Research ArticleDerivation of Asymptotic Dynamical Systems withPartial Lie Symmetry Groups

Masatomo Iwasa

General Education Department Aichi Institute of Technology 1247 Yachigusa Yakusacho Toyota 470-0392 Japan

Correspondence should be addressed to Masatomo Iwasa miwasaaitechacjp

Received 16 July 2015 Revised 12 September 2015 Accepted 14 September 2015

Academic Editor Wan-Tong Li

Copyright copy 2015 Masatomo Iwasa This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Lie group analysis has been applied to singular perturbation problems in both ordinary differential and difference equations and hasallowed us to find the reduced dynamics describing the asymptotic behavior of the dynamical system The present study providesan extended method that is also applicable to partial differential equations The main characteristic of the extended method is therestriction of the manifold by some constraint equations on which we search for a Lie symmetry group This extension makes itpossible to find a partial Lie symmetry group which leads to a reduced dynamics describing the asymptotic behavior

1 Introduction

Appropriate reduction of a dynamical system is one ofthe important approximation approaches when the systemis too complicated to be solved analytically The deriva-tion of a reduced dynamical system which describes theasymptotic behavior of the solution has been achieved byusing various singular perturbation methods developed forordinary differential equations partial differential equationsand difference equations [1ndash11] On the other hand the Liegroup analysis a systematic method to construct solutionswith Lie symmetry groups that leave the manifold formedby the dynamical system invariant [12] has been extended inorder to obtain approximation solutions or reduced dynamicsby considering approximate symmetries [13ndash19] asymptoticsymmetries [20ndash25] and renormalization group symmetries[26ndash30]

While the above singular perturbation methods and theLie symmetry group analysis had been developed indepen-dently relations between them have been studied recentlyIt has been shown that the asymptotic dynamics derivedwith the singular perturbation methods can also be derivedwith the Lie group analysis when it is applied to singularperturbation problems of ordinary differential equations ordifference equations [31ndash33] In other words the Lie groupanalysis plays also a role of a singular perturbation method

While the method works for ordinary differential equationsand difference equations it has not been clear if it works ornot also for partial differential equationsThe straightforwardapplication of the method does not work because due tothe increase of independent variables in the case of partialdifferential equations systems have no Lie symmetry groupsuitable for the construction of asymptotic dynamics

This paper reports that by extending the method appro-priately we can reduce partial differential equations toasymptotic dynamics The main characteristic of the exten-sion is to add some constraint equations to the equationsoriginally considered in order to find partial Lie symmetrygroups The extended method is applied to one ordinary dif-ferential equation and two partial differential equations in thepaperWhile examples taken up here are quite simple in orderto make it easy to catch the essence of the procedure thismethod works for all the same types of singular perturbationproblems in which secular terms are included in the naıveexpansion of the solution

2 Outline of the Extended Method

The singular perturbationmethodwith the Lie group analysis[31ndash33] is briefly reviewed For the perturbed dynamicalsystem under consideration here referred to as the originalsystem inwhat follows we find an approximate Lie symmetry

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2015 Article ID 601657 8 pageshttpdxdoiorg1011552015601657

2 Journal of Applied Mathematics

group which includes the transformation of the perturbationparameter that is a constant in terms of the dynamics Anapproximate solution is generated from the nonperturbativesolution by the transformation of the group in other wordsthe approximate solution is given by the group-invariantsolutionThedifferential equation which the group-invariantsolution satisfies provides a reduced system describing theasymptotic behavior of the solution when we consider itslong-time dynamics It should be noted that another methodpresented by Gazizov and Khalique [19] also considerstransformations of the perturbation parameter while the typeof the transformation is different from that presented hereWith their method the regular expansion of the solution ofthe original system is successfully derived

The method is extended in this study The procedure isbriefly summarized as follows (1) Specify beforehand a non-perturbative solution on which we would like to see the effectof the perturbation (2) Construct equations composed of theindependent and dependent variables and their derivativesbased on the nonperturbative solution specified in step (1)Those equations are referred to as constraint equations inwhat follows (3) Find an approximate Lie symmetry groupof the original equations on the manifold formed by theconstraint equations constructed in the previous step (4) Forthe approximate Lie symmetry group found in step (3) con-struct the differential equation that is satisfied by the group-invariant solution and make an appropriate approximationwhen necessary As a consequence this differential equationgenerates the reduced systemwhich describes the asymptoticbehavior of the original system

The procedure is almost the same as that presented beforefor ordinary differential equations and difference equations[31ndash33] The main difference is in the second step that isto say the addition of the constraint equations for findingan approximate Lie symmetry group In general additionalLie symmetry groups can be found by the addition of sucha constraint equation because the groups do not need tobe admitted by the entire manifold formed by the originalsystem in the jet space but need to be admitted only by thatpart of themanifold restricted by the added equations In thissense a Lie symmetry group found by using this procedureis interpreted as a partial Lie symmetry group [24] Becauseof the increase in the number of the admitted Lie symmetrygroups in some cases this method allows us to find a Liesymmetry group when it is impossible to find such a groupwith the normal Lie group analysis

3 Application to OrdinaryDifferential Equation

Themethod presented in this study is particularlymeaningfulwhen it is applied to partial differential equations Howeverfirst we apply the method to an ordinary differential equa-tion because the calculation is simpler and therefore servesto clarify the essence of the method

Consider the following ordinary differential equation[34]

2119906 + 2= 120576 (1)

Here 119906 denotes the dependent variable the overdot denotesthe derivative with respect to the independent variable 119905 and120576 denotes a perturbation parameter supposed to be smallsufficiently

The two independent solutions of the nonperturbativeequation 2119906 + 2 = 0 are

119906(0)= 11986011990523 119861 (2)

where 119860 and 119861 are constantsFocusing on one of these solutions 119906(0) = 11986011990523 we inves-

tigate the effect of the perturbation on this nonperturbativesolution For this solution

(0)=2

3

119906(0)

119905

(3)

is satisfied From this relation an equation is constructed

=2

3

119906

119905 (4)

which is used as a constraint equation in finding a Liesymmetry group

We find a Lie symmetry group admitted by (1) Let

119883 = 120597120576+ 120578 (119905 119906) 120597

119906 (5)

be the infinitesimal generator of a Lie symmetry group Itssecond prolongation119883lowast is given by

119883lowast= 119883 + 120578

(119905 119906 ) 120597

+ 120578(119905 119906 ) 120597

120578= (120597119905+ 120597119906) 120578

120578= (120597119905+ 120597119906+ 120597) 120578

(6)

When we adopt the normal procedure followed by Lie groupanalysis [12] we find a Lie symmetry group by solving thedifferential equation for 120578(119905 119906)

119883lowast[2119906 +

2minus 120576]100381610038161003816100381610038162119906+2=120576

= 0 (7)

which indicates the criterion for the invariance of the entiremanifold formed by the original differential equation in thejet space In the extended method presented in this studyinstead we find a Lie symmetry group by solving

119883lowast[2119906 +

2minus 120576]10038161003816100381610038161003816=(23)(119906119905)

= 0 (8)

Note that = (23)(119906119905) comes from (4) Then (8) reads

[21199061205972

119905+4

3

1199062

119905120597119905120597119906+8

9

1199063

11990521205972

119906+4

3

119906

119905120597119905+4

9

1199062

1199052120597119906minus4

9

119906

1199052]

sdot 120578 minus 1 = 0

(9)

Noting the homogeneity of the equation we find an exactsolution of this differential equation

120578 = minus9

20

1199052

119906 (10)

Journal of Applied Mathematics 3

For this Lie symmetry group the group-invariant solu-tion 119906 = 119906(120576 119905) satisfies

119883 [119906 minus 119906 (120576 119905)]|119906=119906(120576119905) = 0 (11)

which reads120597119906

120597120576= minus9

20

1199052

119906 (12)

This equation which can be analytically solved has thefollowing solution

119906 (120576 119905) = plusmn11990523(119860 minus

9

1012057611990523)

12

(13)

Here we have used 119906(0 119905) = 11986011990523 which is the nonpertur-bative solution as the boundary condition in the integrationThis solution corresponds exactly to the solution derivedwiththe renormalization group method [34]

The original differential equation (1) admits no exact Liesymmetry group when we adopt the normal procedure (seethe Appendix) However the addition of constraint equation(4) has allowed us to find a Lie symmetry group

4 Application to Partial Differential Equation

In this section the present method is applied to two partialdifferential equations In general in the case of partialdifferential equations because of the increase of independentvariables it often happens that a partial differential equa-tion admits no approximate Lie symmetry group suitablefor the derivation of a system describing the asymptoticdynamics However it is in some cases possible to find anapproximate Lie symmetry group suitable for deriving theasymptotic dynamic properties by adding some constraintequations constructed from the nonperturbative solutionthereby restricting the manifold on which we find a Liesymmetry group The reduced systems obtained here areconsistent with those derived with the singular perturbationmethods presented before

41 First Example 119906119905+ 119860(120576119906)119906

119909= 0 Consider a system of

weakly nonlinear first-order partial differential equations

119906119905+ 119860 (120576119906) 119906

119909= 0 (14)

Here 119906 is a real vector that has 119899 components and depends ontwo independent variables 119905 and119909119860 is an 119899times119899nondegeneratematrix that depends on 119906 120576 is a perturbation parameterThe subscripts of 119906 denote the derivative with respect to theindependent variables

Similar to the approach followed in the previous sectionwe find a Lie symmetry group such that it is not admitted bythe entire manifold formed by the original equation ratherit is admitted by a part of the manifold determined from asolution of the nonperturbative system In the following wefocus on such a nonperturbative solution 119906(0)(119905 119909) that is anarbitrary function of 119909 minus 120582119905 where 120582 is an eigenvalue of119860

0=

119860(0) namely we set

119906(0)(119905 119909) = 119880 (119909 minus 120582119905) (15)

This solution represents a traveling wave of which the velo-city is 120582 As shown by the direct substitution into the non-perturbative system 119906(0)

119909is an eigenvector of 119860

0

1198600119906(0)

119909= 120582119906(0)

119909 (16)

In addition

119906(0)

119905+ 120582119906(0)

119909= 0 (17)

is also satisfied Based on these relations we find a Liesymmetry group with constraints of

1198600119906119909= 120582119906119909

119906119905+ 120582119906119909= 0

(18)

in the followingLet

119883 = 120597120576+ 120578 (119905 119909 119906 120576) 120597

119906 (19)

be the infinitesimal generator of a Lie symmetry group Itsfirst prolongation119883lowast is given by

119883lowast= 119883 + 120578

119906119905120597119906119905

+ 120578119906119909120597119906119909

120578119906119905= (120597119905+ 119906119905120597119906) 120578

120578119906119909= (120597119909+ 119906119909120597119906) 120578

(20)

The criterion for the invariance of the original differentialequation (14) on the manifold restricted by the constraintequations (18) is given by

119883lowast[119906119905+ 119860 (120576119906) 119906

119909]10038161003816100381610038161198600119906119909=120582119906119909 119906119905+120582119906119909=0= 0 (21)

This equation reads

[120578119906119905+ 119860 (120576119906) 120578

119906119909+ (119906 sdot nabla)119860 (120576119906) 119906

119909

+ 120576 (120578 sdot nabla)119860 (120576119906) 119906119909]10038161003816100381610038161198600119906119909=120582119906119909 119906119905+120582119906119909=0= 0

(22)

Here (119906 sdot nabla)119860(120576119906) and (120578 sdot nabla)119860(120576119906) are defined by matrices ofwhich the 119894119895 components are given by respectively

[sum

119896

119906119896120597119903119896

119886119894119895(119903)]

1003816100381610038161003816100381610038161003816100381610038161003816119903=120576119906

[sum

119896

120578119896120597119903119896

119886119894119895(119903)]

1003816100381610038161003816100381610038161003816100381610038161003816119903=120576119906

(23)

where 119886119894119895denotes the 119894119895 component of the matrix 119860

Suppose 120578(119905 119909 119906 120576) and 119860(120576119906) can be expanded as apower series of 120576 Their expanded forms

120578 =

infin

sum

119896=0

120576119896120578119896(119905 119909 119906)

119860 (120576119906) =

infin

sum

119896=0

120576119896119860119896(119906)

(24)

4 Journal of Applied Mathematics

are substituted into (22) which is then rewritten as

[120578119906119905

0+ 1198600120578119906119909

0+ (119906 sdot nabla)119860

0119906119909]10038161003816100381610038161198600119906119909=120582119906119909 119906119905+120582119906119909=0= 0 (25)

[

[

120578119906119905

119894+ 1198600120578119906119909

119894+ (119906 sdot nabla)119860

119894119906119909

+

119894minus1

sum

119895=0

(120578119894minus119895minus1sdot nabla)119860

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119906=0

119906119909]

]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161198600119906119909=120582119906119909 119906119905+120582119906119909=0

= 0

(119894 = 1 2 )

(26)

We consider only the lowest order of 120576 namely (25) which isrewritten as

(120597119905+ 119906119905120597119906) 1205780+ 1198600(120597119909+ 119906119909120597119906) 1205780

minus [(119906 sdot nabla)1198600] 119906119909

10038161003816100381610038161198600119906119909=120582119906119909 119906119905+120582119906119909=0= 0

(27)

and reads

(120597119905+ 1198600120597119909) 1205780= [minus (119906 sdot nabla) 120582] 119906119909 (28)

The vector on the right-hand side of this equation is separatedinto the following two components

(120597119905+ 1198600120597119909) 1205780= minus (119890

sdot [(119906 sdot nabla) 120582] 119906119909) 119890

minus (119890perpsdot [(119906 sdot nabla) 120582] 119906119909) 119890perp

(29)

where 119890denotes the unit vector with the same direction

as 1199060(119905 119909) in R119899 and 119890

perpis the unit vector parallel to the

projection of [minus(119906 sdot nabla)120582]119906119909on a 119899 minus 1 dimensional subspace

in R119899 spanned by all the eigenvectors of 1198600except 119890

The

solution corresponding to the second term is given by a linearcombination of the 119899 minus 1 eigenvectors of 119860

0whose linear

coefficients are a function of 119909 minus 120582119905 On the other hand aterm proportional to 119905 appears in the solution correspondingto the first term because every vector proportional to 119890

whose proportional coefficient is a function of 119909minus120582119905 is a zeroeigenvector of the linear operator 120597

119905+ 1198600120597119909 For this reason

we can write a solution of (29) as follows

1205780= minus119905 (119890

sdot [(119906 sdot nabla) 120582] 119906119909) 119890

+ (terms not proportional to 119905) (30)

The group-invariant solution 119906 = 119906(119905 119909) for thisapproximate Lie symmetry group satisfies

119883 [119906 minus 119906 (119905 119909)]|119906=119906(119905119909) = 0 (31)

which reads120597119906 (119905 119909)

120597120576= minus119905 (119890

sdot [(119906 sdot nabla)119860

0] 119906119909) 119890

+ (terms not proportional to 119905) (32)

The long-time behavior of the solution is obtained by consid-ering the case in which 119905 ≫ 1 and by neglecting the termsother than the first one Then (32) is reduced to

119889

119889120591119906 (119905 119909) = minus (119890

sdot [(119906 sdot nabla)119860

0] 119906119909) 119890 (33)

where 120591 is a long time-scale defined by 120591 = 120576119905 As aresult we have obtained a dynamical system which describesasymptotic behavior of the original system This reduceddynamics corresponds to that presented before with thereductive perturbation method [3] or the renormalizationgroup method [7]

42 Second Example 119906119905119905minus 119906119909119909+ 119906 minus 120576119906

3= 0 We consider a

perturbed Klein-Gordon equation represented by

119906119905119905minus 119906119909119909+ 119906 minus 120576119906

3= 0 (34)

Here 119906 is a scalar function which depends on two indepen-dent variables 119905 and 119909 and 120576 is a perturbation parameterThe subscripts of 119906 denote the derivative with respect to theindependent variables By introducing new dependent vari-ables as

1199061= 119906119905

1199062= 119906119909

(35)

we rewrite (34) in terms of four first-order partial differentialequations for three dependent variables 119906 119906

1 and 119906

2as

follows

1199061119905minus 1199062119909+ 119906 minus 120576119906

3= 0

119906119905minus 1199061= 0

119906119909minus 1199062= 0

1199061119909minus 1199062119905= 0

(36)

The last equation implies the integrable conditionAs in the previous examples a specific nonperturbative

solution 119906(0)(119905 119909) is prespecified on which we investigate theeffect of the perturbationHerewe focus on a nonperturbativesolution describing a traveling wave with a constant velocityV The solution is represented by

119906(0)(119905 119909) = 119880 (119909 minus V119905) (37)

where 119880(120585) = 120572119890i120581120585 + 120572119890minusi120581120585 for 120581 = (V2 minus 1)minus12 and anarbitrary constant 120572 For this solution

119906(0)

119905= minusV119906(0)

119909

119906(0)

119905119905= minusV119906(0)

119905119909= minusV119906(0)

119909119905= V2119906(0)

119909119909

(38)

therefore

119906(0)

119905119905=119906(0)

119905

119906(0)

119909

119906(0)

119909119905=119906(0)

