdquo Journal of Experimental andTheoreticalPhysics vol 35 no 1 pp 44ndash47 1972

[7] KM Leung ldquoP-polarized nonlinear surface polaritons inmate-rials with intensity-dependent dielectric functionsrdquo PhysicalReview B vol 32 no 8 pp 5093ndash5101 1985

[8] R I Joseph and D N Christodoulides ldquoExact field decompo-sition for tm waves in nonlinear mediardquo Optics Letters vol 12no 10 pp 826ndash828 1987

[9] D V Valovik ldquoPropagation of tmwaves in a layer with arbitrarynonlinearityrdquo Computational Mathematics and MathematicalPhysics vol 51 no 9 pp 1622ndash1632 2011

[10] D V Valovik ldquoPropagation of electromagnetic TE waves in anonlinear mediumwith saturationrdquo Journal of CommunicationsTechnology and Electronics vol 56 no 11 pp 1311ndash1316 2011

[11] G Yu Smirnov and D V Valovik Electromagnetic WavePropagation inNon-Linear LayeredWaveguide Structures PenzaState University Press Penza Russia 2011

[12] H W Schurmann V S Serov and Y V Shestopalov ldquoTE-polarized waves guided by a lossless nonlinear three-layerstructurerdquo Physical Review E vol 58 no 1 pp 1040ndash1050 1998

[13] D V Valovik ldquoPropagation of electromagnetic waves in anonlinearmetamaterial layerrdquo Journal of Communications Tech-nology and Electronics vol 56 no 5 pp 544ndash556 2011

[14] HW SchurmannV S Serov andYuV Shestopalov ldquoSolutionsto the Helmholtz equation for TE-guided waves in a three-layerstructure with Kerr-type nonlinearityrdquo Journal of Physics A vol35 no 50 pp 10789ndash10801 2002

[15] D V Valovik and Yu G Smirnov ldquoPropagation of tm wavesin a kerr nonlinear layerrdquo Computational Mathematics andMathematical Physics vol 48 no 12 pp 2217ndash2225 2008

[16] DVValovik andYG Smirnov ldquoCalculation of the propagationconstants of TM electromagnetic waves in a nonlinear layerrdquoJournal of Communications Technology and Electronics vol 53no 8 pp 883ndash889 2008

[17] DVValovik andYG Smirnov ldquoCalculation of the propagationconstants and fields of polarized electromagnetic TM wavesin a nonlinear anisotropic layerrdquo Journal of CommunicationsTechnology and Electronics vol 54 no 4 pp 391ndash398 2009

[18] G Yu Smirnov and D V Valovik ldquoBoundary eigenvalueproblem for maxwell equations in a nonlinear dielectric layerrdquoApplied Mathematics no 1 pp 29ndash36 2010

[19] D V Valovik and Y G Smirnov ldquoNonlinear effects in theproblem of propagation of TM electromagnetic waves in a Kerrnonlinear layerrdquo Journal of Communications Technology andElectronics vol 56 no 3 pp 283ndash288 2011

[20] K A Yuskaeva V S Serov and H W Schurmann ldquoTm-electromagnetic guided waves in a (kerr-) nonlinear three-layerstructurerdquo in Proceedings of the Progress in ElectromagneticsResearch Symposium (PIERS rsquo09) pp 364ndash369 Moscow RussiaAugust 2009

[21] H W Schurmann G Yu Smirnov and V Yu ShestopalovldquoPropagation of te-waves in cylindrical nonlinear dielectricwaveguidesrdquo Physical Review E vol 71 no 1 Article ID 0166142005

[22] Y Smirnov H W Schurmann and Y Shestopalov ldquoIntegralequation approach for the propagation of TE-waves in a non-linear dielectric cylindrical waveguiderdquo Journal of NonlinearMathematical Physics vol 11 no 2 pp 256ndash268 2004

[23] Yu G Smirnov and S N Kupriyanova ldquoPropagation of electro-magnetic waves in cylindrical dielectric waveguides filled witha nonlinear mediumrdquo Computational Mathematics and Mathe-matical Physics vol 44 no 10 pp 1850ndash1860 2004

[24] G Yu Smirnov and E A Horosheva ldquoOn the solvabilityof the nonlinear boundary eigenvalue problem for tm wavespropagation in a circle cylyindrical nonlinear waveguiderdquoIzvestiya Vysshikh Uchebnykh Zavedenij Povolzh Region Fiziki-Matematicheskie Nauki no 3 pp 55ndash70 2010 (Russian)

[25] G Yu Smirnov andD V Valovik ldquoCoupled electromagnetic te-tm wave propagation in a layer with kerr nonlinearityrdquo Journalof Mathematical Physics vol 53 no 12 Article ID 123530 pp1ndash24 2012

[26] V S Serov K A Yuskaeva and H W Schurmann ldquoIntegralequations approach to tm-electromagnetic waves guided bya (linearnonlinear) dielectric film with a spatially varyingpermittivityrdquo in Proceedings of the Progress in ElectromagneticsResearch Symposium (PIERS rsquo09) pp 1915ndash1919 Moscow Rus-sia August 2009

[27] J A Stretton Electromagnetic Theory McGraw Hill New YorkNY USA 1941

[28] L D Landau E M Lifshitz and L P Pitaevskii Course ofTheoretical Physics vol 8 of Electrodynamics of ContinuousMedia Butterworth-Heinemann Oxford UK 1993

[29] M A Naimark Linear Differential Operators Part I ElementaryTheory of Linear Differential Operators Frederick Ungar NewYork NY USA 1967

[30] M A Naimark Linear Differential Operators Part II LinearDifferential Operators in Hilbertspace Frederick Ungar NewYork NY USA 1968

[31] I Stakgold Greenrsquos Functions and Boundary Value ProblemsJohn Wiley amp Sons New York NY USA 1979

[32] V A Trenogin Functional Analysis Nauka Moscow Russia1993

[33] Y G Smirnov ldquoPropagation of electromagnetic waves in cylin-drical dielectric waveguides filled with a nonlinear mediumrdquoJournal of Communications Technology and Electronics vol 50no 2 pp 179ndash185 2005

[34] P I Lizorkin Course of Differential and Integral Equations withSupplementary Chapters of Calculus Nauka Moscow Russia1981

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 ISRNMathematical Physics

Tangential components of electromagnetic field areknown to be continuous atmedia interfaces Hence we obtain

119864120593(119877 + 0) = 119864

120593(119877 minus 0) 119867

119911(119877 + 0) = 119867

119911(119877 minus 0)

(20)

Further we have 119867119911(120588) = (1119894120596120583)(1120588)119864

120593(120588) + 119864

1015840

120593(120588))

Since 119864120593(120588) and 119867

119911(120588) are continuous at the point 120588 = 119877

therefore 1198641015840120593(120588) is continuous at 120588 = 119877 These conditions

imply the transmission conditions for functions 119906(120588) and1199061015840(120588)

[119906]120588=119877 = 0 [1199061015840]120588=119877= 0 (21)

where [119891]119909=1199090= lim119909rarr1199090minus0

119891(119909) minus lim119909rarr1199090+0

119891(119909)Let us formulate the transmission eigenvalue problem

(problem P) It is necessary to find eigenvalues 120574 andcorrespond to them nonzero eigenfunctions 119906(120588) such that119906(120588) satisfy (18) (19) transmission conditions (21) and theradiation condition at infinity eigenfunctions exponentiallydecay as 120588 rarr infin

The general solution of (18) is taken in the followingform 119906(120588) = 119887119867(1)

1(1198961120588) + 119887

1119867(2)

1(1198961120588) where 119867(1)

1and 119867(2)

1

are the Hankel functions of the first and the second kindsrespectively In accordance with the radiation condition weobtain that 119887

1= 0 then the solution has the form 119906(120588) =

119887119867(1)

1(1198961120588) 120588 gt 119877 where 119887 is a constant If Re 119896

1= 0 then

119906 (120588) =1198871198701(10038161003816100381610038161198961

1003816100381610038161003816120588) 120588 gt 119877 (22)

as 119867(1)1(119894119911) = minus(2120587)119870

1(119911) and 119870

1(119911) is the Macdonald

functionThe radiation condition is fulfilled since 119870

1(|1198961|120588) rarr 0

as 120588 rarr infin

3 Nonlinear Integral Equation andDispersion Equation

Consider nonlinear equation (19) written in the form

(1205881199061015840)

1015840

+ (1198962(120588) 120588 minus

1

120588

) 119906 + 1205721205881199063= 0 (23)

and the linear equation

(1205881199061015840)

1015840

+ (1198962(120588) 120588 minus

1

120588

) 119906 = 0 (24)

The latter equation can be written in the operator form as

119871119896119906 = 0 119871

119896=

119889

119889120588

(120588

119889

119889120588

) + (1198962(120588) 120588 minus

1

120588

) (25)

(here we place index 119896 in order to stress that the operator andthe Green function depend on 119896(120588))

Suppose that the Green function119866119896(120588 1205880 120582) exists for the

following boundary value problem

119871119896119866119896= minus120575 (120588 minus 120588

0) 119866|120588=0 = 119866

101584010038161003816100381610038161003816120588=119877= 0 (0 lt 120588

0lt 119877)

(26)

In this case the Green function has the representation(see eg [29 30])

119866119896(120588 1205880 120582) = minus

V119894(120588) V119894(1205880)