119905

119906(0)

119909

119906(0)

119905119909= (119906(0)

119905

119906(0)

119909

)

2

119906(0)

119909119909(39)

Journal of Applied Mathematics 5

are satisfied By combining these relations with the nonper-turbative system 119906(0)

119905119905minus 119906(0)

119909119909+ 119906(0)= 0 we obtain

119906(0)

119905119905=

[119906(0)

119905]2

119906(0)

[119906(0)

119909 ]2

minus [119906(0)

119905]2

119906(0)

119909119905= 119906(0)

119905119909=119906(0)

119905119906(0)

119909119906(0)

[119906(0)

119909 ]2

minus [119906(0)

119905]2

119906(0)

119909119909=

[119906(0)

119909]2

119906(0)

[119906(0)

119909 ]2

minus [119906(0)

119905]2

(40)

From these relations we construct the constraint equations

1199061119905=1199062

1119906

1199062

2minus 1199062

1

1199062119905= 1199061119909=11990611199062119906

1199062

2minus 1199062

1

1199062119909=1199062

2119906

1199062

2minus 1199062

1

(41)

which are used in the finding of Lie symmetry groupLet

119883(119905 119909 119906 1199061 1199062 120576) = 120597

120576+ 120578 (119905 119909 119906 119906

1 1199062 120576) 120597119906

+ 1205781199061(119905 119909 119906 119906

1 1199062 120576) 1205971199061

+ 1205781199062(119905 119909 119906 119906

1 1199062 120576) 1205971199062

(42)

be the infinitesimal generator of a Lie symmetry group of thefour partial differential equations (36) Its first prolongation119883lowast is given by

119883lowast(119905 119909 119906 119906

1 1199062 120576)

= 119883 + 120578119905(119905 119909 119906 119906

1 1199062 11990611 11990612 11990622 120576) 120597119906119905

+ 120578119909(119905 119909 119906 119906

1 1199062 11990611 11990612 11990622 120576) 120597119906119909

+ 1205781199061119905(119905 119909 119906 119906

1 1199062 11990611 11990612 11990622 120576) 1205971199061119905

+ 1205781199062119909(119905 119909 119906 119906

1 1199062 11990611 11990612 11990622 120576) 1205971199062119909

(43)

where 120578119905 120578119909 1205781199061119905 and 1205781199062119909 are defined as follows

120578119905(119905 119909 119906 119906

1 1199062 119906119905 1199061119905 1199062119905 120576)

= 119863119905120578 (119905 119909 119906 119906

1 1199062 120576)

120578119909(119905 119909 119906 119906

1 1199062 119906119909 1199061119909 1199062119909 120576)

= 119863119909120578 (119905 119909 119906 119906

1 1199062 120576)

1205781199061119905(119905 119909 119906 119906

1 1199062 119906119905 1199061119905 1199062119905 120576)

= 1198631199051205781199061(119905 119909 119906 119906

1 1199062 120576)

1205781199062119909(119905 119909 119906 119906

1 1199062 119906119909 1199061119909 1199062119909 120576)

= 1198631199091205781199062(119905 119909 119906 119906

1 1199062 120576)

119863119905= 120597119905+ 1199061120597119906+ 11990611199051205971199061

+ 11990621199051205971199062

119863119909= 120597119909+ 119906119909120597119906+ 11990611199091205971199061

+ 11990621199091205971199062

(44)

The criteria for the invariance of the original system onthe manifold formed by the constraint equations (41) aregiven by

119883lowast[1199061119905minus 1199062119909+ 119906 minus 120576119906

3]10038161003816100381610038161003816(41)= 0

119883lowast[119906119905minus 1199061]1003816100381610038161003816(41)= 0

119883lowast[119906119909minus 1199062]1003816100381610038161003816(41)= 0

119883lowast[1199061119909minus 1199062119905]1003816100381610038161003816(41)= 0

(45)

With the identities

1205971199061199062

1119906

1199062

2minus 1199062

1

=1199062

1

1199062

2minus 1199062

1

1205971199061

1199062

1119906

1199062

2minus 1199062

1

=211990611199062

2119906

(1199062

2minus 1199062

1)2

1205971199062

1199062

1119906

1199062

2minus 1199062

1

=minus21199062

11199062119906

(1199062

2minus 1199062

1)2

12059711990611990611199062119906

1199062

2minus 1199062

1

=11990611199062

1199062

2minus 1199062

1

1205971199061

11990611199062119906

1199062

2minus 1199062

1

=

(1199062

1+ 1199062

2) 1199062119906

(1199062

2minus 1199062

1)2

1205971199062

11990611199062119906

1199062

2minus 1199062

1

=

minus (1199062

1+ 1199062

2) 1199061119906

(1199062

2minus 1199062

1)2

1205971199061199062

2119906

1199062

2minus 1199062

1

=1199062

2

1199062

2minus 1199062

1

1205971199061

1199062

2119906

1199062

2minus 1199062

1

=211990611199062

2119906

(1199062

2minus 1199062

1)2

1205971199062

1199062

2119906

1199062

2minus 1199062

1

=minus21199062

11199062119906

(1199062

2minus 1199062

1)2

(46)

6 Journal of Applied Mathematics

(45) read in the lowest order of 120576

(1199062

2minus 1199062

1) (120578119905119905minus 120578119909119909) minus (119906

2

2minus 1199062

1)2

120578119906119906minus 1199062

1119906212057811990611199061

minus 1199062

1119906212057811990621199062

+ 21199061119906 (1199062

2minus 1199062

1) 1205781199061199061

minus 21199062119906 (1199062

2minus 1199062

1) 1205781199061199062

minus 211990611199062119906212057811990611199062

+ 21199061(1199062

2minus 1199062

1) 120578119905119906+ 21199062

11199061205781199051199061

+ 2119906111990621199061205781199051199062

minus 21199062(1199062

2minus 1199062

1) 120578119909119906minus 2119906111990621199061205781199091199061

minus 21199062

21199061205781199091199062

minus 1199061(1199062

2minus 1199062

1) 1205781199061

minus 1199062(1199062

2minus 1199062

1) 1205781199062

minus 119906 (1199062

2minus 1199062

1) 120578119906+ (1199062

2minus 1199062

1) 120578 minus 119906

3(1199062

2minus 1199062

1)

= 0

(47)

Noting the homogeneity of the equation we find a solution120578 (119905 119909 119906 119906

1 1199062)

= minus3

8(1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2) (119886

119905

1199061

minus (1 minus 119886)119909

1199062

)

minus1

32119906 (1199062minus 1199062

1minus 1199062

2)

(48)

where 119886 is an arbitrary constant For this infinitesimalgenerator of the approximate Lie symmetry group the group-invariant solution satisfies

119883 [119906 minus 119906 (119905 119909)]|119906=119906(119905119909) = 0 (49)which reads

120597119906 (119905 119909)

120597120576= 120578 (119905 119909 119906 (119905 119909) 119906

1(119905 119909) 119906

2(119905 119909)) (50)

Asymptotic dynamic behavior can be obtained by consider-ing the cases 119905 ≫ 1 and 119909 ≫ 1 and by neglecting the termswhich are not proportional to 119909 or 119905 By introducing 120591 = 120576119905and 120585 = 120576119909 we obtain

120597119906

120597120591= minus

3 (1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2)

81199061

119886

120597119906

120597120585=

3 (1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2)

81199062

(1 minus 119886)

(51)

By eliminating 119886 the equations are reduced to

1199061

120597119906

120597120591minus 1199062

120597119906

120597120585= minus3

8(1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2) (52)

This reduced equation describes the asymptotic behavior ofthe original systemThis fact can be confirmed by comparingwith the result derived before By introducing a transforma-tion of the variables

|119860|2=1

4(1199062+ 1199062

1minus 1199062

2)

120596 = 1199061(1199062

1minus 1199062

2)minus12

119896 = 1199062(1199062

1minus 1199062

2)minus12

(53)

the reduced equation (52) is transformed into

[120597

120597120591+119896

120596

120597

120597120585]119860 = minus

3

2120596i |119860|2 119860 (54)

This is the nonlinear Schrodinger equation which corre-sponds to the reduced equation derived with the renormal-ization group method [9]

5 Concluding Remarks

In summary this paper presents an investigation of perturba-tion problems which involves the application of a particulartype of Lie group method to partial as well as ordinarydifferential equations As a consequence we have obtainedthe dynamics which describes the asymptotic behavior ofthe original system The main characteristic of the presentmethod is the addition of some constraint equations whensearching for a Lie symmetry group an approach whichallows us to find a partial Lie symmetry groupThe constraintequations are determined from the nonperturbative solutionon which we try to investigate the effect of the perturbationThe reason why it is necessary to specify a nonperturbativesolution beforehand for the partial differential equations isthat unlike for ordinary differential equations the non-perturbative system has an infinite number of independentsolutions In terms of technique those constraint equationsare constructed in such a way that the arbitrary constantsor functions included in the nonperturbative solution areeliminated as we have done in deriving (4) (18) and (41) Inthis study we have investigated the effect of the perturbationof its lowest order however extending our work to higher-order approximations would require us to use higher-orderconstraint equations instead

Examples towhich themethod is applied here are singularperturbation problems in which secular terms appear in thenaıve expansion of the solution [3 6 8 10 11 34] Termsproportional to the independent variables emerge in theinfinitesimal generator of the Lie symmetry group as seen in(10) (30) and (48) corresponding to the emergence of thesecular terms in the naıve expansion As inferred from thismathematical similarity the presentmethod is also applicableto other systems which exhibit the same type of singularperturbation problems In the singular perturbationmethodsproposed thus far it is only after the construction of the naıveexpansion that it becomes clear whether the method shouldbe used or not However the present method can be appliedwithout making the expansion enabling the appropriatereduced equations to be derived

Future work should involve a clarification of the relationsbetween the present method and other types of singularperturbation methods presented thus far such as the WKBor boundary layer problem methods In addition there areexisting methods that have enabled the successful deriva-tion of the asymptotic behavior for various other partialdifferential equations [13ndash18 20ndash30 35ndash38] Therefore itshould be clarified whether or not the present method canbe appropriately extended and applied to the problems thosemethods have resolved

Journal of Applied Mathematics 7

Appendix

Searching for Lie Symmetry withthe Normal Procedure

Let us search for the Lie symmetry group of (1) withoutconstraint equations The criterion for the invariance of theequation is given by (7) namely

119883lowast[2119906 + minus 120576]

10038161003816100381610038162119906+minus120576=0= 0 (A1)

which reads

21199062120578119905119905minus 119906 + (4119906

2120578119906119905+ 2119906120578

119905)

+ 2(21199062120578119906119906+ 119906120578 minus 120578) + 120576 (120578 + 119906120578

119906) = 0

(A2)

In the lowest order of 120576 120578 has to satisfy the followingdifferential equations

21199062120578119905119905minus 119906 = 0

41199062120578119906119905+ 2119906120578

119905= 0

21199062120578119906119906+ 119906120578 minus 120578 = 0

(A3)

However we can easily find out that there is no solution thatsatisfies all the above equations which implies that there is noLie symmetry group

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by JSPS KAKENHI Grant no26800208

References

[1] N N Bogoliubov and C M Place Asymptotic Methods in theTheory of Non-Linear Oscillations Gordon and Break 1961

[2] GHori ldquoTheory of general perturbationwith unspecified cano-nical variablerdquo Publications of the Astronomical Society of Japanvol 18 p 287 1966

[3] T Taniuchi ldquoReductive perturbation method and far fields ofwave equationsrdquo Progress ofTheoretical Physics Supplement vol55 pp 1ndash35 1974

[4] J Guckenheimer andPHolmesNonlinearOscillationsDynam-ical systems and Bifurcations of Vector Fields Springer NewYork NY USA 1983

[5] A H Nayfeh Method of Normal Forms John Wiley amp SonsNew York NY USA 1993

[6] T Kunihiro ldquoA geometrical formulation of the renormalizationgroup method for global analysisrdquo Progress of TheoreticalPhysics vol 94 no 4 pp 503ndash514 1995

[7] L-Y ChenNGoldenfeld YOono andG Paquette ldquoSelectionstability and renormalizationrdquo Physica A Statistical Mechanicsand its Applications vol 204 no 1ndash4 pp 111ndash133 1994

[8] L Y Chen N Goldenfeld and Y Oono ldquoRenormalizationgroup and singular perturbations multiple scales boundarylayers and reductive perturbation theoryrdquo Physical Review Evol 54 no 1 pp 376ndash394 1996

[9] S-I Goto Y Masutomi and K Nozaki ldquoLie-group approachto perturbative renormalization group methodrdquo Progress ofTheoretical Physics vol 102 no 3 pp 471ndash497 1999

[10] R E L DeVille A Harkin M Holzer K Josic and T J KaperldquoAnalysis of a renormalization group method and normal formtheory for perturbed ordinary differential equationsrdquoPhysicaDvol 237 no 8 pp 1029ndash1052 2008

[11] H Chiba ldquo1198621 approximation of vector fields based on therenormalization group methodrdquo SIAM Journal on AppliedDynamical Systems vol 7 no 3 pp 895ndash932 2008

[12] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1986

[13] V A Baıkov R K Gazizov and N H Ibragimov ldquoApproximatesymmetriesrdquoMathematics of the USSR-Sbornik vol 64 no 2 p427 1989

[14] W I Fushchich andWM Shtelen ldquoOn approximate symmetryand approximate solutions of the nonlinear wave equation witha small parameterrdquo Journal of Physics A Mathematical andGeneral vol 22 no 18 pp L887ndashL890 1989

[15] N Eular MW Shulga andW-H Steeb ldquoApproximate symme-tries and approximate solutions for amultidimensional Landau-Ginzburg equationrdquo Journal of Physics A Mathematical andGeneral vol 25 no 18 Article ID L1095 1992

[16] Y Y Bagderina and R K Gazizov ldquoEquivalence of ordinarydifferential equations 11991010158401015840 = 119877(119909 119910)11991010158402 + 2119876(119909 119910)1199101015840 + 119875(119909 119910)rdquoDifferential Equations vol 43 no 5 pp 595ndash604 2007

[17] V N Grebenev andM Oberlack ldquoApproximate Lie symmetriesof theNavier-Stokes equationsrdquo Journal ofNonlinearMathemat-ical Physics vol 14 no 2 pp 157ndash163 2007

[18] R K Gazizov and N K Ibragimov ldquoApproximate symmetriesand solutions of the Kompaneets equationrdquo Journal of AppliedMechanics and Technical Physics vol 55 no 2 pp 220ndash2242014

[19] R K Gazizov and C M Khalique ldquoApproximate transforma-tions for van der Pol-type equationsrdquoMathematical Problems inEngineering vol 2006 Article ID 68753 11 pages 2006

[20] G Gaeta ldquoAsymptotic symmetries and asymptotically symmet-ric solutions of partial differential equationsrdquo Journal of PhysicsA Mathematical and General vol 27 no 2 pp 437ndash451 1994

[21] G Gaeta ldquoAsymptotic symmetries in an optical latticerdquo PhysicalReview A vol 72 no 3 Article ID 033419 2005

[22] G Gaeta D Levi and R Mancinelli ldquoAsymptotic symmetriesof difference equations on a latticerdquo Journal of NonlinearMathe-matical Physics vol 12 supplement 2 pp 137ndash146 2005

[23] G Gaeta and RMancinelli ldquoAsymptotic scaling symmetries fornonlinear PDEsrdquo International Journal of Geometric Methods inModern Physics vol 2 no 6 pp 1081ndash1114 2005

[24] G Gaeta and R Mancinelli ldquoAsymptotic scaling in a modelclass of anomalous reaction-diffusion equationsrdquo Journal ofNonlinear Mathematical Physics vol 12 no 4 pp 550ndash5662005

[25] D Levi and M A Rodrıguez ldquoAsymptotic symmetries andintegrability the KdV caserdquo Europhysics Letters vol 80 no 6Article ID 60005 2007

[26] V F Kovalev V V Pustovalov and D V Shirkov ldquoGroup anal-ysis and renormgroup symmetriesrdquo Journal of MathematicalPhysics vol 39 no 2 pp 1170ndash1188 1998

8 Journal of Applied Mathematics

[27] V F Kovalev ldquoRenormalization group analysis for singularitiesin the wave beam self-focusing problemrdquo Theoretical andMathematical Physics vol 119 no 3 pp 719ndash730 1999

[28] V F Kovalev and D V Shirkov ldquoFunctional self-similarityand renormalization group symmetry inmathematical physicsrdquoTheoretical and Mathematical Physics vol 121 no 1 pp 1315ndash1332 1999

[29] D V Shirkov and V F Kovalev ldquoThe Bogoliubov renormaliza-tion group and solution symmetry in mathematical physicsrdquoPhysics Report vol 352 no 4ndash6 pp 219ndash249 2001

[30] V F Kovalev and D V Shirkov ldquoThe renormalization groupsymmetry for solution of integral equationsrdquo Proceedings of theInstitute of Mathematics of NAS of Ukraine vol 50 no 2 pp850ndash861 2004

[31] M Iwasa and K Nozaki ldquoA method to construct asymptoticsolutions invariant under the renormalization grouprdquo Progressof Theoretical Physics vol 116 no 4 pp 605ndash613 2006

[32] M Iwasa and K Nozaki ldquoRenormalization group in differencesystemsrdquo Journal of Physics A Mathematical and Theoreticalvol 41 no 8 Article ID 085204 2008

[33] H Chiba andM Iwasa ldquoLie equations for asymptotic solutionsof perturbation problems of ordinary differential equationsrdquoJournal of Mathematical Physics vol 50 no 4 2009

[34] Y Y Yamaguchi and Y Nambu ldquoRenormalization group equa-tions and integrability in hamiltonian systemsrdquo Progress ofTheoretical Physics vol 100 no 1 pp 199ndash204 1998

[35] N D Goldenfeld Lectures on Phase Transitions and the Renor-malization Group Addison-Wesley 1992

[36] J Bricmont A Kupiainen and G Lin ldquoRenormalization groupand asymptotics of solutions of nonlinear parabolic equationsrdquoCommunications on Pure and Applied Mathematics vol 47 no6 pp 893ndash922 1994

[37] N V Antonov and J Honkonen ldquoField-theoretic renormaliza-tion group for a nonlinear diffusion equationrdquo Physical ReviewE vol 66 no 4 Article ID 046105 7 pages 2002

[38] S Yoshida andT Fukui ldquoExact renormalization group approachto a nonlinear diffusion equationrdquo Physical Review E vol 72 no4 Article ID 046136 2005

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Mathematical Problems in Engineering

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Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

group which includes the transformation of the perturbationparameter that is a constant in terms of the dynamics Anapproximate solution is generated from the nonperturbativesolution by the transformation of the group in other wordsthe approximate solution is given by the group-invariantsolutionThedifferential equation which the group-invariantsolution satisfies provides a reduced system describing theasymptotic behavior of the solution when we consider itslong-time dynamics It should be noted that another methodpresented by Gazizov and Khalique [19] also considerstransformations of the perturbation parameter while the typeof the transformation is different from that presented hereWith their method the regular expansion of the solution ofthe original system is successfully derived

The method is extended in this study The procedure isbriefly summarized as follows (1) Specify beforehand a non-perturbative solution on which we would like to see the effectof the perturbation (2) Construct equations composed of theindependent and dependent variables and their derivativesbased on the nonperturbative solution specified in step (1)Those equations are referred to as constraint equations inwhat follows (3) Find an approximate Lie symmetry groupof the original equations on the manifold formed by theconstraint equations constructed in the previous step (4) Forthe approximate Lie symmetry group found in step (3) con-struct the differential equation that is satisfied by the group-invariant solution and make an appropriate approximationwhen necessary As a consequence this differential equationgenerates the reduced systemwhich describes the asymptoticbehavior of the original system

The procedure is almost the same as that presented beforefor ordinary differential equations and difference equations[31ndash33] The main difference is in the second step that isto say the addition of the constraint equations for findingan approximate Lie symmetry group In general additionalLie symmetry groups can be found by the addition of sucha constraint equation because the groups do not need tobe admitted by the entire manifold formed by the originalsystem in the jet space but need to be admitted only by thatpart of themanifold restricted by the added equations In thissense a Lie symmetry group found by using this procedureis interpreted as a partial Lie symmetry group [24] Becauseof the increase in the number of the admitted Lie symmetrygroups in some cases this method allows us to find a Liesymmetry group when it is impossible to find such a groupwith the normal Lie group analysis

3 Application to OrdinaryDifferential Equation

Themethod presented in this study is particularlymeaningfulwhen it is applied to partial differential equations Howeverfirst we apply the method to an ordinary differential equa-tion because the calculation is simpler and therefore servesto clarify the essence of the method

Consider the following ordinary differential equation[34]

2119906 + 2= 120576 (1)

Here 119906 denotes the dependent variable the overdot denotesthe derivative with respect to the independent variable 119905 and120576 denotes a perturbation parameter supposed to be smallsufficiently

The two independent solutions of the nonperturbativeequation 2119906 + 2 = 0 are

119906(0)= 11986011990523 119861 (2)

where 119860 and 119861 are constantsFocusing on one of these solutions 119906(0) = 11986011990523 we inves-

tigate the effect of the perturbation on this nonperturbativesolution For this solution

(0)=2

3

119906(0)

119905

(3)

is satisfied From this relation an equation is constructed

=2

3

119906

119905 (4)

which is used as a constraint equation in finding a Liesymmetry group

We find a Lie symmetry group admitted by (1) Let

119883 = 120597120576+ 120578 (119905 119906) 120597

119906 (5)

be the infinitesimal generator of a Lie symmetry group Itssecond prolongation119883lowast is given by

119883lowast= 119883 + 120578

(119905 119906 ) 120597

+ 120578(119905 119906 ) 120597

120578= (120597119905+ 120597119906) 120578

120578= (120597119905+ 120597119906+ 120597) 120578

(6)

When we adopt the normal procedure followed by Lie groupanalysis [12] we find a Lie symmetry group by solving thedifferential equation for 120578(119905 119906)

119883lowast[2119906 +

2minus 120576]100381610038161003816100381610038162119906+2=120576

= 0 (7)

which indicates the criterion for the invariance of the entiremanifold formed by the original differential equation in thejet space In the extended method presented in this studyinstead we find a Lie symmetry group by solving

119883lowast[2119906 +

2minus 120576]10038161003816100381610038161003816=(23)(119906119905)

= 0 (8)

Note that = (23)(119906119905) comes from (4) Then (8) reads

[21199061205972

119905+4

3

1199062

119905120597119905120597119906+8

9

1199063

11990521205972

119906+4

3

119906

119905120597119905+4

9

1199062

1199052120597119906minus4

9

119906

1199052]

sdot 120578 minus 1 = 0

(9)

Noting the homogeneity of the equation we find an exactsolution of this differential equation

120578 = minus9

20

1199052

119906 (10)

Journal of Applied Mathematics 3

For this Lie symmetry group the group-invariant solu-tion 119906 = 119906(120576 119905) satisfies

119883 [119906 minus 119906 (120576 119905)]|119906=119906(120576119905) = 0 (11)

which reads120597119906

120597120576= minus9

20

1199052

119906 (12)

This equation which can be analytically solved has thefollowing solution

119906 (120576 119905) = plusmn11990523(119860 minus

9

1012057611990523)

12

(13)

Here we have used 119906(0 119905) = 11986011990523 which is the nonpertur-bative solution as the boundary condition in the integrationThis solution corresponds exactly to the solution derivedwiththe renormalization group method [34]

The original differential equation (1) admits no exact Liesymmetry group when we adopt the normal procedure (seethe Appendix) However the addition of constraint equation(4) has allowed us to find a Lie symmetry group

4 Application to Partial Differential Equation

In this section the present method is applied to two partialdifferential equations In general in the case of partialdifferential equations because of the increase of independentvariables it often happens that a partial differential equa-tion admits no approximate Lie symmetry group suitablefor the derivation of a system describing the asymptoticdynamics However it is in some cases possible to find anapproximate Lie symmetry group suitable for deriving theasymptotic dynamic properties by adding some constraintequations constructed from the nonperturbative solutionthereby restricting the manifold on which we find a Liesymmetry group The reduced systems obtained here areconsistent with those derived with the singular perturbationmethods presented before

41 First Example 119906119905+ 119860(120576119906)119906

119909= 0 Consider a system of

weakly nonlinear first-order partial differential equations

119906119905+ 119860 (120576119906) 119906

119909= 0 (14)

Here 119906 is a real vector that has 119899 components and depends ontwo independent variables 119905 and119909119860 is an 119899times119899nondegeneratematrix that depends on 119906 120576 is a perturbation parameterThe subscripts of 119906 denote the derivative with respect to theindependent variables

Similar to the approach followed in the previous sectionwe find a Lie symmetry group such that it is not admitted bythe entire manifold formed by the original equation ratherit is admitted by a part of the manifold determined from asolution of the nonperturbative system In the following wefocus on such a nonperturbative solution 119906(0)(119905 119909) that is anarbitrary function of 119909 minus 120582119905 where 120582 is an eigenvalue of119860

0=

119860(0) namely we set

119906(0)(119905 119909) = 119880 (119909 minus 120582119905) (15)

This solution represents a traveling wave of which the velo-city is 120582 As shown by the direct substitution into the non-perturbative system 119906(0)

119909is an eigenvector of 119860

0

1198600119906(0)

119909= 120582119906(0)

119909 (16)

In addition

119906(0)

119905+ 120582119906(0)

119909= 0 (17)

is also satisfied Based on these relations we find a Liesymmetry group with constraints of

1198600119906119909= 120582119906119909

119906119905+ 120582119906119909= 0

(18)

in the followingLet

119883 = 120597120576+ 120578 (119905 119909 119906 120576) 120597

119906 (19)

be the infinitesimal generator of a Lie symmetry group Itsfirst prolongation119883lowast is given by

119883lowast= 119883 + 120578

119906119905120597119906119905

+ 120578119906119909120597119906119909

120578119906119905= (120597119905+ 119906119905120597119906) 120578

120578119906119909= (120597119909+ 119906119909120597119906) 120578

(20)

The criterion for the invariance of the original differentialequation (14) on the manifold restricted by the constraintequations (18) is given by

119883lowast[119906119905+ 119860 (120576119906) 119906

119909]10038161003816100381610038161198600119906119909=120582119906119909 119906119905+120582119906119909=0= 0 (21)

This equation reads

[120578119906119905+ 119860 (120576119906) 120578

119906119909+ (119906 sdot nabla)119860 (120576119906) 119906

119909

+ 120576 (120578 sdot nabla)119860 (120576119906) 119906119909]10038161003816100381610038161198600119906119909=120582119906119909 119906119905+120582119906119909=0= 0

(22)

Here (119906 sdot nabla)119860(120576119906) and (120578 sdot nabla)119860(120576119906) are defined by matrices ofwhich the 119894119895 components are given by respectively

[sum

119896

119906119896120597119903119896

119886119894119895(119903)]

1003816100381610038161003816100381610038161003816100381610038161003816119903=120576119906

[sum

119896

120578119896120597119903119896

119886119894119895(119903)]

1003816100381610038161003816100381610038161003816100381610038161003816119903=120576119906

(23)

where 119886119894119895denotes the 119894119895 component of the matrix 119860

Suppose 120578(119905 119909 119906 120576) and 119860(120576119906) can be expanded as apower series of 120576 Their expanded forms

120578 =

infin

sum

119896=0

120576119896120578119896(119905 119909 119906)

119860 (120576119906) =

infin

sum

119896=0

120576119896119860119896(119906)

(24)

4 Journal of Applied Mathematics

are substituted into (22) which is then rewritten as

[120578119906119905

0+ 1198600120578119906119909

0+ (119906 sdot nabla)119860

0119906119909]10038161003816100381610038161198600119906119909=120582119906119909 119906119905+120582119906119909=0= 0 (25)

[

[

120578119906119905

119894+ 1198600120578119906119909

119894+ (119906 sdot nabla)119860

119894119906119909

+

119894minus1

sum

119895=0

(120578119894minus119895minus1sdot nabla)119860

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119906=0

119906119909]

]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161198600119906119909=120582119906119909 119906119905+120582119906119909=0

= 0

(119894 = 1 2 )

(26)

We consider only the lowest order of 120576 namely (25) which isrewritten as

(120597119905+ 119906119905120597119906) 1205780+ 1198600(120597119909+ 119906119909120597119906) 1205780

minus [(119906 sdot nabla)1198600] 119906119909

10038161003816100381610038161198600119906119909=120582119906119909 119906119905+120582119906119909=0= 0

(27)

and reads

(120597119905+ 1198600120597119909) 1205780= [minus (119906 sdot nabla) 120582] 119906119909 (28)

The vector on the right-hand side of this equation is separatedinto the following two components

(120597119905+ 1198600120597119909) 1205780= minus (119890

sdot [(119906 sdot nabla) 120582] 119906119909) 119890

minus (119890perpsdot [(119906 sdot nabla) 120582] 119906119909) 119890perp

(29)

where 119890denotes the unit vector with the same direction

as 1199060(119905 119909) in R119899 and 119890

perpis the unit vector parallel to the

projection of [minus(119906 sdot nabla)120582]119906119909on a 119899 minus 1 dimensional subspace

in R119899 spanned by all the eigenvectors of 1198600except 119890

The

solution corresponding to the second term is given by a linearcombination of the 119899 minus 1 eigenvectors of 119860

0whose linear

coefficients are a function of 119909 minus 120582119905 On the other hand aterm proportional to 119905 appears in the solution correspondingto the first term because every vector proportional to 119890

whose proportional coefficient is a function of 119909minus120582119905 is a zeroeigenvector of the linear operator 120597

119905+ 1198600120597119909 For this reason

we can write a solution of (29) as follows

1205780= minus119905 (119890

sdot [(119906 sdot nabla) 120582] 119906119909) 119890

+ (terms not proportional to 119905) (30)

The group-invariant solution 119906 = 119906(119905 119909) for thisapproximate Lie symmetry group satisfies

119883 [119906 minus 119906 (119905 119909)]|119906=119906(119905119909) = 0 (31)

which reads120597119906 (119905 119909)

120597120576= minus119905 (119890

sdot [(119906 sdot nabla)119860

0] 119906119909) 119890

+ (terms not proportional to 119905) (32)

The long-time behavior of the solution is obtained by consid-ering the case in which 119905 ≫ 1 and by neglecting the termsother than the first one Then (32) is reduced to

119889

119889120591119906 (119905 119909) = minus (119890

sdot [(119906 sdot nabla)119860

0] 119906119909) 119890 (33)

where 120591 is a long time-scale defined by 120591 = 120576119905 As aresult we have obtained a dynamical system which describesasymptotic behavior of the original system This reduceddynamics corresponds to that presented before with thereductive perturbation method [3] or the renormalizationgroup method [7]

42 Second Example 119906119905119905minus 119906119909119909+ 119906 minus 120576119906

3= 0 We consider a

perturbed Klein-Gordon equation represented by

119906119905119905minus 119906119909119909+ 119906 minus 120576119906

3= 0 (34)

Here 119906 is a scalar function which depends on two indepen-dent variables 119905 and 119909 and 120576 is a perturbation parameterThe subscripts of 119906 denote the derivative with respect to theindependent variables By introducing new dependent vari-ables as

1199061= 119906119905

1199062= 119906119909

(35)

we rewrite (34) in terms of four first-order partial differentialequations for three dependent variables 119906 119906

1 and 119906

2as

follows

1199061119905minus 1199062119909+ 119906 minus 120576119906

3= 0

119906119905minus 1199061= 0

119906119909minus 1199062= 0

1199061119909minus 1199062119905= 0

(36)

The last equation implies the integrable conditionAs in the previous examples a specific nonperturbative

solution 119906(0)(119905 119909) is prespecified on which we investigate theeffect of the perturbationHerewe focus on a nonperturbativesolution describing a traveling wave with a constant velocityV The solution is represented by

119906(0)(119905 119909) = 119880 (119909 minus V119905) (37)

where 119880(120585) = 120572119890i120581120585 + 120572119890minusi120581120585 for 120581 = (V2 minus 1)minus12 and anarbitrary constant 120572 For this solution

119906(0)

119905= minusV119906(0)

119909

119906(0)

119905119905= minusV119906(0)

119905119909= minusV119906(0)

119909119905= V2119906(0)

119909119909

(38)

therefore

119906(0)

119905119905=119906(0)

119905

119906(0)

119909

119906(0)

119909119905=119906(0)

119905

119906(0)

119909

119906(0)

119905119909= (119906(0)

119905

119906(0)

119909

)

2

119906(0)

119909119909(39)

Journal of Applied Mathematics 5

are satisfied By combining these relations with the nonper-turbative system 119906(0)

119905119905minus 119906(0)

119909119909+ 119906(0)= 0 we obtain

119906(0)

119905119905=

[119906(0)

119905]2

119906(0)

[119906(0)

119909 ]2

minus [119906(0)

119905]2

119906(0)

119909119905= 119906(0)

119905119909=119906(0)

119905119906(0)

119909119906(0)

[119906(0)

119909 ]2

minus [119906(0)

119905]2

119906(0)

119909119909=

[119906(0)

119909]2

119906(0)

[119906(0)

119909 ]2

minus [119906(0)

119905]2

(40)

From these relations we construct the constraint equations

1199061119905=1199062

1119906

1199062

2minus 1199062

1

1199062119905= 1199061119909=11990611199062119906

1199062

2minus 1199062

1

1199062119909=1199062

2119906

1199062

2minus 1199062

1

(41)

which are used in the finding of Lie symmetry groupLet

119883(119905 119909 119906 1199061 1199062 120576) = 120597

120576+ 120578 (119905 119909 119906 119906

1 1199062 120576) 120597119906

+ 1205781199061(119905 119909 119906 119906

1 1199062 120576) 1205971199061

+ 1205781199062(119905 119909 119906 119906

1 1199062 120576) 1205971199062

(42)

be the infinitesimal generator of a Lie symmetry group of thefour partial differential equations (36) Its first prolongation119883lowast is given by