120582 minus 120582119894

+ 1198661(120588 1205880 120582) (27)

in the vicinity of eigenvalue 120582119894 Here 120582 = 1205742 and 119866

1(120588 1205880 120582)

is regular with respect to 120582 in the vicinity of 120582119894 120582119899 V119899(120588) are

complete orthonormal (real) eigenvalues and eigenfunctionssystems of boundary eigenvalue problem

(120588V1015840

119899)

1015840

+ (1198962

2(120588) 120588 minus

1

120588

) V119899= 120582119899120588V119899

V119899

1003816100381610038161003816120588=0= V1015840

119899

10038161003816100381610038161003816 120588=119877

= 0

(28)

The Green function exists if 120582 = 120582119894

For 1205762equiv const explicit form of theGreen function is given

in [21]Let us write (19) in the operator form

119871119896119906 + 120572119861 (119906) = 0 119861 (119906) = 120588119906

3(120588) (29)

Using the second Green formula [31]

int

119877

0

(V119871119896119906 minus 119906119871

119896V) 119889120588 = int

119877

0

(V(1205881199061015840)

1015840

minus 119906(120588V1015840)

1015840

) 119889120588

= 119877 (1199061015840(119877) V (119877) minus V

1015840(119877) 119906 (119877))

(30)

and assuming that V = 119866 we obtain that

int

119877

0

(119866119896119871119896119906 minus 119906119871

119896119866119896) 119889119901

= 119877 (1199061015840(119877 minus 0) 119866

119896(119877 1205880) minus 1198661015840

119896(119877 1205880) 119906 (119877 minus 0))

= 1198771199061015840(119877 minus 0) 119866

119896(119877 1205880)

(31)

From the previous formulae we obtain

int

119877

0

119906119871119896119866119896119889120588 = minusint

119877

0

119906 (120588) 120575 (120588 minus 1205880) 119889120588 = minus119906 (120588

0)

int

119877

0

119866119896119871119896119906119889120588 = minus120572int

119877

0

119866119896(120588 1205880) 1205881199063(120588) 119889120588

(32)

Taking into account these results and using (29) weobtain the nonlinear integral representation of solution 119906(120588

0)

of (19) on the segment [0 119877]

119906 (1205880) = 120572int

119877

0

119866119896(120588 1205880) 1205881199063(120588) 119889120588

+ 1198771199061015840(119877 minus 0) 119866

119896(119877 1205880) 0 le 120588

0le 119877

(33)

Using transmission conditions 1199061015840(119877 minus 0) = 1199061015840(119877 + 0) wecan rewrite (33)

119906 (1205880) = 120572int

119877

0

119866119896(120588 1205880) 1205881199063(120588) 119889120588 + 119891 (120588

0) 0 le 120588

0le 119877

(34)

where 119891(1205880) = 119877119906

1015840(119877 + 0)119866

119896(119877 1205880)

ISRNMathematical Physics 5

Using (34) and transmission condition119906(119877minus0) = 119906(119877+0)we obtain the dispersion equation (DE) with respect to thepropagation constant

119906 (119877 + 0) = 120572int

119877

0

119866119896(120588 119877) 120588119906

3(120588) 119889120588 + 119877119906

1015840(119877 + 0) 119866 (119877 119877)

(35)

Let us denote by 119873(120588 1205880 120582) = 120572119866

119896(120588 1205880 120582)120588 and

consider integral equation (34)

119906 (1205880) = int

119877

0

119873(120588 1205880) 1199063(120588) 119889120588 + 119891 (120588

0) (36)

in 119862[0 119877] [32] It is assumed that 119891 isin 119862[0 119877] and 120582 = 120582119894

The kernel 119873(120588 1205880) is continuous in the square 0 le

120588 1205880le 119877

Let us consider linear integral operator 119873119908 =

int

119877

0119873(120588 120588

0)119908(120588)119889120588 in 119862[0 119877] It is bounded completely

continuous and 119873 = max1205880isin[0119877]

int

119877

0|119873(120588 120588

0)|119889120588

Since nonlinear operator 1198610(119906) = 119906

3(120588) is bounded and

continuous in 119862[0 119877] therefore nonlinear operator 119865(119906) =int

119877

0119873(120588 120588

0)1199063(120588)119889120588 + 119891(120588

0) is completely continuous in any

bounded set in 119862[0 119877]The following theorems (about existence of a unique

solution and continuous dependence of the solution on theparameter) can be proved in the same way as for the case of ahomogeneous nonlinear cylindrical waveguide (for details ofproofs see [22 33])

Proposition 1 If 120572 le 1198602 where

119860 =

2

3

1

10038171003817100381710038171198911003817100381710038171003817radic310038171003817100381710038171198731

1003817100381710038171003817

10038171003817100381710038171198731

1003817100381710038171003817= max1205880isin[0119877]

int

119877

0

1003816100381610038161003816120588119866119896(120588 1205880)1003816100381610038161003816119889120588

(37)

then (36) has a unique continuous solution 119906 isin 119862[0 119877] suchthat 119906 le 119903

lowast where

119903lowast= minus

2

radic3 119873

cos(13

arccos(3radic3

2

10038171003817100381710038171198911003817100381710038171003817radic119873) minus

2120587

3

)

(38)

is a root of the equation 1198731199033 + 119891 = 119903

Note that 119860 gt 0 does not depend on 120572

Proposition 2 Let the kernel119873 and the right-hand side 119891 ofequation (36) depend continuously on the parameter 120582 isin Λ

0

119873(120588 1205880 120582) sub 119862([0 119877]times[0 119877]timesΛ

0)119891(1205880 120582) sub 119862([0 119877]timesΛ

0)

on some segment Λ0of the real number axis Let also

0 lt1003817100381710038171003817119891 (120582)

1003817100381710038171003817lt

2

3radic3 119873 (120582)

(39)

Then for 120582 isin Λ0 a unique solution 119906(120588 120582) of (36) exists

and depends continuously on 120582 119906(120588 120582) sub 119862([0 119877] times Λ0)

4 Iteration Method

Approximate solutions 119906119899of integral equation (36) repre-

sented in the form 119906 = 119882(119906) can be found by means of theiteration process 119906

119899+1= 119882(119906

119899) 119899 = 0 1

1199060= 0 119906

119899+1= 120572int

119877

0

119866119896(120588 1205880) 1205881199063

119899119889120588 + 119891

119899 = 0 1

(40)

The sequence119906119899converges uniformly to solution119906 of (36)

by virtue of the fact that 119865(119906) is a contracting operator Theestimate of the convergence rate of iteration process (40) isalso known Let us formulate these results as the following(for proof see [22])

Proposition 3 The sequence of approximate solutions 119906119899of

(36) obtained by means of iteration process (40) converges inthe norm of space 119862[0 119877] to (unique) exact solution 119906 of thisequationThe following estimate of the convergence rate is valid119906119899minus 119906 le (119902

119899(1 minus 119902))119891(119906

0) 119899 rarr infin where 119902 = 31198731199032

lowastlt 1

is the coefficient of contraction of mapping 119865

5 Theorem of Existence

Taking into account formula (22) DE (35) can be representedin the form

1198701(10038161003816100381610038161198961

1003816100381610038161003816119877) minus

10038161003816100381610038161198961

10038161003816100381610038161198771198701015840

1(10038161003816100381610038161198961

1003816100381610038161003816119877) 119866119896(119877 119877 120582)

=

120572

119887

int

119877

0

119866119896(120588 119877 120582) 120588119906

3(120588) 119889120588

(41)

As it can be seen DE (41) depends on 119887 Here 119887 is an initialcondition This is the peculiarity of this (and not only this)nonlinear problem For the linear problem (if 120572 = 0) weobtain as it is expected the DE that does not depend on theinitial condition

From the properties of Bessel functions it follows that

minus10038161003816100381610038161198961

10038161003816100381610038161198771198701015840

1(10038161003816100381610038161198961

1003816100381610038161003816119877) =

10038161003816100381610038161198961

10038161003816100381610038161198771198700(10038161003816100381610038161198961

1003816100381610038161003816119877) + 119870

1(10038161003816100381610038161198961

1003816100381610038161003816119877) (42)

Now we can rewrite DE (41) in the following form

119892 (120582) = 120572119865 (120582) (43)

where

119892 (120582) = 1198701(10038161003816100381610038161198961

1003816100381610038161003816119877)

+ (10038161003816100381610038161198961

10038161003816100381610038161198771198700(10038161003816100381610038161198961

1003816100381610038161003816119877) + 119870

1(10038161003816100381610038161198961

1003816100381610038161003816119877))119866

119896

(119877 119877 120582)

119865 (120582) = int

119877

0

119866119896(120588 119877 120582) 120588119906

3(120588) 119889120588

(44)We should note that DE (41) depends on frequency 120596

implicitly If one obtains 120582lowast for chosen119877lowast (radius of the innercore) such that 119892(120582lowast) = 120572119865(120582lowast) is satisfied then one cancalculate 120596lowast which satisfies the propagation constants 120574lowast =radic120582lowast using formulae in the beginning of this section

6 ISRNMathematical Physics

The zeros of the function Φ(120574) equiv 119892(120582) minus 120572119865(120582) are thosevalues of 120582 for which a nonzero solution of the problem Pexists The following assertion gives us sufficient conditionsfor the existence of zeros of the function Φ