119883lowast(119905 119909 119906 119906

1 1199062 120576)

= 119883 + 120578119905(119905 119909 119906 119906

1 1199062 11990611 11990612 11990622 120576) 120597119906119905

+ 120578119909(119905 119909 119906 119906

1 1199062 11990611 11990612 11990622 120576) 120597119906119909

+ 1205781199061119905(119905 119909 119906 119906

1 1199062 11990611 11990612 11990622 120576) 1205971199061119905

+ 1205781199062119909(119905 119909 119906 119906

1 1199062 11990611 11990612 11990622 120576) 1205971199062119909

(43)

where 120578119905 120578119909 1205781199061119905 and 1205781199062119909 are defined as follows

120578119905(119905 119909 119906 119906

1 1199062 119906119905 1199061119905 1199062119905 120576)

= 119863119905120578 (119905 119909 119906 119906

1 1199062 120576)

120578119909(119905 119909 119906 119906

1 1199062 119906119909 1199061119909 1199062119909 120576)

= 119863119909120578 (119905 119909 119906 119906

1 1199062 120576)

1205781199061119905(119905 119909 119906 119906

1 1199062 119906119905 1199061119905 1199062119905 120576)

= 1198631199051205781199061(119905 119909 119906 119906

1 1199062 120576)

1205781199062119909(119905 119909 119906 119906

1 1199062 119906119909 1199061119909 1199062119909 120576)

= 1198631199091205781199062(119905 119909 119906 119906

1 1199062 120576)

119863119905= 120597119905+ 1199061120597119906+ 11990611199051205971199061

+ 11990621199051205971199062

119863119909= 120597119909+ 119906119909120597119906+ 11990611199091205971199061

+ 11990621199091205971199062

(44)

The criteria for the invariance of the original system onthe manifold formed by the constraint equations (41) aregiven by

119883lowast[1199061119905minus 1199062119909+ 119906 minus 120576119906

3]10038161003816100381610038161003816(41)= 0

119883lowast[119906119905minus 1199061]1003816100381610038161003816(41)= 0

119883lowast[119906119909minus 1199062]1003816100381610038161003816(41)= 0

119883lowast[1199061119909minus 1199062119905]1003816100381610038161003816(41)= 0

(45)

With the identities

1205971199061199062

1119906

1199062

2minus 1199062

1

=1199062

1

1199062

2minus 1199062

1

1205971199061

1199062

1119906

1199062

2minus 1199062

1

=211990611199062

2119906

(1199062

2minus 1199062

1)2

1205971199062

1199062

1119906

1199062

2minus 1199062

1

=minus21199062

11199062119906

(1199062

2minus 1199062

1)2

12059711990611990611199062119906

1199062

2minus 1199062

1

=11990611199062

1199062

2minus 1199062

1

1205971199061

11990611199062119906

1199062

2minus 1199062

1

=

(1199062

1+ 1199062

2) 1199062119906

(1199062

2minus 1199062

1)2

1205971199062

11990611199062119906

1199062

2minus 1199062

1

=

minus (1199062

1+ 1199062

2) 1199061119906

(1199062

2minus 1199062

1)2

1205971199061199062

2119906

1199062

2minus 1199062

1

=1199062

2

1199062

2minus 1199062

1

1205971199061

1199062

2119906

1199062

2minus 1199062

1

=211990611199062

2119906

(1199062

2minus 1199062

1)2

1205971199062

1199062

2119906

1199062

2minus 1199062

1

=minus21199062

11199062119906

(1199062

2minus 1199062

1)2

(46)

6 Journal of Applied Mathematics

(45) read in the lowest order of 120576

(1199062

2minus 1199062

1) (120578119905119905minus 120578119909119909) minus (119906

2

2minus 1199062

1)2

120578119906119906minus 1199062

1119906212057811990611199061

minus 1199062

1119906212057811990621199062

+ 21199061119906 (1199062

2minus 1199062

1) 1205781199061199061

minus 21199062119906 (1199062

2minus 1199062

1) 1205781199061199062

minus 211990611199062119906212057811990611199062

+ 21199061(1199062

2minus 1199062

1) 120578119905119906+ 21199062

11199061205781199051199061

+ 2119906111990621199061205781199051199062

minus 21199062(1199062

2minus 1199062

1) 120578119909119906minus 2119906111990621199061205781199091199061

minus 21199062

21199061205781199091199062

minus 1199061(1199062

2minus 1199062

1) 1205781199061

minus 1199062(1199062

2minus 1199062

1) 1205781199062

minus 119906 (1199062

2minus 1199062

1) 120578119906+ (1199062

2minus 1199062

1) 120578 minus 119906

3(1199062

2minus 1199062

1)

= 0

(47)

Noting the homogeneity of the equation we find a solution120578 (119905 119909 119906 119906

1 1199062)

= minus3

8(1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2) (119886

119905

1199061

minus (1 minus 119886)119909

1199062

)

minus1

32119906 (1199062minus 1199062

1minus 1199062

2)

(48)

where 119886 is an arbitrary constant For this infinitesimalgenerator of the approximate Lie symmetry group the group-invariant solution satisfies

119883 [119906 minus 119906 (119905 119909)]|119906=119906(119905119909) = 0 (49)which reads

120597119906 (119905 119909)

120597120576= 120578 (119905 119909 119906 (119905 119909) 119906

1(119905 119909) 119906

2(119905 119909)) (50)

Asymptotic dynamic behavior can be obtained by consider-ing the cases 119905 ≫ 1 and 119909 ≫ 1 and by neglecting the termswhich are not proportional to 119909 or 119905 By introducing 120591 = 120576119905and 120585 = 120576119909 we obtain

120597119906

120597120591= minus

3 (1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2)

81199061

119886

120597119906

120597120585=

3 (1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2)

81199062

(1 minus 119886)

(51)

By eliminating 119886 the equations are reduced to

1199061

120597119906

120597120591minus 1199062

120597119906

120597120585= minus3

8(1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2) (52)

This reduced equation describes the asymptotic behavior ofthe original systemThis fact can be confirmed by comparingwith the result derived before By introducing a transforma-tion of the variables

|119860|2=1

4(1199062+ 1199062

1minus 1199062

2)

120596 = 1199061(1199062

1minus 1199062

2)minus12

119896 = 1199062(1199062

1minus 1199062

2)minus12

(53)

the reduced equation (52) is transformed into

[120597

120597120591+119896

120596

120597

120597120585]119860 = minus

3

2120596i |119860|2 119860 (54)

This is the nonlinear Schrodinger equation which corre-sponds to the reduced equation derived with the renormal-ization group method [9]

5 Concluding Remarks

In summary this paper presents an investigation of perturba-tion problems which involves the application of a particulartype of Lie group method to partial as well as ordinarydifferential equations As a consequence we have obtainedthe dynamics which describes the asymptotic behavior ofthe original system The main characteristic of the presentmethod is the addition of some constraint equations whensearching for a Lie symmetry group an approach whichallows us to find a partial Lie symmetry groupThe constraintequations are determined from the nonperturbative solutionon which we try to investigate the effect of the perturbationThe reason why it is necessary to specify a nonperturbativesolution beforehand for the partial differential equations isthat unlike for ordinary differential equations the non-perturbative system has an infinite number of independentsolutions In terms of technique those constraint equationsare constructed in such a way that the arbitrary constantsor functions included in the nonperturbative solution areeliminated as we have done in deriving (4) (18) and (41) Inthis study we have investigated the effect of the perturbationof its lowest order however extending our work to higher-order approximations would require us to use higher-orderconstraint equations instead

Examples towhich themethod is applied here are singularperturbation problems in which secular terms appear in thenaıve expansion of the solution [3 6 8 10 11 34] Termsproportional to the independent variables emerge in theinfinitesimal generator of the Lie symmetry group as seen in(10) (30) and (48) corresponding to the emergence of thesecular terms in the naıve expansion As inferred from thismathematical similarity the presentmethod is also applicableto other systems which exhibit the same type of singularperturbation problems In the singular perturbationmethodsproposed thus far it is only after the construction of the naıveexpansion that it becomes clear whether the method shouldbe used or not However the present method can be appliedwithout making the expansion enabling the appropriatereduced equations to be derived

Future work should involve a clarification of the relationsbetween the present method and other types of singularperturbation methods presented thus far such as the WKBor boundary layer problem methods In addition there areexisting methods that have enabled the successful deriva-tion of the asymptotic behavior for various other partialdifferential equations [13ndash18 20ndash30 35ndash38] Therefore itshould be clarified whether or not the present method canbe appropriately extended and applied to the problems thosemethods have resolved

Journal of Applied Mathematics 7

Appendix

Searching for Lie Symmetry withthe Normal Procedure

Let us search for the Lie symmetry group of (1) withoutconstraint equations The criterion for the invariance of theequation is given by (7) namely

119883lowast[2119906 + minus 120576]

10038161003816100381610038162119906+minus120576=0= 0 (A1)

which reads

21199062120578119905119905minus 119906 + (4119906

2120578119906119905+ 2119906120578

119905)

+ 2(21199062120578119906119906+ 119906120578 minus 120578) + 120576 (120578 + 119906120578

119906) = 0

(A2)

In the lowest order of 120576 120578 has to satisfy the followingdifferential equations

21199062120578119905119905minus 119906 = 0

41199062120578119906119905+ 2119906120578

119905= 0

21199062120578119906119906+ 119906120578 minus 120578 = 0

(A3)

However we can easily find out that there is no solution thatsatisfies all the above equations which implies that there is noLie symmetry group

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by JSPS KAKENHI Grant no26800208

References

[1] N N Bogoliubov and C M Place Asymptotic Methods in theTheory of Non-Linear Oscillations Gordon and Break 1961

[2] GHori ldquoTheory of general perturbationwith unspecified cano-nical variablerdquo Publications of the Astronomical Society of Japanvol 18 p 287 1966

[3] T Taniuchi ldquoReductive perturbation method and far fields ofwave equationsrdquo Progress ofTheoretical Physics Supplement vol55 pp 1ndash35 1974

[4] J Guckenheimer andPHolmesNonlinearOscillationsDynam-ical systems and Bifurcations of Vector Fields Springer NewYork NY USA 1983

[5] A H Nayfeh Method of Normal Forms John Wiley amp SonsNew York NY USA 1993

[6] T Kunihiro ldquoA geometrical formulation of the renormalizationgroup method for global analysisrdquo Progress of TheoreticalPhysics vol 94 no 4 pp 503ndash514 1995

[7] L-Y ChenNGoldenfeld YOono andG Paquette ldquoSelectionstability and renormalizationrdquo Physica A Statistical Mechanicsand its Applications vol 204 no 1ndash4 pp 111ndash133 1994

[8] L Y Chen N Goldenfeld and Y Oono ldquoRenormalizationgroup and singular perturbations multiple scales boundarylayers and reductive perturbation theoryrdquo Physical Review Evol 54 no 1 pp 376ndash394 1996

[9] S-I Goto Y Masutomi and K Nozaki ldquoLie-group approachto perturbative renormalization group methodrdquo Progress ofTheoretical Physics vol 102 no 3 pp 471ndash497 1999

[10] R E L DeVille A Harkin M Holzer K Josic and T J KaperldquoAnalysis of a renormalization group method and normal formtheory for perturbed ordinary differential equationsrdquoPhysicaDvol 237 no 8 pp 1029ndash1052 2008

[11] H Chiba ldquo1198621 approximation of vector fields based on therenormalization group methodrdquo SIAM Journal on AppliedDynamical Systems vol 7 no 3 pp 895ndash932 2008

[12] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1986

[13] V A Baıkov R K Gazizov and N H Ibragimov ldquoApproximatesymmetriesrdquoMathematics of the USSR-Sbornik vol 64 no 2 p427 1989

[14] W I Fushchich andWM Shtelen ldquoOn approximate symmetryand approximate solutions of the nonlinear wave equation witha small parameterrdquo Journal of Physics A Mathematical andGeneral vol 22 no 18 pp L887ndashL890 1989

[15] N Eular MW Shulga andW-H Steeb ldquoApproximate symme-tries and approximate solutions for amultidimensional Landau-Ginzburg equationrdquo Journal of Physics A Mathematical andGeneral vol 25 no 18 Article ID L1095 1992

[16] Y Y Bagderina and R K Gazizov ldquoEquivalence of ordinarydifferential equations 11991010158401015840 = 119877(119909 119910)11991010158402 + 2119876(119909 119910)1199101015840 + 119875(119909 119910)rdquoDifferential Equations vol 43 no 5 pp 595ndash604 2007

[17] V N Grebenev andM Oberlack ldquoApproximate Lie symmetriesof theNavier-Stokes equationsrdquo Journal ofNonlinearMathemat-ical Physics vol 14 no 2 pp 157ndash163 2007

[18] R K Gazizov and N K Ibragimov ldquoApproximate symmetriesand solutions of the Kompaneets equationrdquo Journal of AppliedMechanics and Technical Physics vol 55 no 2 pp 220ndash2242014

[19] R K Gazizov and C M Khalique ldquoApproximate transforma-tions for van der Pol-type equationsrdquoMathematical Problems inEngineering vol 2006 Article ID 68753 11 pages 2006

[20] G Gaeta ldquoAsymptotic symmetries and asymptotically symmet-ric solutions of partial differential equationsrdquo Journal of PhysicsA Mathematical and General vol 27 no 2 pp 437ndash451 1994

[21] G Gaeta ldquoAsymptotic symmetries in an optical latticerdquo PhysicalReview A vol 72 no 3 Article ID 033419 2005

[22] G Gaeta D Levi and R Mancinelli ldquoAsymptotic symmetriesof difference equations on a latticerdquo Journal of NonlinearMathe-matical Physics vol 12 supplement 2 pp 137ndash146 2005

[23] G Gaeta and RMancinelli ldquoAsymptotic scaling symmetries fornonlinear PDEsrdquo International Journal of Geometric Methods inModern Physics vol 2 no 6 pp 1081ndash1114 2005

[24] G Gaeta and R Mancinelli ldquoAsymptotic scaling in a modelclass of anomalous reaction-diffusion equationsrdquo Journal ofNonlinear Mathematical Physics vol 12 no 4 pp 550ndash5662005

[25] D Levi and M A Rodrıguez ldquoAsymptotic symmetries andintegrability the KdV caserdquo Europhysics Letters vol 80 no 6Article ID 60005 2007

[26] V F Kovalev V V Pustovalov and D V Shirkov ldquoGroup anal-ysis and renormgroup symmetriesrdquo Journal of MathematicalPhysics vol 39 no 2 pp 1170ndash1188 1998

8 Journal of Applied Mathematics

[27] V F Kovalev ldquoRenormalization group analysis for singularitiesin the wave beam self-focusing problemrdquo Theoretical andMathematical Physics vol 119 no 3 pp 719ndash730 1999

[28] V F Kovalev and D V Shirkov ldquoFunctional self-similarityand renormalization group symmetry inmathematical physicsrdquoTheoretical and Mathematical Physics vol 121 no 1 pp 1315ndash1332 1999

[29] D V Shirkov and V F Kovalev ldquoThe Bogoliubov renormaliza-tion group and solution symmetry in mathematical physicsrdquoPhysics Report vol 352 no 4ndash6 pp 219ndash249 2001

[30] V F Kovalev and D V Shirkov ldquoThe renormalization groupsymmetry for solution of integral equationsrdquo Proceedings of theInstitute of Mathematics of NAS of Ukraine vol 50 no 2 pp850ndash861 2004

[31] M Iwasa and K Nozaki ldquoA method to construct asymptoticsolutions invariant under the renormalization grouprdquo Progressof Theoretical Physics vol 116 no 4 pp 605ndash613 2006

[32] M Iwasa and K Nozaki ldquoRenormalization group in differencesystemsrdquo Journal of Physics A Mathematical and Theoreticalvol 41 no 8 Article ID 085204 2008

[33] H Chiba andM Iwasa ldquoLie equations for asymptotic solutionsof perturbation problems of ordinary differential equationsrdquoJournal of Mathematical Physics vol 50 no 4 2009

[34] Y Y Yamaguchi and Y Nambu ldquoRenormalization group equa-tions and integrability in hamiltonian systemsrdquo Progress ofTheoretical Physics vol 100 no 1 pp 199ndash204 1998

[35] N D Goldenfeld Lectures on Phase Transitions and the Renor-malization Group Addison-Wesley 1992

[36] J Bricmont A Kupiainen and G Lin ldquoRenormalization groupand asymptotics of solutions of nonlinear parabolic equationsrdquoCommunications on Pure and Applied Mathematics vol 47 no6 pp 893ndash922 1994

[37] N V Antonov and J Honkonen ldquoField-theoretic renormaliza-tion group for a nonlinear diffusion equationrdquo Physical ReviewE vol 66 no 4 Article ID 046105 7 pages 2002

[38] S Yoshida andT Fukui ldquoExact renormalization group approachto a nonlinear diffusion equationrdquo Physical Review E vol 72 no4 Article ID 046136 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

For this Lie symmetry group the group-invariant solu-tion 119906 = 119906(120576 119905) satisfies

119883 [119906 minus 119906 (120576 119905)]|119906=119906(120576119905) = 0 (11)

which reads120597119906

120597120576= minus9

20

1199052

119906 (12)

This equation which can be analytically solved has thefollowing solution

119906 (120576 119905) = plusmn11990523(119860 minus

9

1012057611990523)

12

(13)

Here we have used 119906(0 119905) = 11986011990523 which is the nonpertur-bative solution as the boundary condition in the integrationThis solution corresponds exactly to the solution derivedwiththe renormalization group method [34]

The original differential equation (1) admits no exact Liesymmetry group when we adopt the normal procedure (seethe Appendix) However the addition of constraint equation(4) has allowed us to find a Lie symmetry group

4 Application to Partial Differential Equation

In this section the present method is applied to two partialdifferential equations In general in the case of partialdifferential equations because of the increase of independentvariables it often happens that a partial differential equa-tion admits no approximate Lie symmetry group suitablefor the derivation of a system describing the asymptoticdynamics However it is in some cases possible to find anapproximate Lie symmetry group suitable for deriving theasymptotic dynamic properties by adding some constraintequations constructed from the nonperturbative solutionthereby restricting the manifold on which we find a Liesymmetry group The reduced systems obtained here areconsistent with those derived with the singular perturbationmethods presented before

41 First Example 119906119905+ 119860(120576119906)119906

119909= 0 Consider a system of

weakly nonlinear first-order partial differential equations

119906119905+ 119860 (120576119906) 119906

119909= 0 (14)

Here 119906 is a real vector that has 119899 components and depends ontwo independent variables 119905 and119909119860 is an 119899times119899nondegeneratematrix that depends on 119906 120576 is a perturbation parameterThe subscripts of 119906 denote the derivative with respect to theindependent variables

Similar to the approach followed in the previous sectionwe find a Lie symmetry group such that it is not admitted bythe entire manifold formed by the original equation ratherit is admitted by a part of the manifold determined from asolution of the nonperturbative system In the following wefocus on such a nonperturbative solution 119906(0)(119905 119909) that is anarbitrary function of 119909 minus 120582119905 where 120582 is an eigenvalue of119860

0=

119860(0) namely we set

119906(0)(119905 119909) = 119880 (119909 minus 120582119905) (15)

This solution represents a traveling wave of which the velo-city is 120582 As shown by the direct substitution into the non-perturbative system 119906(0)

119909is an eigenvector of 119860

0

1198600119906(0)

119909= 120582119906(0)

119909 (16)

In addition

119906(0)

119905+ 120582119906(0)

119909= 0 (17)

is also satisfied Based on these relations we find a Liesymmetry group with constraints of

1198600119906119909= 120582119906119909

119906119905+ 120582119906119909= 0

(18)

in the followingLet

119883 = 120597120576+ 120578 (119905 119909 119906 120576) 120597

119906 (19)

be the infinitesimal generator of a Lie symmetry group Itsfirst prolongation119883lowast is given by

119883lowast= 119883 + 120578

119906119905120597119906119905

+ 120578119906119909120597119906119909

120578119906119905= (120597119905+ 119906119905120597119906) 120578

120578119906119909= (120597119909+ 119906119909120597119906) 120578

(20)

The criterion for the invariance of the original differentialequation (14) on the manifold restricted by the constraintequations (18) is given by

119883lowast[119906119905+ 119860 (120576119906) 119906

119909]10038161003816100381610038161198600119906119909=120582119906119909 119906119905+120582119906119909=0= 0 (21)

This equation reads

[120578119906119905+ 119860 (120576119906) 120578

119906119909+ (119906 sdot nabla)119860 (120576119906) 119906

119909

+ 120576 (120578 sdot nabla)119860 (120576119906) 119906119909]10038161003816100381610038161198600119906119909=120582119906119909 119906119905+120582119906119909=0= 0

(22)

Here (119906 sdot nabla)119860(120576119906) and (120578 sdot nabla)119860(120576119906) are defined by matrices ofwhich the 119894119895 components are given by respectively

[sum

119896

119906119896120597119903119896

119886119894119895(119903)]

1003816100381610038161003816100381610038161003816100381610038161003816119903=120576119906

[sum

119896

120578119896120597119903119896

119886119894119895(119903)]

1003816100381610038161003816100381610038161003816100381610038161003816119903=120576119906

(23)

where 119886119894119895denotes the 119894119895 component of the matrix 119860

Suppose 120578(119905 119909 119906 120576) and 119860(120576119906) can be expanded as apower series of 120576 Their expanded forms

120578 =

infin

sum

119896=0

120576119896120578119896(119905 119909 119906)

119860 (120576119906) =

infin

sum

119896=0

120576119896119860119896(119906)

(24)

4 Journal of Applied Mathematics

are substituted into (22) which is then rewritten as

[120578119906119905

0+ 1198600120578119906119909

0+ (119906 sdot nabla)119860

0119906119909]10038161003816100381610038161198600119906119909=120582119906119909 119906119905+120582119906119909=0= 0 (25)

[

[

120578119906119905

119894+ 1198600120578119906119909

119894+ (119906 sdot nabla)119860

119894119906119909

+

119894minus1

sum

119895=0

(120578119894minus119895minus1sdot nabla)119860

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119906=0

119906119909]

]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161198600119906119909=120582119906119909 119906119905+120582119906119909=0

= 0

(119894 = 1 2 )

(26)

We consider only the lowest order of 120576 namely (25) which isrewritten as

(120597119905+ 119906119905120597119906) 1205780+ 1198600(120597119909+ 119906119909120597119906) 1205780

minus [(119906 sdot nabla)1198600] 119906119909

10038161003816100381610038161198600119906119909=120582119906119909 119906119905+120582119906119909=0= 0

(27)

and reads

(120597119905+ 1198600120597119909) 1205780= [minus (119906 sdot nabla) 120582] 119906119909 (28)

The vector on the right-hand side of this equation is separatedinto the following two components

(120597119905+ 1198600120597119909) 1205780= minus (119890

sdot [(119906 sdot nabla) 120582] 119906119909) 119890

minus (119890perpsdot [(119906 sdot nabla) 120582] 119906119909) 119890perp

(29)

where 119890denotes the unit vector with the same direction

as 1199060(119905 119909) in R119899 and 119890

perpis the unit vector parallel to the

projection of [minus(119906 sdot nabla)120582]119906119909on a 119899 minus 1 dimensional subspace

in R119899 spanned by all the eigenvectors of 1198600except 119890

The

solution corresponding to the second term is given by a linearcombination of the 119899 minus 1 eigenvectors of 119860

0whose linear

coefficients are a function of 119909 minus 120582119905 On the other hand aterm proportional to 119905 appears in the solution correspondingto the first term because every vector proportional to 119890

whose proportional coefficient is a function of 119909minus120582119905 is a zeroeigenvector of the linear operator 120597

119905+ 1198600120597119909 For this reason

we can write a solution of (29) as follows

1205780= minus119905 (119890

sdot [(119906 sdot nabla) 120582] 119906119909) 119890

+ (terms not proportional to 119905) (30)

The group-invariant solution 119906 = 119906(119905 119909) for thisapproximate Lie symmetry group satisfies

119883 [119906 minus 119906 (119905 119909)]|119906=119906(119905119909) = 0 (31)

which reads120597119906 (119905 119909)

120597120576= minus119905 (119890

sdot [(119906 sdot nabla)119860

0] 119906119909) 119890

+ (terms not proportional to 119905) (32)

The long-time behavior of the solution is obtained by consid-ering the case in which 119905 ≫ 1 and by neglecting the termsother than the first one Then (32) is reduced to

119889

119889120591119906 (119905 119909) = minus (119890

sdot [(119906 sdot nabla)119860

0] 119906119909) 119890 (33)

where 120591 is a long time-scale defined by 120591 = 120576119905 As aresult we have obtained a dynamical system which describesasymptotic behavior of the original system This reduceddynamics corresponds to that presented before with thereductive perturbation method [3] or the renormalizationgroup method [7]

42 Second Example 119906119905119905minus 119906119909119909+ 119906 minus 120576119906

3= 0 We consider a

perturbed Klein-Gordon equation represented by

119906119905119905minus 119906119909119909+ 119906 minus 120576119906

3= 0 (34)

Here 119906 is a scalar function which depends on two indepen-dent variables 119905 and 119909 and 120576 is a perturbation parameterThe subscripts of 119906 denote the derivative with respect to theindependent variables By introducing new dependent vari-ables as

1199061= 119906119905

1199062= 119906119909

(35)

we rewrite (34) in terms of four first-order partial differentialequations for three dependent variables 119906 119906

1 and 119906

2as

follows

1199061119905minus 1199062119909+ 119906 minus 120576119906

3= 0

119906119905minus 1199061= 0

119906119909minus 1199062= 0

1199061119909minus 1199062119905= 0

(36)

The last equation implies the integrable conditionAs in the previous examples a specific nonperturbative

solution 119906(0)(119905 119909) is prespecified on which we investigate theeffect of the perturbationHerewe focus on a nonperturbativesolution describing a traveling wave with a constant velocityV The solution is represented by

119906(0)(119905 119909) = 119880 (119909 minus V119905) (37)

where 119880(120585) = 120572119890i120581120585 + 120572119890minusi120581120585 for 120581 = (V2 minus 1)minus12 and anarbitrary constant 120572 For this solution

119906(0)

119905= minusV119906(0)

119909

119906(0)

119905119905= minusV119906(0)

119905119909= minusV119906(0)

119909119905= V2119906(0)

119909119909

(38)

therefore

119906(0)

119905119905=119906(0)

119905

119906(0)

119909

119906(0)

119909119905=119906(0)

119905

119906(0)

119909

119906(0)

119905119909= (119906(0)

119905

119906(0)

119909

)

2

119906(0)

119909119909(39)

Journal of Applied Mathematics 5

are satisfied By combining these relations with the nonper-turbative system 119906(0)

119905119905minus 119906(0)

119909119909+ 119906(0)= 0 we obtain

119906(0)

119905119905=

[119906(0)

119905]2

119906(0)

[119906(0)

119909 ]2

minus [119906(0)

119905]2

119906(0)

119909119905= 119906(0)

119905119909=119906(0)

119905119906(0)

119909119906(0)

[119906(0)

119909 ]2

minus [119906(0)

119905]2

119906(0)

119909119909=

[119906(0)

119909]2

119906(0)

[119906(0)

119909 ]2

minus [119906(0)

119905]2

(40)

From these relations we construct the constraint equations

1199061119905=1199062

1119906

1199062

2minus 1199062

1

1199062119905= 1199061119909=11990611199062119906

1199062

2minus 1199062

1

1199062119909=1199062

2119906

1199062

2minus 1199062

1

(41)

which are used in the finding of Lie symmetry groupLet

119883(119905 119909 119906 1199061 1199062 120576) = 120597

120576+ 120578 (119905 119909 119906 119906

1 1199062 120576) 120597119906

+ 1205781199061(119905 119909 119906 119906

1 1199062 120576) 1205971199061

+ 1205781199062(119905 119909 119906 119906

1 1199062 120576) 1205971199062

(42)

be the infinitesimal generator of a Lie symmetry group of thefour partial differential equations (36) Its first prolongation119883lowast is given by

119883lowast(119905 119909 119906 119906

1 1199062 120576)

= 119883 + 120578119905(119905 119909 119906 119906

1 1199062 11990611 11990612 11990622 120576) 120597119906119905

+ 120578119909(119905 119909 119906 119906

1 1199062 11990611 11990612 11990622 120576) 120597119906119909

+ 1205781199061119905(119905 119909 119906 119906

1 1199062 11990611 11990612 11990622 120576) 1205971199061119905

+ 1205781199062119909(119905 119909 119906 119906

1 1199062 11990611 11990612 11990622 120576) 1205971199062119909

(43)

where 120578119905 120578119909 1205781199061119905 and 1205781199062119909 are defined as follows

120578119905(119905 119909 119906 119906

1 1199062 119906119905 1199061119905 1199062119905 120576)

= 119863119905120578 (119905 119909 119906 119906

1 1199062 120576)

120578119909(119905 119909 119906 119906

1 1199062 119906119909 1199061119909 1199062119909 120576)

= 119863119909120578 (119905 119909 119906 119906

1 1199062 120576)

1205781199061119905(119905 119909 119906 119906

1 1199062 119906119905 1199061119905 1199062119905 120576)

= 1198631199051205781199061(119905 119909 119906 119906

1 1199062 120576)

1205781199062119909(119905 119909 119906 119906

1 1199062 119906119909 1199061119909 1199062119909 120576)

= 1198631199091205781199062(119905 119909 119906 119906

1 1199062 120576)

119863119905= 120597119905+ 1199061120597119906+ 11990611199051205971199061

+ 11990621199051205971199062

119863119909= 120597119909+ 119906119909120597119906+ 11990611199091205971199061

+ 11990621199091205971199062

(44)

The criteria for the invariance of the original system onthe manifold formed by the constraint equations (41) aregiven by

119883lowast[1199061119905minus 1199062119909+ 119906 minus 120576119906

3]10038161003816100381610038161003816(41)= 0

119883lowast[119906119905minus 1199061]1003816100381610038161003816(41)= 0

119883lowast[119906119909minus 1199062]1003816100381610038161003816(41)= 0

119883lowast[1199061119909minus 1199062119905]1003816100381610038161003816(41)= 0

(45)

With the identities

1205971199061199062

1119906

1199062

2minus 1199062

1

=1199062

1

1199062

2minus 1199062

1

1205971199061

1199062

1119906

1199062

2minus 1199062

1

=211990611199062

2119906

(1199062

2minus 1199062

1)2

1205971199062

1199062

1119906

1199062

2minus 1199062

1

=minus21199062

11199062119906

(1199062

2minus 1199062

1)2

12059711990611990611199062119906

1199062

2minus 1199062

1

=11990611199062

1199062

2minus 1199062

1

1205971199061

11990611199062119906

1199062

2minus 1199062

1

=

(1199062

1+ 1199062

2) 1199062119906

(1199062

2minus 1199062

1)2

1205971199062

11990611199062119906

1199062

2minus 1199062

1

=

minus (1199062

1+ 1199062

2) 1199061119906

(1199062

2minus 1199062

1)2

1205971199061199062

2119906

1199062

2minus 1199062

1

=1199062

2

1199062

2minus 1199062

1

1205971199061

1199062

2119906

1199062

2minus 1199062

1

=211990611199062

2119906

(1199062

2minus 1199062

1)2

1205971199062

1199062

2119906

1199062

2minus 1199062

1

=minus21199062

11199062119906

(1199062

2minus 1199062

1)2

(46)

6 Journal of Applied Mathematics

(45) read in the lowest order of 120576

(1199062

2minus 1199062

1) (120578119905119905minus 120578119909119909) minus (119906

2

2minus 1199062

1)2

120578119906119906minus 1199062

1119906212057811990611199061

minus 1199062

1119906212057811990621199062

+ 21199061119906 (1199062

2minus 1199062

1) 1205781199061199061

minus 21199062119906 (1199062

2minus 1199062

1) 1205781199061199062

minus 211990611199062119906212057811990611199062

+ 21199061(1199062

2minus 1199062

1) 120578119905119906+ 21199062

11199061205781199051199061

+ 2119906111990621199061205781199051199062

minus 21199062(1199062

2minus 1199062

1) 120578119909119906minus 2119906111990621199061205781199091199061

minus 21199062

21199061205781199091199062

minus 1199061(1199062

2minus 1199062

1) 1205781199061

minus 1199062(1199062

2minus 1199062

1) 1205781199062

minus 119906 (1199062

2minus 1199062

1) 120578119906+ (1199062

2minus 1199062

1) 120578 minus 119906

3(1199062

2minus 1199062

1)

= 0

(47)

Noting the homogeneity of the equation we find a solution120578 (119905 119909 119906 119906

1 1199062)

= minus3

8(1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2) (119886

119905

1199061

minus (1 minus 119886)119909

1199062

)

minus1

32119906 (1199062minus 1199062

1minus 1199062

2)

(48)

where 119886 is an arbitrary constant For this infinitesimalgenerator of the approximate Lie symmetry group the group-invariant solution satisfies

119883 [119906 minus 119906 (119905 119909)]|119906=119906(119905119909) = 0 (49)which reads

120597119906 (119905 119909)

120597120576= 120578 (119905 119909 119906 (119905 119909) 119906

1(119905 119909) 119906

2(119905 119909)) (50)

Asymptotic dynamic behavior can be obtained by consider-ing the cases 119905 ≫ 1 and 119909 ≫ 1 and by neglecting the termswhich are not proportional to 119909 or 119905 By introducing 120591 = 120576119905and 120585 = 120576119909 we obtain

120597119906

120597120591= minus

3 (1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2)

81199061

119886

120597119906

120597120585=

3 (1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2)

81199062

(1 minus 119886)

(51)

By eliminating 119886 the equations are reduced to

1199061

120597119906

120597120591minus 1199062

120597119906

120597120585= minus3

8(1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2) (52)

This reduced equation describes the asymptotic behavior ofthe original systemThis fact can be confirmed by comparingwith the result derived before By introducing a transforma-tion of the variables

|119860|2=1

4(1199062+ 1199062

1minus 1199062

2)