Let us consider the question about existence of solutionsof the linear problem 119892(120582) = 0

This equation can be rewritten in the form

119866119896(119877 119877 120582) = minus

1198701(10038161003816100381610038161198961

1003816100381610038161003816119877)

10038161003816100381610038161198961

10038161003816100381610038161198771198700(10038161003816100381610038161198961

1003816100381610038161003816119877) + 119870

1(10038161003816100381610038161198961

1003816100381610038161003816119877)

(45)

From expression 119866119896(119877 119877 120582) = minusV2

119894(119877)(120582 minus 120582

119894) +

1198661(119877 119877 120582) it follows that 119866

119896(119877 119877 120582) continuously varies

from minusinfin to +infin when 120582 varies from 120582119894to 120582119894+1

As value 119870

1(|1198961|119877)(|119896

1|1198771198700(|1198961|119877) + 119870

1(|1198961|119877)) is

bounded then there is at least one root of equation 119892(120582) = 0and this root lies between 120582

119894and 120582

119894+1

Finally it is necessary to prove that term V119894(119877) does

not vanish in expression 119866119896(119877 119877 120582) We prove this fact by

contradiction Let V119894(119877) = 0 Consider a Cauchy problem

for equation 120588V10158401015840119894+ V1015840119894+ (1198962

2(120588)120588 minus 1120588)V

119894= 120582119894120588V119894with

initial conditions V119894|120588=119877= V1015840119894|120588=119877= 0 as 120588 isin [120575 119877] where

120575 gt 0 From the general theory of ordinary differentialequations (see eg [34]) it is known that solution V

119894(120588) of

consideredCauchy problem exists and is unique as 120588 isin [120575 119877]In this case this solution coincides with function V

119894(119877) as

120588 isin [120575 119877] Function V119894(119877) is the function which is contained

in Greenrsquos function representation (27) On the other handa solution of the Cauchy problem for a linear equation withzero initial condition is the trivial solution This contradictswith representation (27) ofGreenrsquos function119866

119896(120588 1205880 120582) in the

vicinity of 120582 = 120582119894

Consider nonlinear problem Let inequalities

1205761lt 1205820lt 1205821lt sdot sdot sdot lt 120582

119896minus1lt 120582119896lt 1205762 (46)

hold where 119896 ge 1 and 1205762= min

120588isin[0119877]1205762(120588)

We can choose sufficiently small 120575119894gt 0 such that the

Green function 119866119896(120588 1205880 120582) exists and is continuous on Γ =

⋃119896

119894=1Γ119894 where

Γ119894= [radic120582

119894minus1+ 120575119894minus1 radic120582119894minus 120575119894] 119894 = 1 119896 (47)

and the following inequality 119892(radic120582119894minus1+ 120575119894minus1)119892(radic120582

119894minus 120575119894) lt 0

is satisfiedIt follows from the choice of 120575

119894that 119865(120582) is bounded

Moreover product 120572119865(120582) can be made sufficiently small bychoosing appropriate 120572 (the estimation is given at the end ofthis section) Let us consider DE Φ(120582) = 0 As it is shownbefore function 119892(120582) is continuous and reverse sign when 120582varies from 120582

119894minus1+120575119894minus1

to 120582119894minus120575119894 As function 119865(120582) is bounded

then it is clear that equation Φ(120582) = 0 has at least 119896 roots 120582119894

119894 = 1 119896 if we choose appropriate 120572 Here 120582119894isin (120582119894minus1+120575119894minus1 120582119894minus

120575119894) 119894 = 1 119896On the basis of previous consideration we can formulate

the main result of this paper

Theorem 4 Let the values 1205761 1205762= min

120588isin[0119877]1205762(120588) 120572 satisfy

condition 1205762gt 1205761gt 0 and let the following inequalities

1205761lt 1205820lt 1205821lt sdot sdot sdot lt 120582

119896minus1lt 120582119896lt 1205762hold where 119896 ge 1 is

an integerThen there is a value 1205720gt 0 such that for any 120572 le 120572

0

at least 119896 values 120574119894 119894 = 1 119896 exist such that the problem P has a

nonzero solution and 120574119894isin (radic120582

119894minus1+ 120575119894minus1 radic120582119894minus 120575119894)

Proof The Green function exists for all 120574 isin Γ It is also clearthat function 119860(120574) = 23119891(120574)radic3119873

1(120574) is continuous as

120574 isin Γ Let 1198601= min

120574isinΓ119860(120574) and 120572 lt 1198602

1 In accordance

with Proposition 1 there is a unique solution 119906 = 119906(120574) of(36) for any 120574 isin Γ This solution is continuous and 119906 le119903lowast= 119903lowast(120574) Let 119903

00= max

120574isinΓ119903lowast(120574) The following estimation

|119865(120582)| le 1198621199033

00is valid where 119862 is a constant

Function 119892(120574) is continuous and equation 119892(120574) = 0 has atleast one root 120574

119894inside segment Γ

119894 that isradic120582

119894minus1+ 120575119894minus1lt 120574119894lt

radic120582119894minus 120575119894 Let us denote119872

1= min

0leile119896minus1|119892(radic120582119894 + 120575119894)|1198722 =min1le119894le119896|119892(radic120582

119894minus 120575119894)| Value = min119872

11198722 is positive

and does not depend on 120572If 120572 le 1198621199033

00 then

(119892(radic120582119894minus1+ 120575119894minus1) minus 120572119865(radic120582

119894minus1+ 120575119894minus1))

times (119892(radic120582119894minus 120575119894) minus 120572119865(radic120582

119894minus 120575119894)) lt 0

(48)

As 119892(120582) minus 120572119865(120582) is continuous it follows that equation119892(120582)minus120572119865(120582) = 0 has a root 120574

119894inside Γ

119894 that isradic120582

119894+120575119894lt 120574119894lt

radic120582119894+1minus 120575119894+1

We can choose 1205720= min1198602

1 119862119903

3

00

From Theorem 4 it follows that under the previousassumptions there exist axially symmetrical propagating TEwaves in cylindrical dielectric waveguides of circular cross-section filled with a nonmagnetic isotropic inhomogeneousmedium with Kerr nonlinearity This result generalizes thewell-known similar statement for dielectric waveguides ofcircular cross-section filled with a linearmedium (ie 120572 = 0)

It should be noticed that the value 1205720can be effectively

estimated

6 Conclusion

In this study we suggest and develop a method to inves-tigate the problem of existence of electromagnetic wavesthat propagate along axis of an inhomogeneous nonlinearcylindrical waveguideThe nonlinearity inside the waveguideis described by the Kerr law the inhomogeneity is describedby a function that depends on radius of the waveguide

Here we show that the integral equation approach allowsus to investigate quite general problem for nonlinear inhomo-geneous waveguides

We should say that this method can be used to proveexistence of guided waves in a nonlinear inhomogeneouswaveguide for TM waves

Numerical results can be obtained with the help ofiteration procedure from Section 4

A separate paper will be devoted to development of acouple of numerical methods for this problem

ISRNMathematical Physics 7

Acknowledgments

This work is partially supported by the RFBR (Grants nos 11-07-00330-A 12-07-97010-RA) theMinistry of Education andScience of the Russian Federation (Grant no 14B37211950)

References

[1] M J AdamsAn Introduction To OpticalWaveguide JohnWileyamp Sons Chichester UK 1981

[2] A Snyder and J LoveOpticalWaveguideTheory Chapman andHall London UK 1983

[3] N N Akhmediev and A Ankevich Solitons Nonlinear Pulsesand Beams Chapman and Hall London UK 1997

[4] Y R Shen The Principles of Nonlinear Optics John Wiley ampSons New York NY USA 1984

[5] H M Gibbs Optical Bistability Controlling Light with LightAcademic Press New York NY USA 1985

[6] P N Eleonskii L G Oganersquosyants and V P Silin ldquoCylindricalnonlinear waveguidesrdquo Journal of Experimental andTheoreticalPhysics vol 35 no 1 pp 44ndash47 1972

[7] KM Leung ldquoP-polarized nonlinear surface polaritons inmate-rials with intensity-dependent dielectric functionsrdquo PhysicalReview B vol 32 no 8 pp 5093ndash5101 1985

[8] R I Joseph and D N Christodoulides ldquoExact field decompo-sition for tm waves in nonlinear mediardquo Optics Letters vol 12no 10 pp 826ndash828 1987

[9] D V Valovik ldquoPropagation of tmwaves in a layer with arbitrarynonlinearityrdquo Computational Mathematics and MathematicalPhysics vol 51 no 9 pp 1622ndash1632 2011

[10] D V Valovik ldquoPropagation of electromagnetic TE waves in anonlinear mediumwith saturationrdquo Journal of CommunicationsTechnology and Electronics vol 56 no 11 pp 1311ndash1316 2011

[11] G Yu Smirnov and D V Valovik Electromagnetic WavePropagation inNon-Linear LayeredWaveguide Structures PenzaState University Press Penza Russia 2011

[12] H W Schurmann V S Serov and Y V Shestopalov ldquoTE-polarized waves guided by a lossless nonlinear three-layerstructurerdquo Physical Review E vol 58 no 1 pp 1040ndash1050 1998

[13] D V Valovik ldquoPropagation of electromagnetic waves in anonlinearmetamaterial layerrdquo Journal of Communications Tech-nology and Electronics vol 56 no 5 pp 544ndash556 2011

[14] HW SchurmannV S Serov andYuV Shestopalov ldquoSolutionsto the Helmholtz equation for TE-guided waves in a three-layerstructure with Kerr-type nonlinearityrdquo Journal of Physics A vol35 no 50 pp 10789ndash10801 2002

[15] D V Valovik and Yu G Smirnov ldquoPropagation of tm wavesin a kerr nonlinear layerrdquo Computational Mathematics andMathematical Physics vol 48 no 12 pp 2217ndash2225 2008

[16] DVValovik andYG Smirnov ldquoCalculation of the propagationconstants of TM electromagnetic waves in a nonlinear layerrdquoJournal of Communications Technology and Electronics vol 53no 8 pp 883ndash889 2008

[17] DVValovik andYG Smirnov ldquoCalculation of the propagationconstants and fields of polarized electromagnetic TM wavesin a nonlinear anisotropic layerrdquo Journal of CommunicationsTechnology and Electronics vol 54 no 4 pp 391ndash398 2009

[18] G Yu Smirnov and D V Valovik ldquoBoundary eigenvalueproblem for maxwell equations in a nonlinear dielectric layerrdquoApplied Mathematics no 1 pp 29ndash36 2010

[19] D V Valovik and Y G Smirnov ldquoNonlinear effects in theproblem of propagation of TM electromagnetic waves in a Kerrnonlinear layerrdquo Journal of Communications Technology andElectronics vol 56 no 3 pp 283ndash288 2011

[20] K A Yuskaeva V S Serov and H W Schurmann ldquoTm-electromagnetic guided waves in a (kerr-) nonlinear three-layerstructurerdquo in Proceedings of the Progress in ElectromagneticsResearch Symposium (PIERS rsquo09) pp 364ndash369 Moscow RussiaAugust 2009

[21] H W Schurmann G Yu Smirnov and V Yu ShestopalovldquoPropagation of te-waves in cylindrical nonlinear dielectricwaveguidesrdquo Physical Review E vol 71 no 1 Article ID 0166142005

[22] Y Smirnov H W Schurmann and Y Shestopalov ldquoIntegralequation approach for the propagation of TE-waves in a non-linear dielectric cylindrical waveguiderdquo Journal of NonlinearMathematical Physics vol 11 no 2 pp 256ndash268 2004

[23] Yu G Smirnov and S N Kupriyanova ldquoPropagation of electro-magnetic waves in cylindrical dielectric waveguides filled witha nonlinear mediumrdquo Computational Mathematics and Mathe-matical Physics vol 44 no 10 pp 1850ndash1860 2004

[24] G Yu Smirnov and E A Horosheva ldquoOn the solvabilityof the nonlinear boundary eigenvalue problem for tm wavespropagation in a circle cylyindrical nonlinear waveguiderdquoIzvestiya Vysshikh Uchebnykh Zavedenij Povolzh Region Fiziki-Matematicheskie Nauki no 3 pp 55ndash70 2010 (Russian)

[25] G Yu Smirnov andD V Valovik ldquoCoupled electromagnetic te-tm wave propagation in a layer with kerr nonlinearityrdquo Journalof Mathematical Physics vol 53 no 12 Article ID 123530 pp1ndash24 2012

[26] V S Serov K A Yuskaeva and H W Schurmann ldquoIntegralequations approach to tm-electromagnetic waves guided bya (linearnonlinear) dielectric film with a spatially varyingpermittivityrdquo in Proceedings of the Progress in ElectromagneticsResearch Symposium (PIERS rsquo09) pp 1915ndash1919 Moscow Rus-sia August 2009

[27] J A Stretton Electromagnetic Theory McGraw Hill New YorkNY USA 1941

[28] L D Landau E M Lifshitz and L P Pitaevskii Course ofTheoretical Physics vol 8 of Electrodynamics of ContinuousMedia Butterworth-Heinemann Oxford UK 1993

[29] M A Naimark Linear Differential Operators Part I ElementaryTheory of Linear Differential Operators Frederick Ungar NewYork NY USA 1967

[30] M A Naimark Linear Differential Operators Part II LinearDifferential Operators in Hilbertspace Frederick Ungar NewYork NY USA 1968

[31] I Stakgold Greenrsquos Functions and Boundary Value ProblemsJohn Wiley amp Sons New York NY USA 1979

[32] V A Trenogin Functional Analysis Nauka Moscow Russia1993

[33] Y G Smirnov ldquoPropagation of electromagnetic waves in cylin-drical dielectric waveguides filled with a nonlinear mediumrdquoJournal of Communications Technology and Electronics vol 50no 2 pp 179ndash185 2005

[34] P I Lizorkin Course of Differential and Integral Equations withSupplementary Chapters of Calculus Nauka Moscow Russia1981

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRNMathematical Physics 5

Using (34) and transmission condition119906(119877minus0) = 119906(119877+0)we obtain the dispersion equation (DE) with respect to thepropagation constant

119906 (119877 + 0) = 120572int

119877

0

119866119896(120588 119877) 120588119906

3(120588) 119889120588 + 119877119906

1015840(119877 + 0) 119866 (119877 119877)

(35)

Let us denote by 119873(120588 1205880 120582) = 120572119866

119896(120588 1205880 120582)120588 and

consider integral equation (34)

119906 (1205880) = int

119877

0

119873(120588 1205880) 1199063(120588) 119889120588 + 119891 (120588

0) (36)

in 119862[0 119877] [32] It is assumed that 119891 isin 119862[0 119877] and 120582 = 120582119894

The kernel 119873(120588 1205880) is continuous in the square 0 le

120588 1205880le 119877

Let us consider linear integral operator 119873119908 =

int

119877

0119873(120588 120588

0)119908(120588)119889120588 in 119862[0 119877] It is bounded completely

continuous and 119873 = max1205880isin[0119877]

int

119877

0|119873(120588 120588

0)|119889120588

Since nonlinear operator 1198610(119906) = 119906

3(120588) is bounded and

continuous in 119862[0 119877] therefore nonlinear operator 119865(119906) =int

119877

0119873(120588 120588

0)1199063(120588)119889120588 + 119891(120588

0) is completely continuous in any

bounded set in 119862[0 119877]The following theorems (about existence of a unique

solution and continuous dependence of the solution on theparameter) can be proved in the same way as for the case of ahomogeneous nonlinear cylindrical waveguide (for details ofproofs see [22 33])

Proposition 1 If 120572 le 1198602 where

119860 =

2

3

1

10038171003817100381710038171198911003817100381710038171003817radic310038171003817100381710038171198731

1003817100381710038171003817

10038171003817100381710038171198731

1003817100381710038171003817= max1205880isin[0119877]

int

119877

0

1003816100381610038161003816120588119866119896(120588 1205880)1003816100381610038161003816119889120588

(37)

then (36) has a unique continuous solution 119906 isin 119862[0 119877] suchthat 119906 le 119903

lowast where

119903lowast= minus

2

radic3 119873

cos(13

arccos(3radic3

2

10038171003817100381710038171198911003817100381710038171003817radic119873) minus

2120587

3

)

(38)

is a root of the equation 1198731199033 + 119891 = 119903

Note that 119860 gt 0 does not depend on 120572

Proposition 2 Let the kernel119873 and the right-hand side 119891 ofequation (36) depend continuously on the parameter 120582 isin Λ

0

119873(120588 1205880 120582) sub 119862([0 119877]times[0 119877]timesΛ

0)119891(1205880 120582) sub 119862([0 119877]timesΛ

0)

on some segment Λ0of the real number axis Let also

0 lt1003817100381710038171003817119891 (120582)

1003817100381710038171003817lt

2

3radic3 119873 (120582)

(39)

Then for 120582 isin Λ0 a unique solution 119906(120588 120582) of (36) exists

and depends continuously on 120582 119906(120588 120582) sub 119862([0 119877] times Λ0)

4 Iteration Method

Approximate solutions 119906119899of integral equation (36) repre-

sented in the form 119906 = 119882(119906) can be found by means of theiteration process 119906

119899+1= 119882(119906

119899) 119899 = 0 1

1199060= 0 119906

119899+1= 120572int

119877

0

119866119896(120588 1205880) 1205881199063

119899119889120588 + 119891

119899 = 0 1

(40)

The sequence119906119899converges uniformly to solution119906 of (36)

by virtue of the fact that 119865(119906) is a contracting operator Theestimate of the convergence rate of iteration process (40) isalso known Let us formulate these results as the following(for proof see [22])

Proposition 3 The sequence of approximate solutions 119906119899of

(36) obtained by means of iteration process (40) converges inthe norm of space 119862[0 119877] to (unique) exact solution 119906 of thisequationThe following estimate of the convergence rate is valid119906119899minus 119906 le (119902

119899(1 minus 119902))119891(119906

0) 119899 rarr infin where 119902 = 31198731199032

lowastlt 1

is the coefficient of contraction of mapping 119865

5 Theorem of Existence

Taking into account formula (22) DE (35) can be representedin the form

1198701(10038161003816100381610038161198961

1003816100381610038161003816119877) minus

10038161003816100381610038161198961

10038161003816100381610038161198771198701015840

1(10038161003816100381610038161198961

1003816100381610038161003816119877) 119866119896(119877 119877 120582)

=

120572

119887

int

119877

0

119866119896(120588 119877 120582) 120588119906

3(120588) 119889120588

(41)