120596 = 1199061(1199062

1minus 1199062

2)minus12

119896 = 1199062(1199062

1minus 1199062

2)minus12

(53)

the reduced equation (52) is transformed into

[120597

120597120591+119896

120596

120597

120597120585]119860 = minus

3

2120596i |119860|2 119860 (54)

This is the nonlinear Schrodinger equation which corre-sponds to the reduced equation derived with the renormal-ization group method [9]

5 Concluding Remarks

In summary this paper presents an investigation of perturba-tion problems which involves the application of a particulartype of Lie group method to partial as well as ordinarydifferential equations As a consequence we have obtainedthe dynamics which describes the asymptotic behavior ofthe original system The main characteristic of the presentmethod is the addition of some constraint equations whensearching for a Lie symmetry group an approach whichallows us to find a partial Lie symmetry groupThe constraintequations are determined from the nonperturbative solutionon which we try to investigate the effect of the perturbationThe reason why it is necessary to specify a nonperturbativesolution beforehand for the partial differential equations isthat unlike for ordinary differential equations the non-perturbative system has an infinite number of independentsolutions In terms of technique those constraint equationsare constructed in such a way that the arbitrary constantsor functions included in the nonperturbative solution areeliminated as we have done in deriving (4) (18) and (41) Inthis study we have investigated the effect of the perturbationof its lowest order however extending our work to higher-order approximations would require us to use higher-orderconstraint equations instead

Examples towhich themethod is applied here are singularperturbation problems in which secular terms appear in thenaıve expansion of the solution [3 6 8 10 11 34] Termsproportional to the independent variables emerge in theinfinitesimal generator of the Lie symmetry group as seen in(10) (30) and (48) corresponding to the emergence of thesecular terms in the naıve expansion As inferred from thismathematical similarity the presentmethod is also applicableto other systems which exhibit the same type of singularperturbation problems In the singular perturbationmethodsproposed thus far it is only after the construction of the naıveexpansion that it becomes clear whether the method shouldbe used or not However the present method can be appliedwithout making the expansion enabling the appropriatereduced equations to be derived

Future work should involve a clarification of the relationsbetween the present method and other types of singularperturbation methods presented thus far such as the WKBor boundary layer problem methods In addition there areexisting methods that have enabled the successful deriva-tion of the asymptotic behavior for various other partialdifferential equations [13ndash18 20ndash30 35ndash38] Therefore itshould be clarified whether or not the present method canbe appropriately extended and applied to the problems thosemethods have resolved

Journal of Applied Mathematics 7

Appendix

Searching for Lie Symmetry withthe Normal Procedure

Let us search for the Lie symmetry group of (1) withoutconstraint equations The criterion for the invariance of theequation is given by (7) namely

119883lowast[2119906 + minus 120576]

10038161003816100381610038162119906+minus120576=0= 0 (A1)

which reads

21199062120578119905119905minus 119906 + (4119906

2120578119906119905+ 2119906120578

119905)

+ 2(21199062120578119906119906+ 119906120578 minus 120578) + 120576 (120578 + 119906120578

119906) = 0

(A2)

In the lowest order of 120576 120578 has to satisfy the followingdifferential equations

21199062120578119905119905minus 119906 = 0

41199062120578119906119905+ 2119906120578

119905= 0

21199062120578119906119906+ 119906120578 minus 120578 = 0

(A3)

However we can easily find out that there is no solution thatsatisfies all the above equations which implies that there is noLie symmetry group

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by JSPS KAKENHI Grant no26800208

References

[1] N N Bogoliubov and C M Place Asymptotic Methods in theTheory of Non-Linear Oscillations Gordon and Break 1961

[2] GHori ldquoTheory of general perturbationwith unspecified cano-nical variablerdquo Publications of the Astronomical Society of Japanvol 18 p 287 1966

[3] T Taniuchi ldquoReductive perturbation method and far fields ofwave equationsrdquo Progress ofTheoretical Physics Supplement vol55 pp 1ndash35 1974

[4] J Guckenheimer andPHolmesNonlinearOscillationsDynam-ical systems and Bifurcations of Vector Fields Springer NewYork NY USA 1983

[5] A H Nayfeh Method of Normal Forms John Wiley amp SonsNew York NY USA 1993

[6] T Kunihiro ldquoA geometrical formulation of the renormalizationgroup method for global analysisrdquo Progress of TheoreticalPhysics vol 94 no 4 pp 503ndash514 1995

[7] L-Y ChenNGoldenfeld YOono andG Paquette ldquoSelectionstability and renormalizationrdquo Physica A Statistical Mechanicsand its Applications vol 204 no 1ndash4 pp 111ndash133 1994

[8] L Y Chen N Goldenfeld and Y Oono ldquoRenormalizationgroup and singular perturbations multiple scales boundarylayers and reductive perturbation theoryrdquo Physical Review Evol 54 no 1 pp 376ndash394 1996

[9] S-I Goto Y Masutomi and K Nozaki ldquoLie-group approachto perturbative renormalization group methodrdquo Progress ofTheoretical Physics vol 102 no 3 pp 471ndash497 1999

[10] R E L DeVille A Harkin M Holzer K Josic and T J KaperldquoAnalysis of a renormalization group method and normal formtheory for perturbed ordinary differential equationsrdquoPhysicaDvol 237 no 8 pp 1029ndash1052 2008

[11] H Chiba ldquo1198621 approximation of vector fields based on therenormalization group methodrdquo SIAM Journal on AppliedDynamical Systems vol 7 no 3 pp 895ndash932 2008

[12] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1986

[13] V A Baıkov R K Gazizov and N H Ibragimov ldquoApproximatesymmetriesrdquoMathematics of the USSR-Sbornik vol 64 no 2 p427 1989

[14] W I Fushchich andWM Shtelen ldquoOn approximate symmetryand approximate solutions of the nonlinear wave equation witha small parameterrdquo Journal of Physics A Mathematical andGeneral vol 22 no 18 pp L887ndashL890 1989

[15] N Eular MW Shulga andW-H Steeb ldquoApproximate symme-tries and approximate solutions for amultidimensional Landau-Ginzburg equationrdquo Journal of Physics A Mathematical andGeneral vol 25 no 18 Article ID L1095 1992

[16] Y Y Bagderina and R K Gazizov ldquoEquivalence of ordinarydifferential equations 11991010158401015840 = 119877(119909 119910)11991010158402 + 2119876(119909 119910)1199101015840 + 119875(119909 119910)rdquoDifferential Equations vol 43 no 5 pp 595ndash604 2007

[17] V N Grebenev andM Oberlack ldquoApproximate Lie symmetriesof theNavier-Stokes equationsrdquo Journal ofNonlinearMathemat-ical Physics vol 14 no 2 pp 157ndash163 2007

[18] R K Gazizov and N K Ibragimov ldquoApproximate symmetriesand solutions of the Kompaneets equationrdquo Journal of AppliedMechanics and Technical Physics vol 55 no 2 pp 220ndash2242014

[19] R K Gazizov and C M Khalique ldquoApproximate transforma-tions for van der Pol-type equationsrdquoMathematical Problems inEngineering vol 2006 Article ID 68753 11 pages 2006

[20] G Gaeta ldquoAsymptotic symmetries and asymptotically symmet-ric solutions of partial differential equationsrdquo Journal of PhysicsA Mathematical and General vol 27 no 2 pp 437ndash451 1994

[21] G Gaeta ldquoAsymptotic symmetries in an optical latticerdquo PhysicalReview A vol 72 no 3 Article ID 033419 2005

[22] G Gaeta D Levi and R Mancinelli ldquoAsymptotic symmetriesof difference equations on a latticerdquo Journal of NonlinearMathe-matical Physics vol 12 supplement 2 pp 137ndash146 2005

[23] G Gaeta and RMancinelli ldquoAsymptotic scaling symmetries fornonlinear PDEsrdquo International Journal of Geometric Methods inModern Physics vol 2 no 6 pp 1081ndash1114 2005

[24] G Gaeta and R Mancinelli ldquoAsymptotic scaling in a modelclass of anomalous reaction-diffusion equationsrdquo Journal ofNonlinear Mathematical Physics vol 12 no 4 pp 550ndash5662005

[25] D Levi and M A Rodrıguez ldquoAsymptotic symmetries andintegrability the KdV caserdquo Europhysics Letters vol 80 no 6Article ID 60005 2007

[26] V F Kovalev V V Pustovalov and D V Shirkov ldquoGroup anal-ysis and renormgroup symmetriesrdquo Journal of MathematicalPhysics vol 39 no 2 pp 1170ndash1188 1998

8 Journal of Applied Mathematics

[27] V F Kovalev ldquoRenormalization group analysis for singularitiesin the wave beam self-focusing problemrdquo Theoretical andMathematical Physics vol 119 no 3 pp 719ndash730 1999

[28] V F Kovalev and D V Shirkov ldquoFunctional self-similarityand renormalization group symmetry inmathematical physicsrdquoTheoretical and Mathematical Physics vol 121 no 1 pp 1315ndash1332 1999

[29] D V Shirkov and V F Kovalev ldquoThe Bogoliubov renormaliza-tion group and solution symmetry in mathematical physicsrdquoPhysics Report vol 352 no 4ndash6 pp 219ndash249 2001

[30] V F Kovalev and D V Shirkov ldquoThe renormalization groupsymmetry for solution of integral equationsrdquo Proceedings of theInstitute of Mathematics of NAS of Ukraine vol 50 no 2 pp850ndash861 2004

[31] M Iwasa and K Nozaki ldquoA method to construct asymptoticsolutions invariant under the renormalization grouprdquo Progressof Theoretical Physics vol 116 no 4 pp 605ndash613 2006

[32] M Iwasa and K Nozaki ldquoRenormalization group in differencesystemsrdquo Journal of Physics A Mathematical and Theoreticalvol 41 no 8 Article ID 085204 2008

[33] H Chiba andM Iwasa ldquoLie equations for asymptotic solutionsof perturbation problems of ordinary differential equationsrdquoJournal of Mathematical Physics vol 50 no 4 2009

[34] Y Y Yamaguchi and Y Nambu ldquoRenormalization group equa-tions and integrability in hamiltonian systemsrdquo Progress ofTheoretical Physics vol 100 no 1 pp 199ndash204 1998

[35] N D Goldenfeld Lectures on Phase Transitions and the Renor-malization Group Addison-Wesley 1992

[36] J Bricmont A Kupiainen and G Lin ldquoRenormalization groupand asymptotics of solutions of nonlinear parabolic equationsrdquoCommunications on Pure and Applied Mathematics vol 47 no6 pp 893ndash922 1994

[37] N V Antonov and J Honkonen ldquoField-theoretic renormaliza-tion group for a nonlinear diffusion equationrdquo Physical ReviewE vol 66 no 4 Article ID 046105 7 pages 2002

[38] S Yoshida andT Fukui ldquoExact renormalization group approachto a nonlinear diffusion equationrdquo Physical Review E vol 72 no4 Article ID 046136 2005

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Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

are substituted into (22) which is then rewritten as

[120578119906119905

0+ 1198600120578119906119909

0+ (119906 sdot nabla)119860

0119906119909]10038161003816100381610038161198600119906119909=120582119906119909 119906119905+120582119906119909=0= 0 (25)

[

[

120578119906119905

119894+ 1198600120578119906119909

119894+ (119906 sdot nabla)119860

119894119906119909

+

119894minus1

sum

119895=0

(120578119894minus119895minus1sdot nabla)119860

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119906=0

119906119909]

]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161198600119906119909=120582119906119909 119906119905+120582119906119909=0

= 0

(119894 = 1 2 )

(26)

We consider only the lowest order of 120576 namely (25) which isrewritten as

(120597119905+ 119906119905120597119906) 1205780+ 1198600(120597119909+ 119906119909120597119906) 1205780

minus [(119906 sdot nabla)1198600] 119906119909

10038161003816100381610038161198600119906119909=120582119906119909 119906119905+120582119906119909=0= 0

(27)

and reads

(120597119905+ 1198600120597119909) 1205780= [minus (119906 sdot nabla) 120582] 119906119909 (28)

The vector on the right-hand side of this equation is separatedinto the following two components

(120597119905+ 1198600120597119909) 1205780= minus (119890

sdot [(119906 sdot nabla) 120582] 119906119909) 119890

minus (119890perpsdot [(119906 sdot nabla) 120582] 119906119909) 119890perp

(29)

where 119890denotes the unit vector with the same direction

as 1199060(119905 119909) in R119899 and 119890

perpis the unit vector parallel to the

projection of [minus(119906 sdot nabla)120582]119906119909on a 119899 minus 1 dimensional subspace

in R119899 spanned by all the eigenvectors of 1198600except 119890

The

solution corresponding to the second term is given by a linearcombination of the 119899 minus 1 eigenvectors of 119860

0whose linear

coefficients are a function of 119909 minus 120582119905 On the other hand aterm proportional to 119905 appears in the solution correspondingto the first term because every vector proportional to 119890

whose proportional coefficient is a function of 119909minus120582119905 is a zeroeigenvector of the linear operator 120597

119905+ 1198600120597119909 For this reason

we can write a solution of (29) as follows

1205780= minus119905 (119890

sdot [(119906 sdot nabla) 120582] 119906119909) 119890

+ (terms not proportional to 119905) (30)

The group-invariant solution 119906 = 119906(119905 119909) for thisapproximate Lie symmetry group satisfies

119883 [119906 minus 119906 (119905 119909)]|119906=119906(119905119909) = 0 (31)

which reads120597119906 (119905 119909)

120597120576= minus119905 (119890

sdot [(119906 sdot nabla)119860

0] 119906119909) 119890

+ (terms not proportional to 119905) (32)

The long-time behavior of the solution is obtained by consid-ering the case in which 119905 ≫ 1 and by neglecting the termsother than the first one Then (32) is reduced to

119889

119889120591119906 (119905 119909) = minus (119890

sdot [(119906 sdot nabla)119860

0] 119906119909) 119890 (33)

where 120591 is a long time-scale defined by 120591 = 120576119905 As aresult we have obtained a dynamical system which describesasymptotic behavior of the original system This reduceddynamics corresponds to that presented before with thereductive perturbation method [3] or the renormalizationgroup method [7]

42 Second Example 119906119905119905minus 119906119909119909+ 119906 minus 120576119906

3= 0 We consider a

perturbed Klein-Gordon equation represented by

119906119905119905minus 119906119909119909+ 119906 minus 120576119906

3= 0 (34)

Here 119906 is a scalar function which depends on two indepen-dent variables 119905 and 119909 and 120576 is a perturbation parameterThe subscripts of 119906 denote the derivative with respect to theindependent variables By introducing new dependent vari-ables as

1199061= 119906119905

1199062= 119906119909

(35)

we rewrite (34) in terms of four first-order partial differentialequations for three dependent variables 119906 119906

1 and 119906

2as

follows

1199061119905minus 1199062119909+ 119906 minus 120576119906

3= 0

119906119905minus 1199061= 0

119906119909minus 1199062= 0

1199061119909minus 1199062119905= 0

(36)

The last equation implies the integrable conditionAs in the previous examples a specific nonperturbative

solution 119906(0)(119905 119909) is prespecified on which we investigate theeffect of the perturbationHerewe focus on a nonperturbativesolution describing a traveling wave with a constant velocityV The solution is represented by

119906(0)(119905 119909) = 119880 (119909 minus V119905) (37)

where 119880(120585) = 120572119890i120581120585 + 120572119890minusi120581120585 for 120581 = (V2 minus 1)minus12 and anarbitrary constant 120572 For this solution

119906(0)

119905= minusV119906(0)

119909

119906(0)

119905119905= minusV119906(0)

119905119909= minusV119906(0)

119909119905= V2119906(0)

119909119909

(38)

therefore

119906(0)

119905119905=119906(0)

119905

119906(0)

119909

119906(0)

119909119905=119906(0)

119905

119906(0)

119909

119906(0)

119905119909= (119906(0)

119905

119906(0)

119909

)

2

119906(0)

119909119909(39)

Journal of Applied Mathematics 5

are satisfied By combining these relations with the nonper-turbative system 119906(0)

119905119905minus 119906(0)

119909119909+ 119906(0)= 0 we obtain

119906(0)

119905119905=

[119906(0)

119905]2

119906(0)

[119906(0)

119909 ]2

minus [119906(0)

119905]2

119906(0)

119909119905= 119906(0)

119905119909=119906(0)

119905119906(0)

119909119906(0)

[119906(0)

119909 ]2

minus [119906(0)

119905]2

119906(0)

119909119909=

[119906(0)

119909]2

119906(0)

[119906(0)

119909 ]2

minus [119906(0)

119905]2

(40)

From these relations we construct the constraint equations

1199061119905=1199062

1119906

1199062

2minus 1199062

1

1199062119905= 1199061119909=11990611199062119906

1199062

2minus 1199062

1

1199062119909=1199062

2119906

1199062

2minus 1199062

1

(41)

which are used in the finding of Lie symmetry groupLet

119883(119905 119909 119906 1199061 1199062 120576) = 120597

120576+ 120578 (119905 119909 119906 119906

1 1199062 120576) 120597119906

+ 1205781199061(119905 119909 119906 119906

1 1199062 120576) 1205971199061

+ 1205781199062(119905 119909 119906 119906

1 1199062 120576) 1205971199062

(42)

be the infinitesimal generator of a Lie symmetry group of thefour partial differential equations (36) Its first prolongation119883lowast is given by

119883lowast(119905 119909 119906 119906

1 1199062 120576)

= 119883 + 120578119905(119905 119909 119906 119906

1 1199062 11990611 11990612 11990622 120576) 120597119906119905

+ 120578119909(119905 119909 119906 119906

1 1199062 11990611 11990612 11990622 120576) 120597119906119909

+ 1205781199061119905(119905 119909 119906 119906

1 1199062 11990611 11990612 11990622 120576) 1205971199061119905

+ 1205781199062119909(119905 119909 119906 119906

1 1199062 11990611 11990612 11990622 120576) 1205971199062119909

(43)

where 120578119905 120578119909 1205781199061119905 and 1205781199062119909 are defined as follows

120578119905(119905 119909 119906 119906

1 1199062 119906119905 1199061119905 1199062119905 120576)

= 119863119905120578 (119905 119909 119906 119906

1 1199062 120576)

120578119909(119905 119909 119906 119906

1 1199062 119906119909 1199061119909 1199062119909 120576)

= 119863119909120578 (119905 119909 119906 119906

1 1199062 120576)

1205781199061119905(119905 119909 119906 119906

1 1199062 119906119905 1199061119905 1199062119905 120576)

= 1198631199051205781199061(119905 119909 119906 119906

1 1199062 120576)

1205781199062119909(119905 119909 119906 119906

1 1199062 119906119909 1199061119909 1199062119909 120576)

= 1198631199091205781199062(119905 119909 119906 119906

1 1199062 120576)

119863119905= 120597119905+ 1199061120597119906+ 11990611199051205971199061

+ 11990621199051205971199062

119863119909= 120597119909+ 119906119909120597119906+ 11990611199091205971199061

+ 11990621199091205971199062

(44)

The criteria for the invariance of the original system onthe manifold formed by the constraint equations (41) aregiven by

119883lowast[1199061119905minus 1199062119909+ 119906 minus 120576119906

3]10038161003816100381610038161003816(41)= 0

119883lowast[119906119905minus 1199061]1003816100381610038161003816(41)= 0

119883lowast[119906119909minus 1199062]1003816100381610038161003816(41)= 0

119883lowast[1199061119909minus 1199062119905]1003816100381610038161003816(41)= 0

(45)

With the identities

1205971199061199062

1119906

1199062

2minus 1199062

1

=1199062

1

1199062

2minus 1199062

1

1205971199061

1199062

1119906

1199062

2minus 1199062

1

=211990611199062

2119906

(1199062

2minus 1199062

1)2

1205971199062

1199062

1119906

1199062

2minus 1199062

1

=minus21199062

11199062119906

(1199062

2minus 1199062

1)2

12059711990611990611199062119906

1199062

2minus 1199062

1

=11990611199062

1199062

2minus 1199062

1

1205971199061

11990611199062119906

1199062

2minus 1199062

1

=

(1199062

1+ 1199062

2) 1199062119906

(1199062

2minus 1199062

1)2

1205971199062

11990611199062119906

1199062

2minus 1199062

1

=

minus (1199062

1+ 1199062

2) 1199061119906

(1199062

2minus 1199062

1)2

1205971199061199062

2119906

1199062

2minus 1199062

1

=1199062

2

1199062

2minus 1199062

1

1205971199061

1199062

2119906

1199062

2minus 1199062

1

=211990611199062

2119906

(1199062

2minus 1199062

1)2

1205971199062

1199062

2119906

1199062

2minus 1199062

1

=minus21199062

11199062119906

(1199062

2minus 1199062

1)2

(46)

6 Journal of Applied Mathematics

(45) read in the lowest order of 120576

(1199062

2minus 1199062

1) (120578119905119905minus 120578119909119909) minus (119906

2

2minus 1199062

1)2

120578119906119906minus 1199062

1119906212057811990611199061

minus 1199062

1119906212057811990621199062

+ 21199061119906 (1199062

2minus 1199062

1) 1205781199061199061

minus 21199062119906 (1199062

2minus 1199062

1) 1205781199061199062

minus 211990611199062119906212057811990611199062

+ 21199061(1199062

2minus 1199062

1) 120578119905119906+ 21199062

11199061205781199051199061

+ 2119906111990621199061205781199051199062

minus 21199062(1199062

2minus 1199062

1) 120578119909119906minus 2119906111990621199061205781199091199061

minus 21199062

21199061205781199091199062

minus 1199061(1199062

2minus 1199062

1) 1205781199061

minus 1199062(1199062

2minus 1199062

1) 1205781199062

minus 119906 (1199062

2minus 1199062

1) 120578119906+ (1199062

2minus 1199062

1) 120578 minus 119906

3(1199062

2minus 1199062

1)

= 0

(47)

Noting the homogeneity of the equation we find a solution120578 (119905 119909 119906 119906

1 1199062)

= minus3

8(1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2) (119886

119905

1199061

minus (1 minus 119886)119909

1199062

)

minus1

32119906 (1199062minus 1199062

1minus 1199062

2)

(48)

where 119886 is an arbitrary constant For this infinitesimalgenerator of the approximate Lie symmetry group the group-invariant solution satisfies

119883 [119906 minus 119906 (119905 119909)]|119906=119906(119905119909) = 0 (49)which reads

120597119906 (119905 119909)

120597120576= 120578 (119905 119909 119906 (119905 119909) 119906

1(119905 119909) 119906

2(119905 119909)) (50)

Asymptotic dynamic behavior can be obtained by consider-ing the cases 119905 ≫ 1 and 119909 ≫ 1 and by neglecting the termswhich are not proportional to 119909 or 119905 By introducing 120591 = 120576119905and 120585 = 120576119909 we obtain

120597119906

120597120591= minus

3 (1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2)

81199061

119886

120597119906

120597120585=

3 (1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2)

81199062

(1 minus 119886)

(51)

By eliminating 119886 the equations are reduced to

1199061

120597119906

120597120591minus 1199062

120597119906

120597120585= minus3

8(1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2) (52)

This reduced equation describes the asymptotic behavior ofthe original systemThis fact can be confirmed by comparingwith the result derived before By introducing a transforma-tion of the variables

|119860|2=1

4(1199062+ 1199062

1minus 1199062

2)

120596 = 1199061(1199062

1minus 1199062

2)minus12

119896 = 1199062(1199062

1minus 1199062

2)minus12

(53)

the reduced equation (52) is transformed into

[120597

120597120591+119896

120596

120597

120597120585]119860 = minus

3

2120596i |119860|2 119860 (54)

This is the nonlinear Schrodinger equation which corre-sponds to the reduced equation derived with the renormal-ization group method [9]

5 Concluding Remarks

In summary this paper presents an investigation of perturba-tion problems which involves the application of a particulartype of Lie group method to partial as well as ordinarydifferential equations As a consequence we have obtainedthe dynamics which describes the asymptotic behavior ofthe original system The main characteristic of the presentmethod is the addition of some constraint equations whensearching for a Lie symmetry group an approach whichallows us to find a partial Lie symmetry groupThe constraintequations are determined from the nonperturbative solutionon which we try to investigate the effect of the perturbationThe reason why it is necessary to specify a nonperturbativesolution beforehand for the partial differential equations isthat unlike for ordinary differential equations the non-perturbative system has an infinite number of independentsolutions In terms of technique those constraint equationsare constructed in such a way that the arbitrary constantsor functions included in the nonperturbative solution areeliminated as we have done in deriving (4) (18) and (41) Inthis study we have investigated the effect of the perturbationof its lowest order however extending our work to higher-order approximations would require us to use higher-orderconstraint equations instead

Examples towhich themethod is applied here are singularperturbation problems in which secular terms appear in thenaıve expansion of the solution [3 6 8 10 11 34] Termsproportional to the independent variables emerge in theinfinitesimal generator of the Lie symmetry group as seen in(10) (30) and (48) corresponding to the emergence of thesecular terms in the naıve expansion As inferred from thismathematical similarity the presentmethod is also applicableto other systems which exhibit the same type of singularperturbation problems In the singular perturbationmethodsproposed thus far it is only after the construction of the naıveexpansion that it becomes clear whether the method shouldbe used or not However the present method can be appliedwithout making the expansion enabling the appropriatereduced equations to be derived

Future work should involve a clarification of the relationsbetween the present method and other types of singularperturbation methods presented thus far such as the WKBor boundary layer problem methods In addition there areexisting methods that have enabled the successful deriva-tion of the asymptotic behavior for various other partialdifferential equations [13ndash18 20ndash30 35ndash38] Therefore itshould be clarified whether or not the present method canbe appropriately extended and applied to the problems thosemethods have resolved

Journal of Applied Mathematics 7

Appendix

Searching for Lie Symmetry withthe Normal Procedure

Let us search for the Lie symmetry group of (1) withoutconstraint equations The criterion for the invariance of theequation is given by (7) namely

119883lowast[2119906 + minus 120576]

10038161003816100381610038162119906+minus120576=0= 0 (A1)

which reads

21199062120578119905119905minus 119906 + (4119906

2120578119906119905+ 2119906120578

119905)

+ 2(21199062120578119906119906+ 119906120578 minus 120578) + 120576 (120578 + 119906120578

119906) = 0

(A2)

In the lowest order of 120576 120578 has to satisfy the followingdifferential equations

21199062120578119905119905minus 119906 = 0

41199062120578119906119905+ 2119906120578

119905= 0

21199062120578119906119906+ 119906120578 minus 120578 = 0

(A3)

However we can easily find out that there is no solution thatsatisfies all the above equations which implies that there is noLie symmetry group

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by JSPS KAKENHI Grant no26800208

References

[1] N N Bogoliubov and C M Place Asymptotic Methods in theTheory of Non-Linear Oscillations Gordon and Break 1961

[2] GHori ldquoTheory of general perturbationwith unspecified cano-nical variablerdquo Publications of the Astronomical Society of Japanvol 18 p 287 1966

[3] T Taniuchi ldquoReductive perturbation method and far fields ofwave equationsrdquo Progress ofTheoretical Physics Supplement vol55 pp 1ndash35 1974

[4] J Guckenheimer andPHolmesNonlinearOscillationsDynam-ical systems and Bifurcations of Vector Fields Springer NewYork NY USA 1983

[5] A H Nayfeh Method of Normal Forms John Wiley amp SonsNew York NY USA 1993

[6] T Kunihiro ldquoA geometrical formulation of the renormalizationgroup method for global analysisrdquo Progress of TheoreticalPhysics vol 94 no 4 pp 503ndash514 1995

[7] L-Y ChenNGoldenfeld YOono andG Paquette ldquoSelectionstability and renormalizationrdquo Physica A Statistical Mechanicsand its Applications vol 204 no 1ndash4 pp 111ndash133 1994

[8] L Y Chen N Goldenfeld and Y Oono ldquoRenormalizationgroup and singular perturbations multiple scales boundarylayers and reductive perturbation theoryrdquo Physical Review Evol 54 no 1 pp 376ndash394 1996

[9] S-I Goto Y Masutomi and K Nozaki ldquoLie-group approachto perturbative renormalization group methodrdquo Progress ofTheoretical Physics vol 102 no 3 pp 471ndash497 1999

[10] R E L DeVille A Harkin M Holzer K Josic and T J KaperldquoAnalysis of a renormalization group method and normal formtheory for perturbed ordinary differential equationsrdquoPhysicaDvol 237 no 8 pp 1029ndash1052 2008

[11] H Chiba ldquo1198621 approximation of vector fields based on therenormalization group methodrdquo SIAM Journal on AppliedDynamical Systems vol 7 no 3 pp 895ndash932 2008

[12] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1986

[13] V A Baıkov R K Gazizov and N H Ibragimov ldquoApproximatesymmetriesrdquoMathematics of the USSR-Sbornik vol 64 no 2 p427 1989

[14] W I Fushchich andWM Shtelen ldquoOn approximate symmetryand approximate solutions of the nonlinear wave equation witha small parameterrdquo Journal of Physics A Mathematical andGeneral vol 22 no 18 pp L887ndashL890 1989

[15] N Eular MW Shulga andW-H Steeb ldquoApproximate symme-tries and approximate solutions for amultidimensional Landau-Ginzburg equationrdquo Journal of Physics A Mathematical andGeneral vol 25 no 18 Article ID L1095 1992

[16] Y Y Bagderina and R K Gazizov ldquoEquivalence of ordinarydifferential equations 11991010158401015840 = 119877(119909 119910)11991010158402 + 2119876(119909 119910)1199101015840 + 119875(119909 119910)rdquoDifferential Equations vol 43 no 5 pp 595ndash604 2007

[17] V N Grebenev andM Oberlack ldquoApproximate Lie symmetriesof theNavier-Stokes equationsrdquo Journal ofNonlinearMathemat-ical Physics vol 14 no 2 pp 157ndash163 2007

[18] R K Gazizov and N K Ibragimov ldquoApproximate symmetriesand solutions of the Kompaneets equationrdquo Journal of AppliedMechanics and Technical Physics vol 55 no 2 pp 220ndash2242014

[19] R K Gazizov and C M Khalique ldquoApproximate transforma-tions for van der Pol-type equationsrdquoMathematical Problems inEngineering vol 2006 Article ID 68753 11 pages 2006

[20] G Gaeta ldquoAsymptotic symmetries and asymptotically symmet-ric solutions of partial differential equationsrdquo Journal of PhysicsA Mathematical and General vol 27 no 2 pp 437ndash451 1994

[21] G Gaeta ldquoAsymptotic symmetries in an optical latticerdquo PhysicalReview A vol 72 no 3 Article ID 033419 2005

[22] G Gaeta D Levi and R Mancinelli ldquoAsymptotic symmetriesof difference equations on a latticerdquo Journal of NonlinearMathe-matical Physics vol 12 supplement 2 pp 137ndash146 2005

[23] G Gaeta and RMancinelli ldquoAsymptotic scaling symmetries fornonlinear PDEsrdquo International Journal of Geometric Methods inModern Physics vol 2 no 6 pp 1081ndash1114 2005

[24] G Gaeta and R Mancinelli ldquoAsymptotic scaling in a modelclass of anomalous reaction-diffusion equationsrdquo Journal ofNonlinear Mathematical Physics vol 12 no 4 pp 550ndash5662005

[25] D Levi and M A Rodrıguez ldquoAsymptotic symmetries andintegrability the KdV caserdquo Europhysics Letters vol 80 no 6Article ID 60005 2007

[26] V F Kovalev V V Pustovalov and D V Shirkov ldquoGroup anal-ysis and renormgroup symmetriesrdquo Journal of MathematicalPhysics vol 39 no 2 pp 1170ndash1188 1998

8 Journal of Applied Mathematics

[27] V F Kovalev ldquoRenormalization group analysis for singularitiesin the wave beam self-focusing problemrdquo Theoretical andMathematical Physics vol 119 no 3 pp 719ndash730 1999

[28] V F Kovalev and D V Shirkov ldquoFunctional self-similarityand renormalization group symmetry inmathematical physicsrdquoTheoretical and Mathematical Physics vol 121 no 1 pp 1315ndash1332 1999

[29] D V Shirkov and V F Kovalev ldquoThe Bogoliubov renormaliza-tion group and solution symmetry in mathematical physicsrdquoPhysics Report vol 352 no 4ndash6 pp 219ndash249 2001

[30] V F Kovalev and D V Shirkov ldquoThe renormalization groupsymmetry for solution of integral equationsrdquo Proceedings of theInstitute of Mathematics of NAS of Ukraine vol 50 no 2 pp850ndash861 2004

[31] M Iwasa and K Nozaki ldquoA method to construct asymptoticsolutions invariant under the renormalization grouprdquo Progressof Theoretical Physics vol 116 no 4 pp 605ndash613 2006

[32] M Iwasa and K Nozaki ldquoRenormalization group in differencesystemsrdquo Journal of Physics A Mathematical and Theoreticalvol 41 no 8 Article ID 085204 2008

[33] H Chiba andM Iwasa ldquoLie equations for asymptotic solutionsof perturbation problems of ordinary differential equationsrdquoJournal of Mathematical Physics vol 50 no 4 2009

[34] Y Y Yamaguchi and Y Nambu ldquoRenormalization group equa-tions and integrability in hamiltonian systemsrdquo Progress ofTheoretical Physics vol 100 no 1 pp 199ndash204 1998

[35] N D Goldenfeld Lectures on Phase Transitions and the Renor-malization Group Addison-Wesley 1992

[36] J Bricmont A Kupiainen and G Lin ldquoRenormalization groupand asymptotics of solutions of nonlinear parabolic equationsrdquoCommunications on Pure and Applied Mathematics vol 47 no6 pp 893ndash922 1994

[37] N V Antonov and J Honkonen ldquoField-theoretic renormaliza-tion group for a nonlinear diffusion equationrdquo Physical ReviewE vol 66 no 4 Article ID 046105 7 pages 2002