As it can be seen DE (41) depends on 119887 Here 119887 is an initialcondition This is the peculiarity of this (and not only this)nonlinear problem For the linear problem (if 120572 = 0) weobtain as it is expected the DE that does not depend on theinitial condition

From the properties of Bessel functions it follows that

minus10038161003816100381610038161198961

10038161003816100381610038161198771198701015840

1(10038161003816100381610038161198961

1003816100381610038161003816119877) =

10038161003816100381610038161198961

10038161003816100381610038161198771198700(10038161003816100381610038161198961

1003816100381610038161003816119877) + 119870

1(10038161003816100381610038161198961

1003816100381610038161003816119877) (42)

Now we can rewrite DE (41) in the following form

119892 (120582) = 120572119865 (120582) (43)

where

119892 (120582) = 1198701(10038161003816100381610038161198961

1003816100381610038161003816119877)

+ (10038161003816100381610038161198961

10038161003816100381610038161198771198700(10038161003816100381610038161198961

1003816100381610038161003816119877) + 119870

1(10038161003816100381610038161198961

1003816100381610038161003816119877))119866

119896

(119877 119877 120582)

119865 (120582) = int

119877

0

119866119896(120588 119877 120582) 120588119906

3(120588) 119889120588

(44)We should note that DE (41) depends on frequency 120596

implicitly If one obtains 120582lowast for chosen119877lowast (radius of the innercore) such that 119892(120582lowast) = 120572119865(120582lowast) is satisfied then one cancalculate 120596lowast which satisfies the propagation constants 120574lowast =radic120582lowast using formulae in the beginning of this section

6 ISRNMathematical Physics

The zeros of the function Φ(120574) equiv 119892(120582) minus 120572119865(120582) are thosevalues of 120582 for which a nonzero solution of the problem Pexists The following assertion gives us sufficient conditionsfor the existence of zeros of the function Φ

Let us consider the question about existence of solutionsof the linear problem 119892(120582) = 0

This equation can be rewritten in the form

119866119896(119877 119877 120582) = minus

1198701(10038161003816100381610038161198961

1003816100381610038161003816119877)

10038161003816100381610038161198961

10038161003816100381610038161198771198700(10038161003816100381610038161198961

1003816100381610038161003816119877) + 119870

1(10038161003816100381610038161198961

1003816100381610038161003816119877)

(45)

From expression 119866119896(119877 119877 120582) = minusV2

119894(119877)(120582 minus 120582

119894) +

1198661(119877 119877 120582) it follows that 119866

119896(119877 119877 120582) continuously varies

from minusinfin to +infin when 120582 varies from 120582119894to 120582119894+1

As value 119870

1(|1198961|119877)(|119896

1|1198771198700(|1198961|119877) + 119870

1(|1198961|119877)) is

bounded then there is at least one root of equation 119892(120582) = 0and this root lies between 120582

119894and 120582

119894+1

Finally it is necessary to prove that term V119894(119877) does

not vanish in expression 119866119896(119877 119877 120582) We prove this fact by

contradiction Let V119894(119877) = 0 Consider a Cauchy problem

for equation 120588V10158401015840119894+ V1015840119894+ (1198962

2(120588)120588 minus 1120588)V

119894= 120582119894120588V119894with

initial conditions V119894|120588=119877= V1015840119894|120588=119877= 0 as 120588 isin [120575 119877] where

120575 gt 0 From the general theory of ordinary differentialequations (see eg [34]) it is known that solution V

119894(120588) of

consideredCauchy problem exists and is unique as 120588 isin [120575 119877]In this case this solution coincides with function V

119894(119877) as

120588 isin [120575 119877] Function V119894(119877) is the function which is contained

in Greenrsquos function representation (27) On the other handa solution of the Cauchy problem for a linear equation withzero initial condition is the trivial solution This contradictswith representation (27) ofGreenrsquos function119866

119896(120588 1205880 120582) in the

vicinity of 120582 = 120582119894

Consider nonlinear problem Let inequalities

1205761lt 1205820lt 1205821lt sdot sdot sdot lt 120582

119896minus1lt 120582119896lt 1205762 (46)

hold where 119896 ge 1 and 1205762= min

120588isin[0119877]1205762(120588)

We can choose sufficiently small 120575119894gt 0 such that the

Green function 119866119896(120588 1205880 120582) exists and is continuous on Γ =

⋃119896

119894=1Γ119894 where

Γ119894= [radic120582

119894minus1+ 120575119894minus1 radic120582119894minus 120575119894] 119894 = 1 119896 (47)

and the following inequality 119892(radic120582119894minus1+ 120575119894minus1)119892(radic120582

119894minus 120575119894) lt 0

is satisfiedIt follows from the choice of 120575

119894that 119865(120582) is bounded

Moreover product 120572119865(120582) can be made sufficiently small bychoosing appropriate 120572 (the estimation is given at the end ofthis section) Let us consider DE Φ(120582) = 0 As it is shownbefore function 119892(120582) is continuous and reverse sign when 120582varies from 120582

119894minus1+120575119894minus1

to 120582119894minus120575119894 As function 119865(120582) is bounded

then it is clear that equation Φ(120582) = 0 has at least 119896 roots 120582119894

119894 = 1 119896 if we choose appropriate 120572 Here 120582119894isin (120582119894minus1+120575119894minus1 120582119894minus

120575119894) 119894 = 1 119896On the basis of previous consideration we can formulate

the main result of this paper

Theorem 4 Let the values 1205761 1205762= min

120588isin[0119877]1205762(120588) 120572 satisfy

condition 1205762gt 1205761gt 0 and let the following inequalities

1205761lt 1205820lt 1205821lt sdot sdot sdot lt 120582

119896minus1lt 120582119896lt 1205762hold where 119896 ge 1 is

an integerThen there is a value 1205720gt 0 such that for any 120572 le 120572

0

at least 119896 values 120574119894 119894 = 1 119896 exist such that the problem P has a

nonzero solution and 120574119894isin (radic120582

119894minus1+ 120575119894minus1 radic120582119894minus 120575119894)

Proof The Green function exists for all 120574 isin Γ It is also clearthat function 119860(120574) = 23119891(120574)radic3119873

1(120574) is continuous as

120574 isin Γ Let 1198601= min

120574isinΓ119860(120574) and 120572 lt 1198602

1 In accordance

with Proposition 1 there is a unique solution 119906 = 119906(120574) of(36) for any 120574 isin Γ This solution is continuous and 119906 le119903lowast= 119903lowast(120574) Let 119903

00= max

120574isinΓ119903lowast(120574) The following estimation

|119865(120582)| le 1198621199033

00is valid where 119862 is a constant

Function 119892(120574) is continuous and equation 119892(120574) = 0 has atleast one root 120574

119894inside segment Γ

119894 that isradic120582

119894minus1+ 120575119894minus1lt 120574119894lt

radic120582119894minus 120575119894 Let us denote119872

1= min

0leile119896minus1|119892(radic120582119894 + 120575119894)|1198722 =min1le119894le119896|119892(radic120582

119894minus 120575119894)| Value = min119872

11198722 is positive

and does not depend on 120572If 120572 le 1198621199033

00 then

(119892(radic120582119894minus1+ 120575119894minus1) minus 120572119865(radic120582

119894minus1+ 120575119894minus1))

times (119892(radic120582119894minus 120575119894) minus 120572119865(radic120582

119894minus 120575119894)) lt 0

(48)

As 119892(120582) minus 120572119865(120582) is continuous it follows that equation119892(120582)minus120572119865(120582) = 0 has a root 120574

119894inside Γ

119894 that isradic120582

119894+120575119894lt 120574119894lt

radic120582119894+1minus 120575119894+1

We can choose 1205720= min1198602

1 119862119903

3

00

From Theorem 4 it follows that under the previousassumptions there exist axially symmetrical propagating TEwaves in cylindrical dielectric waveguides of circular cross-section filled with a nonmagnetic isotropic inhomogeneousmedium with Kerr nonlinearity This result generalizes thewell-known similar statement for dielectric waveguides ofcircular cross-section filled with a linearmedium (ie 120572 = 0)

It should be noticed that the value 1205720can be effectively

estimated

6 Conclusion

In this study we suggest and develop a method to inves-tigate the problem of existence of electromagnetic wavesthat propagate along axis of an inhomogeneous nonlinearcylindrical waveguideThe nonlinearity inside the waveguideis described by the Kerr law the inhomogeneity is describedby a function that depends on radius of the waveguide

Here we show that the integral equation approach allowsus to investigate quite general problem for nonlinear inhomo-geneous waveguides

We should say that this method can be used to proveexistence of guided waves in a nonlinear inhomogeneouswaveguide for TM waves

Numerical results can be obtained with the help ofiteration procedure from Section 4

A separate paper will be devoted to development of acouple of numerical methods for this problem

ISRNMathematical Physics 7

Acknowledgments

This work is partially supported by the RFBR (Grants nos 11-07-00330-A 12-07-97010-RA) theMinistry of Education andScience of the Russian Federation (Grant no 14B37211950)

References

[1] M J AdamsAn Introduction To OpticalWaveguide JohnWileyamp Sons Chichester UK 1981

[2] A Snyder and J LoveOpticalWaveguideTheory Chapman andHall London UK 1983

[3] N N Akhmediev and A Ankevich Solitons Nonlinear Pulsesand Beams Chapman and Hall London UK 1997

[4] Y R Shen The Principles of Nonlinear Optics John Wiley ampSons New York NY USA 1984

[5] H M Gibbs Optical Bistability Controlling Light with LightAcademic Press New York NY USA 1985

[6] P N Eleonskii L G Oganersquosyants and V P Silin ldquoCylindricalnonlinear waveguidesrdquo Journal of Experimental andTheoreticalPhysics vol 35 no 1 pp 44ndash47 1972

[7] KM Leung ldquoP-polarized nonlinear surface polaritons inmate-rials with intensity-dependent dielectric functionsrdquo PhysicalReview B vol 32 no 8 pp 5093ndash5101 1985

[8] R I Joseph and D N Christodoulides ldquoExact field decompo-sition for tm waves in nonlinear mediardquo Optics Letters vol 12no 10 pp 826ndash828 1987

[9] D V Valovik ldquoPropagation of tmwaves in a layer with arbitrarynonlinearityrdquo Computational Mathematics and MathematicalPhysics vol 51 no 9 pp 1622ndash1632 2011

[10] D V Valovik ldquoPropagation of electromagnetic TE waves in anonlinear mediumwith saturationrdquo Journal of CommunicationsTechnology and Electronics vol 56 no 11 pp 1311ndash1316 2011

[11] G Yu Smirnov and D V Valovik Electromagnetic WavePropagation inNon-Linear LayeredWaveguide Structures PenzaState University Press Penza Russia 2011

[12] H W Schurmann V S Serov and Y V Shestopalov ldquoTE-polarized waves guided by a lossless nonlinear three-layerstructurerdquo Physical Review E vol 58 no 1 pp 1040ndash1050 1998

[13] D V Valovik ldquoPropagation of electromagnetic waves in anonlinearmetamaterial layerrdquo Journal of Communications Tech-nology and Electronics vol 56 no 5 pp 544ndash556 2011

[14] HW SchurmannV S Serov andYuV Shestopalov ldquoSolutionsto the Helmholtz equation for TE-guided waves in a three-layerstructure with Kerr-type nonlinearityrdquo Journal of Physics A vol35 no 50 pp 10789ndash10801 2002

[15] D V Valovik and Yu G Smirnov ldquoPropagation of tm wavesin a kerr nonlinear layerrdquo Computational Mathematics andMathematical Physics vol 48 no 12 pp 2217ndash2225 2008

[16] DVValovik andYG Smirnov ldquoCalculation of the propagationconstants of TM electromagnetic waves in a nonlinear layerrdquoJournal of Communications Technology and Electronics vol 53no 8 pp 883ndash889 2008

[17] DVValovik andYG Smirnov ldquoCalculation of the propagationconstants and fields of polarized electromagnetic TM wavesin a nonlinear anisotropic layerrdquo Journal of CommunicationsTechnology and Electronics vol 54 no 4 pp 391ndash398 2009

[18] G Yu Smirnov and D V Valovik ldquoBoundary eigenvalueproblem for maxwell equations in a nonlinear dielectric layerrdquoApplied Mathematics no 1 pp 29ndash36 2010

[19] D V Valovik and Y G Smirnov ldquoNonlinear effects in theproblem of propagation of TM electromagnetic waves in a Kerrnonlinear layerrdquo Journal of Communications Technology andElectronics vol 56 no 3 pp 283ndash288 2011

[20] K A Yuskaeva V S Serov and H W Schurmann ldquoTm-electromagnetic guided waves in a (kerr-) nonlinear three-layerstructurerdquo in Proceedings of the Progress in ElectromagneticsResearch Symposium (PIERS rsquo09) pp 364ndash369 Moscow RussiaAugust 2009

[21] H W Schurmann G Yu Smirnov and V Yu ShestopalovldquoPropagation of te-waves in cylindrical nonlinear dielectricwaveguidesrdquo Physical Review E vol 71 no 1 Article ID 0166142005

[22] Y Smirnov H W Schurmann and Y Shestopalov ldquoIntegralequation approach for the propagation of TE-waves in a non-linear dielectric cylindrical waveguiderdquo Journal of NonlinearMathematical Physics vol 11 no 2 pp 256ndash268 2004

[23] Yu G Smirnov and S N Kupriyanova ldquoPropagation of electro-magnetic waves in cylindrical dielectric waveguides filled witha nonlinear mediumrdquo Computational Mathematics and Mathe-matical Physics vol 44 no 10 pp 1850ndash1860 2004

[24] G Yu Smirnov and E A Horosheva ldquoOn the solvabilityof the nonlinear boundary eigenvalue problem for tm wavespropagation in a circle cylyindrical nonlinear waveguiderdquoIzvestiya Vysshikh Uchebnykh Zavedenij Povolzh Region Fiziki-Matematicheskie Nauki no 3 pp 55ndash70 2010 (Russian)

[25] G Yu Smirnov andD V Valovik ldquoCoupled electromagnetic te-tm wave propagation in a layer with kerr nonlinearityrdquo Journalof Mathematical Physics vol 53 no 12 Article ID 123530 pp1ndash24 2012

[26] V S Serov K A Yuskaeva and H W Schurmann ldquoIntegralequations approach to tm-electromagnetic waves guided bya (linearnonlinear) dielectric film with a spatially varyingpermittivityrdquo in Proceedings of the Progress in ElectromagneticsResearch Symposium (PIERS rsquo09) pp 1915ndash1919 Moscow Rus-sia August 2009

[27] J A Stretton Electromagnetic Theory McGraw Hill New YorkNY USA 1941

[28] L D Landau E M Lifshitz and L P Pitaevskii Course ofTheoretical Physics vol 8 of Electrodynamics of ContinuousMedia Butterworth-Heinemann Oxford UK 1993

[29] M A Naimark Linear Differential Operators Part I ElementaryTheory of Linear Differential Operators Frederick Ungar NewYork NY USA 1967

[30] M A Naimark Linear Differential Operators Part II LinearDifferential Operators in Hilbertspace Frederick Ungar NewYork NY USA 1968

[31] I Stakgold Greenrsquos Functions and Boundary Value ProblemsJohn Wiley amp Sons New York NY USA 1979

[32] V A Trenogin Functional Analysis Nauka Moscow Russia1993

[33] Y G Smirnov ldquoPropagation of electromagnetic waves in cylin-drical dielectric waveguides filled with a nonlinear mediumrdquoJournal of Communications Technology and Electronics vol 50no 2 pp 179ndash185 2005

[34] P I Lizorkin Course of Differential and Integral Equations withSupplementary Chapters of Calculus Nauka Moscow Russia1981

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 ISRNMathematical Physics

The zeros of the function Φ(120574) equiv 119892(120582) minus 120572119865(120582) are thosevalues of 120582 for which a nonzero solution of the problem Pexists The following assertion gives us sufficient conditionsfor the existence of zeros of the function Φ

Let us consider the question about existence of solutionsof the linear problem 119892(120582) = 0

This equation can be rewritten in the form

119866119896(119877 119877 120582) = minus

1198701(10038161003816100381610038161198961

1003816100381610038161003816119877)

10038161003816100381610038161198961

10038161003816100381610038161198771198700(10038161003816100381610038161198961

1003816100381610038161003816119877) + 119870

1(10038161003816100381610038161198961

1003816100381610038161003816119877)

(45)

From expression 119866119896(119877 119877 120582) = minusV2

119894(119877)(120582 minus 120582

119894) +

1198661(119877 119877 120582) it follows that 119866

119896(119877 119877 120582) continuously varies

from minusinfin to +infin when 120582 varies from 120582119894to 120582119894+1

As value 119870

1(|1198961|119877)(|119896

1|1198771198700(|1198961|119877) + 119870

1(|1198961|119877)) is

bounded then there is at least one root of equation 119892(120582) = 0and this root lies between 120582

119894and 120582

119894+1

Finally it is necessary to prove that term V119894(119877) does

not vanish in expression 119866119896(119877 119877 120582) We prove this fact by

contradiction Let V119894(119877) = 0 Consider a Cauchy problem

for equation 120588V10158401015840119894+ V1015840119894+ (1198962

2(120588)120588 minus 1120588)V

119894= 120582119894120588V119894with

initial conditions V119894|120588=119877= V1015840119894|120588=119877= 0 as 120588 isin [120575 119877] where

120575 gt 0 From the general theory of ordinary differentialequations (see eg [34]) it is known that solution V

119894(120588) of

consideredCauchy problem exists and is unique as 120588 isin [120575 119877]In this case this solution coincides with function V

119894(119877) as

120588 isin [120575 119877] Function V119894(119877) is the function which is contained

in Greenrsquos function representation (27) On the other handa solution of the Cauchy problem for a linear equation withzero initial condition is the trivial solution This contradictswith representation (27) ofGreenrsquos function119866

119896(120588 1205880 120582) in the

vicinity of 120582 = 120582119894

Consider nonlinear problem Let inequalities

1205761lt 1205820lt 1205821lt sdot sdot sdot lt 120582

119896minus1lt 120582119896lt 1205762 (46)

hold where 119896 ge 1 and 1205762= min

120588isin[0119877]1205762(120588)

We can choose sufficiently small 120575119894gt 0 such that the

Green function 119866119896(120588 1205880 120582) exists and is continuous on Γ =

⋃119896

119894=1Γ119894 where

Γ119894= [radic120582

119894minus1+ 120575119894minus1 radic120582119894minus 120575119894] 119894 = 1 119896 (47)

and the following inequality 119892(radic120582119894minus1+ 120575119894minus1)119892(radic120582

119894minus 120575119894) lt 0

is satisfiedIt follows from the choice of 120575

119894that 119865(120582) is bounded

Moreover product 120572119865(120582) can be made sufficiently small bychoosing appropriate 120572 (the estimation is given at the end ofthis section) Let us consider DE Φ(120582) = 0 As it is shownbefore function 119892(120582) is continuous and reverse sign when 120582varies from 120582

119894minus1+120575119894minus1

to 120582119894minus120575119894 As function 119865(120582) is bounded

then it is clear that equation Φ(120582) = 0 has at least 119896 roots 120582119894

119894 = 1 119896 if we choose appropriate 120572 Here 120582119894isin (120582119894minus1+120575119894minus1 120582119894minus

120575119894) 119894 = 1 119896On the basis of previous consideration we can formulate

the main result of this paper

Theorem 4 Let the values 1205761 1205762= min

120588isin[0119877]1205762(120588) 120572 satisfy

condition 1205762gt 1205761gt 0 and let the following inequalities

1205761lt 1205820lt 1205821lt sdot sdot sdot lt 120582

119896minus1lt 120582119896lt 1205762hold where 119896 ge 1 is

an integerThen there is a value 1205720gt 0 such that for any 120572 le 120572

0

at least 119896 values 120574119894 119894 = 1 119896 exist such that the problem P has a

nonzero solution and 120574119894isin (radic120582

119894minus1+ 120575119894minus1 radic120582119894minus 120575119894)

Proof The Green function exists for all 120574 isin Γ It is also clearthat function 119860(120574) = 23119891(120574)radic3119873

1(120574) is continuous as

120574 isin Γ Let 1198601= min

120574isinΓ119860(120574) and 120572 lt 1198602

1 In accordance

with Proposition 1 there is a unique solution 119906 = 119906(120574) of(36) for any 120574 isin Γ This solution is continuous and 119906 le119903lowast= 119903lowast(120574) Let 119903

00= max

120574isinΓ119903lowast(120574) The following estimation

|119865(120582)| le 1198621199033

00is valid where 119862 is a constant

Function 119892(120574) is continuous and equation 119892(120574) = 0 has atleast one root 120574

119894inside segment Γ

119894 that isradic120582

119894minus1+ 120575119894minus1lt 120574119894lt

radic120582119894minus 120575119894 Let us denote119872

1= min

0leile119896minus1|119892(radic120582119894 + 120575119894)|1198722 =min1le119894le119896|119892(radic120582

119894minus 120575119894)| Value = min119872

11198722 is positive

and does not depend on 120572If 120572 le 1198621199033

00 then

(119892(radic120582119894minus1+ 120575119894minus1) minus 120572119865(radic120582

119894minus1+ 120575119894minus1))

times (119892(radic120582119894minus 120575119894) minus 120572119865(radic120582

119894minus 120575119894)) lt 0

(48)

As 119892(120582) minus 120572119865(120582) is continuous it follows that equation119892(120582)minus120572119865(120582) = 0 has a root 120574

119894inside Γ

119894 that isradic120582

119894+120575119894lt 120574119894lt

radic120582119894+1minus 120575119894+1

We can choose 1205720= min1198602

1 119862119903

3

00

From Theorem 4 it follows that under the previousassumptions there exist axially symmetrical propagating TEwaves in cylindrical dielectric waveguides of circular cross-section filled with a nonmagnetic isotropic inhomogeneousmedium with Kerr nonlinearity This result generalizes thewell-known similar statement for dielectric waveguides ofcircular cross-section filled with a linearmedium (ie 120572 = 0)

It should be noticed that the value 1205720can be effectively

estimated

6 Conclusion

In this study we suggest and develop a method to inves-tigate the problem of existence of electromagnetic wavesthat propagate along axis of an inhomogeneous nonlinearcylindrical waveguideThe nonlinearity inside the waveguideis described by the Kerr law the inhomogeneity is describedby a function that depends on radius of the waveguide

Here we show that the integral equation approach allowsus to investigate quite general problem for nonlinear inhomo-geneous waveguides

We should say that this method can be used to proveexistence of guided waves in a nonlinear inhomogeneouswaveguide for TM waves

Numerical results can be obtained with the help ofiteration procedure from Section 4

A separate paper will be devoted to development of acouple of numerical methods for this problem

ISRNMathematical Physics 7

Acknowledgments

This work is partially supported by the RFBR (Grants nos 11-07-00330-A 12-07-97010-RA) theMinistry of Education andScience of the Russian Federation (Grant no 14B37211950)

References

[1] M J AdamsAn Introduction To OpticalWaveguide JohnWileyamp Sons Chichester UK 1981

[2] A Snyder and J LoveOpticalWaveguideTheory Chapman andHall London UK 1983

[3] N N Akhmediev and A Ankevich Solitons Nonlinear Pulsesand Beams Chapman and Hall London UK 1997

[4] Y R Shen The Principles of Nonlinear Optics John Wiley ampSons New York NY USA 1984

[5] H M Gibbs Optical Bistability Controlling Light with LightAcademic Press New York NY USA 1985

[6] P N Eleonskii L G Oganersquosyants and V P Silin ldquoCylindricalnonlinear waveguidesrdquo Journal of Experimental andTheoreticalPhysics vol 35 no 1 pp 44ndash47 1972

[7] KM Leung ldquoP-polarized nonlinear surface polaritons inmate-rials with intensity-dependent dielectric functionsrdquo PhysicalReview B vol 32 no 8 pp 5093ndash5101 1985

[8] R I Joseph and D N Christodoulides ldquoExact field decompo-sition for tm waves in nonlinear mediardquo Optics Letters vol 12no 10 pp 826ndash828 1987

[9] D V Valovik ldquoPropagation of tmwaves in a layer with arbitrarynonlinearityrdquo Computational Mathematics and MathematicalPhysics vol 51 no 9 pp 1622ndash1632 2011

[10] D V Valovik ldquoPropagation of electromagnetic TE waves in anonlinear mediumwith saturationrdquo Journal of CommunicationsTechnology and Electronics vol 56 no 11 pp 1311ndash1316 2011

[11] G Yu Smirnov and D V Valovik Electromagnetic WavePropagation inNon-Linear LayeredWaveguide Structures PenzaState University Press Penza Russia 2011

[12] H W Schurmann V S Serov and Y V Shestopalov ldquoTE-polarized waves guided by a lossless nonlinear three-layerstructurerdquo Physical Review E vol 58 no 1 pp 1040ndash1050 1998

[13] D V Valovik ldquoPropagation of electromagnetic waves in anonlinearmetamaterial layerrdquo Journal of Communications Tech-nology and Electronics vol 56 no 5 pp 544ndash556 2011

[14] HW SchurmannV S Serov andYuV Shestopalov ldquoSolutionsto the Helmholtz equation for TE-guided waves in a three-layerstructure with Kerr-type nonlinearityrdquo Journal of Physics A vol35 no 50 pp 10789ndash10801 2002

[15] D V Valovik and Yu G Smirnov ldquoPropagation of tm wavesin a kerr nonlinear layerrdquo Computational Mathematics andMathematical Physics vol 48 no 12 pp 2217ndash2225 2008

[16] DVValovik andYG Smirnov ldquoCalculation of the propagationconstants of TM electromagnetic waves in a nonlinear layerrdquoJournal of Communications Technology and Electronics vol 53no 8 pp 883ndash889 2008

[17] DVValovik andYG Smirnov ldquoCalculation of the propagationconstants and fields of polarized electromagnetic TM wavesin a nonlinear anisotropic layerrdquo Journal of CommunicationsTechnology and Electronics vol 54 no 4 pp 391ndash398 2009

[18] G Yu Smirnov and D V Valovik ldquoBoundary eigenvalueproblem for maxwell equations in a nonlinear dielectric layerrdquoApplied Mathematics no 1 pp 29ndash36 2010

[19] D V Valovik and Y G Smirnov ldquoNonlinear effects in theproblem of propagation of TM electromagnetic waves in a Kerrnonlinear layerrdquo Journal of Communications Technology andElectronics vol 56 no 3 pp 283ndash288 2011

[20] K A Yuskaeva V S Serov and H W Schurmann ldquoTm-electromagnetic guided waves in a (kerr-) nonlinear three-layerstructurerdquo in Proceedings of the Progress in ElectromagneticsResearch Symposium (PIERS rsquo09) pp 364ndash369 Moscow RussiaAugust 2009

[21] H W Schurmann G Yu Smirnov and V Yu ShestopalovldquoPropagation of te-waves in cylindrical nonlinear dielectricwaveguidesrdquo Physical Review E vol 71 no 1 Article ID 0166142005

[22] Y Smirnov H W Schurmann and Y Shestopalov ldquoIntegralequation approach for the propagation of TE-waves in a non-linear dielectric cylindrical waveguiderdquo Journal of NonlinearMathematical Physics vol 11 no 2 pp 256ndash268 2004

[23] Yu G Smirnov and S N Kupriyanova ldquoPropagation of electro-magnetic waves in cylindrical dielectric waveguides filled witha nonlinear mediumrdquo Computational Mathematics and Mathe-matical Physics vol 44 no 10 pp 1850ndash1860 2004

[24] G Yu Smirnov and E A Horosheva ldquoOn the solvabilityof the nonlinear boundary eigenvalue problem for tm wavespropagation in a circle cylyindrical nonlinear waveguiderdquoIzvestiya Vysshikh Uchebnykh Zavedenij Povolzh Region Fiziki-Matematicheskie Nauki no 3 pp 55ndash70 2010 (Russian)

[25] G Yu Smirnov andD V Valovik ldquoCoupled electromagnetic te-tm wave propagation in a layer with kerr nonlinearityrdquo Journalof Mathematical Physics vol 53 no 12 Article ID 123530 pp1ndash24 2012

[26] V S Serov K A Yuskaeva and H W Schurmann ldquoIntegralequations approach to tm-electromagnetic waves guided bya (linearnonlinear) dielectric film with a spatially varyingpermittivityrdquo in Proceedings of the Progress in ElectromagneticsResearch Symposium (PIERS rsquo09) pp 1915ndash1919 Moscow Rus-sia August 2009

[27] J A Stretton Electromagnetic Theory McGraw Hill New YorkNY USA 1941

[28] L D Landau E M Lifshitz and L P Pitaevskii Course ofTheoretical Physics vol 8 of Electrodynamics of ContinuousMedia Butterworth-Heinemann Oxford UK 1993

[29] M A Naimark Linear Differential Operators Part I ElementaryTheory of Linear Differential Operators Frederick Ungar NewYork NY USA 1967

[30] M A Naimark Linear Differential Operators Part II LinearDifferential Operators in Hilbertspace Frederick Ungar NewYork NY USA 1968

[31] I Stakgold Greenrsquos Functions and Boundary Value ProblemsJohn Wiley amp Sons New York NY USA 1979

[32] V A Trenogin Functional Analysis Nauka Moscow Russia1993

[33] Y G Smirnov ldquoPropagation of electromagnetic waves in cylin-drical dielectric waveguides filled with a nonlinear mediumrdquoJournal of Communications Technology and Electronics vol 50no 2 pp 179ndash185 2005

[34] P I Lizorkin Course of Differential and Integral Equations withSupplementary Chapters of Calculus Nauka Moscow Russia1981

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRNMathematical Physics 7

Acknowledgments

This work is partially supported by the RFBR (Grants nos 11-07-00330-A 12-07-97010-RA) theMinistry of Education andScience of the Russian Federation (Grant no 14B37211950)

References

[1] M J AdamsAn Introduction To OpticalWaveguide JohnWileyamp Sons Chichester UK 1981

[2] A Snyder and J LoveOpticalWaveguideTheory Chapman andHall London UK 1983

[3] N N Akhmediev and A Ankevich Solitons Nonlinear Pulsesand Beams Chapman and Hall London UK 1997

[4] Y R Shen The Principles of Nonlinear Optics John Wiley ampSons New York NY USA 1984

[5] H M Gibbs Optical Bistability Controlling Light with LightAcademic Press New York NY USA 1985

[6] P N Eleonskii L G Oganersquosyants and V P Silin ldquoCylindricalnonlinear waveguidesrdquo Journal of Experimental andTheoreticalPhysics vol 35 no 1 pp 44ndash47 1972

[7] KM Leung ldquoP-polarized nonlinear surface polaritons inmate-rials with intensity-dependent dielectric functionsrdquo PhysicalReview B vol 32 no 8 pp 5093ndash5101 1985

[8] R I Joseph and D N Christodoulides ldquoExact field decompo-sition for tm waves in nonlinear mediardquo Optics Letters vol 12no 10 pp 826ndash828 1987

[9] D V Valovik ldquoPropagation of tmwaves in a layer with arbitrarynonlinearityrdquo Computational Mathematics and MathematicalPhysics vol 51 no 9 pp 1622ndash1632 2011

[10] D V Valovik ldquoPropagation of electromagnetic TE waves in anonlinear mediumwith saturationrdquo Journal of CommunicationsTechnology and Electronics vol 56 no 11 pp 1311ndash1316 2011

[11] G Yu Smirnov and D V Valovik Electromagnetic WavePropagation inNon-Linear LayeredWaveguide Structures PenzaState University Press Penza Russia 2011

[12] H W Schurmann V S Serov and Y V Shestopalov ldquoTE-polarized waves guided by a lossless nonlinear three-layerstructurerdquo Physical Review E vol 58 no 1 pp 1040ndash1050 1998

[13] D V Valovik ldquoPropagation of electromagnetic waves in anonlinearmetamaterial layerrdquo Journal of Communications Tech-nology and Electronics vol 56 no 5 pp 544ndash556 2011

[14] HW SchurmannV S Serov andYuV Shestopalov ldquoSolutionsto the Helmholtz equation for TE-guided waves in a three-layerstructure with Kerr-type nonlinearityrdquo Journal of Physics A vol35 no 50 pp 10789ndash10801 2002

[15] D V Valovik and Yu G Smirnov ldquoPropagation of tm wavesin a kerr nonlinear layerrdquo Computational Mathematics andMathematical Physics vol 48 no 12 pp 2217ndash2225 2008

[16] DVValovik andYG Smirnov ldquoCalculation of the propagationconstants of TM electromagnetic waves in a nonlinear layerrdquoJournal of Communications Technology and Electronics vol 53no 8 pp 883ndash889 2008

[17] DVValovik andYG Smirnov ldquoCalculation of the propagationconstants and fields of polarized electromagnetic TM wavesin a nonlinear anisotropic layerrdquo Journal of CommunicationsTechnology and Electronics vol 54 no 4 pp 391ndash398 2009

[18] G Yu Smirnov and D V Valovik ldquoBoundary eigenvalueproblem for maxwell equations in a nonlinear dielectric layerrdquoApplied Mathematics no 1 pp 29ndash36 2010

[19] D V Valovik and Y G Smirnov ldquoNonlinear effects in theproblem of propagation of TM electromagnetic waves in a Kerrnonlinear layerrdquo Journal of Communications Technology andElectronics vol 56 no 3 pp 283ndash288 2011

[20] K A Yuskaeva V S Serov and H W Schurmann ldquoTm-electromagnetic guided waves in a (kerr-) nonlinear three-layerstructurerdquo in Proceedings of the Progress in ElectromagneticsResearch Symposium (PIERS rsquo09) pp 364ndash369 Moscow RussiaAugust 2009

[21] H W Schurmann G Yu Smirnov and V Yu ShestopalovldquoPropagation of te-waves in cylindrical nonlinear dielectricwaveguidesrdquo Physical Review E vol 71 no 1 Article ID 0166142005

[22] Y Smirnov H W Schurmann and Y Shestopalov ldquoIntegralequation approach for the propagation of TE-waves in a non-linear dielectric cylindrical waveguiderdquo Journal of NonlinearMathematical Physics vol 11 no 2 pp 256ndash268 2004

[23] Yu G Smirnov and S N Kupriyanova ldquoPropagation of electro-magnetic waves in cylindrical dielectric waveguides filled witha nonlinear mediumrdquo Computational Mathematics and Mathe-matical Physics vol 44 no 10 pp 1850ndash1860 2004

[24] G Yu Smirnov and E A Horosheva ldquoOn the solvabilityof the nonlinear boundary eigenvalue problem for tm wavespropagation in a circle cylyindrical nonlinear waveguiderdquoIzvestiya Vysshikh Uchebnykh Zavedenij Povolzh Region Fiziki-Matematicheskie Nauki no 3 pp 55ndash70 2010 (Russian)

[25] G Yu Smirnov andD V Valovik ldquoCoupled electromagnetic te-tm wave propagation in a layer with kerr nonlinearityrdquo Journalof Mathematical Physics vol 53 no 12 Article ID 123530 pp1ndash24 2012

[26] V S Serov K A Yuskaeva and H W Schurmann ldquoIntegralequations approach to tm-electromagnetic waves guided bya (linearnonlinear) dielectric film with a spatially varyingpermittivityrdquo in Proceedings of the Progress in ElectromagneticsResearch Symposium (PIERS rsquo09) pp 1915ndash1919 Moscow Rus-sia August 2009

[27] J A Stretton Electromagnetic Theory McGraw Hill New YorkNY USA 1941

[28] L D Landau E M Lifshitz and L P Pitaevskii Course ofTheoretical Physics vol 8 of Electrodynamics of ContinuousMedia Butterworth-Heinemann Oxford UK 1993

[29] M A Naimark Linear Differential Operators Part I ElementaryTheory of Linear Differential Operators Frederick Ungar NewYork NY USA 1967

[30] M A Naimark Linear Differential Operators Part II LinearDifferential Operators in Hilbertspace Frederick Ungar NewYork NY USA 1968

[31] I Stakgold Greenrsquos Functions and Boundary Value ProblemsJohn Wiley amp Sons New York NY USA 1979

[32] V A Trenogin Functional Analysis Nauka Moscow Russia1993

[33] Y G Smirnov ldquoPropagation of electromagnetic waves in cylin-drical dielectric waveguides filled with a nonlinear mediumrdquoJournal of Communications Technology and Electronics vol 50no 2 pp 179ndash185 2005

[34] P I Lizorkin Course of Differential and Integral Equations withSupplementary Chapters of Calculus Nauka Moscow Russia1981

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of