[38] S Yoshida andT Fukui ldquoExact renormalization group approachto a nonlinear diffusion equationrdquo Physical Review E vol 72 no4 Article ID 046136 2005

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

Journal of Applied Mathematics 5

are satisfied By combining these relations with the nonper-turbative system 119906(0)

119905119905minus 119906(0)

119909119909+ 119906(0)= 0 we obtain

119906(0)

119905119905=

[119906(0)

119905]2

119906(0)

[119906(0)

119909 ]2

minus [119906(0)

119905]2

119906(0)

119909119905= 119906(0)

119905119909=119906(0)

119905119906(0)

119909119906(0)

[119906(0)

119909 ]2

minus [119906(0)

119905]2

119906(0)

119909119909=

[119906(0)

119909]2

119906(0)

[119906(0)

119909 ]2

minus [119906(0)

119905]2

(40)

From these relations we construct the constraint equations

1199061119905=1199062

1119906

1199062

2minus 1199062

1

1199062119905= 1199061119909=11990611199062119906

1199062

2minus 1199062

1

1199062119909=1199062

2119906

1199062

2minus 1199062

1

(41)

which are used in the finding of Lie symmetry groupLet

119883(119905 119909 119906 1199061 1199062 120576) = 120597

120576+ 120578 (119905 119909 119906 119906

1 1199062 120576) 120597119906

+ 1205781199061(119905 119909 119906 119906

1 1199062 120576) 1205971199061

+ 1205781199062(119905 119909 119906 119906

1 1199062 120576) 1205971199062

(42)

be the infinitesimal generator of a Lie symmetry group of thefour partial differential equations (36) Its first prolongation119883lowast is given by

119883lowast(119905 119909 119906 119906

1 1199062 120576)

= 119883 + 120578119905(119905 119909 119906 119906

1 1199062 11990611 11990612 11990622 120576) 120597119906119905

+ 120578119909(119905 119909 119906 119906

1 1199062 11990611 11990612 11990622 120576) 120597119906119909

+ 1205781199061119905(119905 119909 119906 119906

1 1199062 11990611 11990612 11990622 120576) 1205971199061119905

+ 1205781199062119909(119905 119909 119906 119906

1 1199062 11990611 11990612 11990622 120576) 1205971199062119909

(43)

where 120578119905 120578119909 1205781199061119905 and 1205781199062119909 are defined as follows

120578119905(119905 119909 119906 119906

1 1199062 119906119905 1199061119905 1199062119905 120576)

= 119863119905120578 (119905 119909 119906 119906

1 1199062 120576)

120578119909(119905 119909 119906 119906

1 1199062 119906119909 1199061119909 1199062119909 120576)

= 119863119909120578 (119905 119909 119906 119906

1 1199062 120576)

1205781199061119905(119905 119909 119906 119906

1 1199062 119906119905 1199061119905 1199062119905 120576)

= 1198631199051205781199061(119905 119909 119906 119906

1 1199062 120576)

1205781199062119909(119905 119909 119906 119906

1 1199062 119906119909 1199061119909 1199062119909 120576)

= 1198631199091205781199062(119905 119909 119906 119906

1 1199062 120576)

119863119905= 120597119905+ 1199061120597119906+ 11990611199051205971199061

+ 11990621199051205971199062

119863119909= 120597119909+ 119906119909120597119906+ 11990611199091205971199061

+ 11990621199091205971199062

(44)

The criteria for the invariance of the original system onthe manifold formed by the constraint equations (41) aregiven by

119883lowast[1199061119905minus 1199062119909+ 119906 minus 120576119906

3]10038161003816100381610038161003816(41)= 0

119883lowast[119906119905minus 1199061]1003816100381610038161003816(41)= 0

119883lowast[119906119909minus 1199062]1003816100381610038161003816(41)= 0

119883lowast[1199061119909minus 1199062119905]1003816100381610038161003816(41)= 0

(45)

With the identities

1205971199061199062

1119906

1199062

2minus 1199062

1

=1199062

1

1199062

2minus 1199062

1

1205971199061

1199062

1119906

1199062

2minus 1199062

1

=211990611199062

2119906

(1199062

2minus 1199062

1)2

1205971199062

1199062

1119906

1199062

2minus 1199062

1

=minus21199062

11199062119906

(1199062

2minus 1199062

1)2

12059711990611990611199062119906

1199062

2minus 1199062

1

=11990611199062

1199062

2minus 1199062

1

1205971199061

11990611199062119906

1199062

2minus 1199062

1

=

(1199062

1+ 1199062

2) 1199062119906

(1199062

2minus 1199062

1)2

1205971199062

11990611199062119906

1199062

2minus 1199062

1

=

minus (1199062

1+ 1199062

2) 1199061119906

(1199062

2minus 1199062

1)2

1205971199061199062

2119906

1199062

2minus 1199062

1

=1199062

2

1199062

2minus 1199062

1

1205971199061

1199062

2119906

1199062

2minus 1199062

1

=211990611199062

2119906

(1199062

2minus 1199062

1)2

1205971199062

1199062

2119906

1199062

2minus 1199062

1

=minus21199062

11199062119906

(1199062

2minus 1199062

1)2

(46)

6 Journal of Applied Mathematics

(45) read in the lowest order of 120576

(1199062

2minus 1199062

1) (120578119905119905minus 120578119909119909) minus (119906

2

2minus 1199062

1)2

120578119906119906minus 1199062

1119906212057811990611199061

minus 1199062

1119906212057811990621199062

+ 21199061119906 (1199062

2minus 1199062

1) 1205781199061199061

minus 21199062119906 (1199062

2minus 1199062

1) 1205781199061199062

minus 211990611199062119906212057811990611199062

+ 21199061(1199062

2minus 1199062

1) 120578119905119906+ 21199062

11199061205781199051199061

+ 2119906111990621199061205781199051199062

minus 21199062(1199062

2minus 1199062

1) 120578119909119906minus 2119906111990621199061205781199091199061

minus 21199062

21199061205781199091199062

minus 1199061(1199062

2minus 1199062

1) 1205781199061

minus 1199062(1199062

2minus 1199062

1) 1205781199062

minus 119906 (1199062

2minus 1199062

1) 120578119906+ (1199062

2minus 1199062

1) 120578 minus 119906

3(1199062

2minus 1199062

1)

= 0

(47)

Noting the homogeneity of the equation we find a solution120578 (119905 119909 119906 119906

1 1199062)

= minus3

8(1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2) (119886

119905

1199061

minus (1 minus 119886)119909

1199062

)

minus1

32119906 (1199062minus 1199062

1minus 1199062

2)

(48)

where 119886 is an arbitrary constant For this infinitesimalgenerator of the approximate Lie symmetry group the group-invariant solution satisfies

119883 [119906 minus 119906 (119905 119909)]|119906=119906(119905119909) = 0 (49)which reads

120597119906 (119905 119909)

120597120576= 120578 (119905 119909 119906 (119905 119909) 119906

1(119905 119909) 119906

2(119905 119909)) (50)

Asymptotic dynamic behavior can be obtained by consider-ing the cases 119905 ≫ 1 and 119909 ≫ 1 and by neglecting the termswhich are not proportional to 119909 or 119905 By introducing 120591 = 120576119905and 120585 = 120576119909 we obtain

120597119906

120597120591= minus

3 (1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2)

81199061

119886

120597119906

120597120585=

3 (1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2)

81199062

(1 minus 119886)

(51)

By eliminating 119886 the equations are reduced to

1199061

120597119906

120597120591minus 1199062

120597119906

120597120585= minus3

8(1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2) (52)

This reduced equation describes the asymptotic behavior ofthe original systemThis fact can be confirmed by comparingwith the result derived before By introducing a transforma-tion of the variables

|119860|2=1

4(1199062+ 1199062

1minus 1199062

2)

120596 = 1199061(1199062

1minus 1199062

2)minus12

119896 = 1199062(1199062

1minus 1199062

2)minus12

(53)

the reduced equation (52) is transformed into

[120597

120597120591+119896

120596

120597

120597120585]119860 = minus

3

2120596i |119860|2 119860 (54)

This is the nonlinear Schrodinger equation which corre-sponds to the reduced equation derived with the renormal-ization group method [9]

5 Concluding Remarks

In summary this paper presents an investigation of perturba-tion problems which involves the application of a particulartype of Lie group method to partial as well as ordinarydifferential equations As a consequence we have obtainedthe dynamics which describes the asymptotic behavior ofthe original system The main characteristic of the presentmethod is the addition of some constraint equations whensearching for a Lie symmetry group an approach whichallows us to find a partial Lie symmetry groupThe constraintequations are determined from the nonperturbative solutionon which we try to investigate the effect of the perturbationThe reason why it is necessary to specify a nonperturbativesolution beforehand for the partial differential equations isthat unlike for ordinary differential equations the non-perturbative system has an infinite number of independentsolutions In terms of technique those constraint equationsare constructed in such a way that the arbitrary constantsor functions included in the nonperturbative solution areeliminated as we have done in deriving (4) (18) and (41) Inthis study we have investigated the effect of the perturbationof its lowest order however extending our work to higher-order approximations would require us to use higher-orderconstraint equations instead

Examples towhich themethod is applied here are singularperturbation problems in which secular terms appear in thenaıve expansion of the solution [3 6 8 10 11 34] Termsproportional to the independent variables emerge in theinfinitesimal generator of the Lie symmetry group as seen in(10) (30) and (48) corresponding to the emergence of thesecular terms in the naıve expansion As inferred from thismathematical similarity the presentmethod is also applicableto other systems which exhibit the same type of singularperturbation problems In the singular perturbationmethodsproposed thus far it is only after the construction of the naıveexpansion that it becomes clear whether the method shouldbe used or not However the present method can be appliedwithout making the expansion enabling the appropriatereduced equations to be derived

Future work should involve a clarification of the relationsbetween the present method and other types of singularperturbation methods presented thus far such as the WKBor boundary layer problem methods In addition there areexisting methods that have enabled the successful deriva-tion of the asymptotic behavior for various other partialdifferential equations [13ndash18 20ndash30 35ndash38] Therefore itshould be clarified whether or not the present method canbe appropriately extended and applied to the problems thosemethods have resolved

Journal of Applied Mathematics 7

Appendix

Searching for Lie Symmetry withthe Normal Procedure

Let us search for the Lie symmetry group of (1) withoutconstraint equations The criterion for the invariance of theequation is given by (7) namely

119883lowast[2119906 + minus 120576]

10038161003816100381610038162119906+minus120576=0= 0 (A1)

which reads

21199062120578119905119905minus 119906 + (4119906

2120578119906119905+ 2119906120578

119905)

+ 2(21199062120578119906119906+ 119906120578 minus 120578) + 120576 (120578 + 119906120578

119906) = 0

(A2)

In the lowest order of 120576 120578 has to satisfy the followingdifferential equations

21199062120578119905119905minus 119906 = 0

41199062120578119906119905+ 2119906120578

119905= 0

21199062120578119906119906+ 119906120578 minus 120578 = 0

(A3)

However we can easily find out that there is no solution thatsatisfies all the above equations which implies that there is noLie symmetry group

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by JSPS KAKENHI Grant no26800208

References

[1] N N Bogoliubov and C M Place Asymptotic Methods in theTheory of Non-Linear Oscillations Gordon and Break 1961

[2] GHori ldquoTheory of general perturbationwith unspecified cano-nical variablerdquo Publications of the Astronomical Society of Japanvol 18 p 287 1966

[3] T Taniuchi ldquoReductive perturbation method and far fields ofwave equationsrdquo Progress ofTheoretical Physics Supplement vol55 pp 1ndash35 1974

[4] J Guckenheimer andPHolmesNonlinearOscillationsDynam-ical systems and Bifurcations of Vector Fields Springer NewYork NY USA 1983

[5] A H Nayfeh Method of Normal Forms John Wiley amp SonsNew York NY USA 1993

[6] T Kunihiro ldquoA geometrical formulation of the renormalizationgroup method for global analysisrdquo Progress of TheoreticalPhysics vol 94 no 4 pp 503ndash514 1995

[7] L-Y ChenNGoldenfeld YOono andG Paquette ldquoSelectionstability and renormalizationrdquo Physica A Statistical Mechanicsand its Applications vol 204 no 1ndash4 pp 111ndash133 1994

[8] L Y Chen N Goldenfeld and Y Oono ldquoRenormalizationgroup and singular perturbations multiple scales boundarylayers and reductive perturbation theoryrdquo Physical Review Evol 54 no 1 pp 376ndash394 1996

[9] S-I Goto Y Masutomi and K Nozaki ldquoLie-group approachto perturbative renormalization group methodrdquo Progress ofTheoretical Physics vol 102 no 3 pp 471ndash497 1999

[10] R E L DeVille A Harkin M Holzer K Josic and T J KaperldquoAnalysis of a renormalization group method and normal formtheory for perturbed ordinary differential equationsrdquoPhysicaDvol 237 no 8 pp 1029ndash1052 2008

[11] H Chiba ldquo1198621 approximation of vector fields based on therenormalization group methodrdquo SIAM Journal on AppliedDynamical Systems vol 7 no 3 pp 895ndash932 2008

[12] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1986

[13] V A Baıkov R K Gazizov and N H Ibragimov ldquoApproximatesymmetriesrdquoMathematics of the USSR-Sbornik vol 64 no 2 p427 1989

[14] W I Fushchich andWM Shtelen ldquoOn approximate symmetryand approximate solutions of the nonlinear wave equation witha small parameterrdquo Journal of Physics A Mathematical andGeneral vol 22 no 18 pp L887ndashL890 1989

[15] N Eular MW Shulga andW-H Steeb ldquoApproximate symme-tries and approximate solutions for amultidimensional Landau-Ginzburg equationrdquo Journal of Physics A Mathematical andGeneral vol 25 no 18 Article ID L1095 1992

[16] Y Y Bagderina and R K Gazizov ldquoEquivalence of ordinarydifferential equations 11991010158401015840 = 119877(119909 119910)11991010158402 + 2119876(119909 119910)1199101015840 + 119875(119909 119910)rdquoDifferential Equations vol 43 no 5 pp 595ndash604 2007

[17] V N Grebenev andM Oberlack ldquoApproximate Lie symmetriesof theNavier-Stokes equationsrdquo Journal ofNonlinearMathemat-ical Physics vol 14 no 2 pp 157ndash163 2007

[18] R K Gazizov and N K Ibragimov ldquoApproximate symmetriesand solutions of the Kompaneets equationrdquo Journal of AppliedMechanics and Technical Physics vol 55 no 2 pp 220ndash2242014

[19] R K Gazizov and C M Khalique ldquoApproximate transforma-tions for van der Pol-type equationsrdquoMathematical Problems inEngineering vol 2006 Article ID 68753 11 pages 2006

[20] G Gaeta ldquoAsymptotic symmetries and asymptotically symmet-ric solutions of partial differential equationsrdquo Journal of PhysicsA Mathematical and General vol 27 no 2 pp 437ndash451 1994

[21] G Gaeta ldquoAsymptotic symmetries in an optical latticerdquo PhysicalReview A vol 72 no 3 Article ID 033419 2005

[22] G Gaeta D Levi and R Mancinelli ldquoAsymptotic symmetriesof difference equations on a latticerdquo Journal of NonlinearMathe-matical Physics vol 12 supplement 2 pp 137ndash146 2005

[23] G Gaeta and RMancinelli ldquoAsymptotic scaling symmetries fornonlinear PDEsrdquo International Journal of Geometric Methods inModern Physics vol 2 no 6 pp 1081ndash1114 2005

[24] G Gaeta and R Mancinelli ldquoAsymptotic scaling in a modelclass of anomalous reaction-diffusion equationsrdquo Journal ofNonlinear Mathematical Physics vol 12 no 4 pp 550ndash5662005

[25] D Levi and M A Rodrıguez ldquoAsymptotic symmetries andintegrability the KdV caserdquo Europhysics Letters vol 80 no 6Article ID 60005 2007

[26] V F Kovalev V V Pustovalov and D V Shirkov ldquoGroup anal-ysis and renormgroup symmetriesrdquo Journal of MathematicalPhysics vol 39 no 2 pp 1170ndash1188 1998

8 Journal of Applied Mathematics

[27] V F Kovalev ldquoRenormalization group analysis for singularitiesin the wave beam self-focusing problemrdquo Theoretical andMathematical Physics vol 119 no 3 pp 719ndash730 1999

[28] V F Kovalev and D V Shirkov ldquoFunctional self-similarityand renormalization group symmetry inmathematical physicsrdquoTheoretical and Mathematical Physics vol 121 no 1 pp 1315ndash1332 1999

[29] D V Shirkov and V F Kovalev ldquoThe Bogoliubov renormaliza-tion group and solution symmetry in mathematical physicsrdquoPhysics Report vol 352 no 4ndash6 pp 219ndash249 2001

[30] V F Kovalev and D V Shirkov ldquoThe renormalization groupsymmetry for solution of integral equationsrdquo Proceedings of theInstitute of Mathematics of NAS of Ukraine vol 50 no 2 pp850ndash861 2004

[31] M Iwasa and K Nozaki ldquoA method to construct asymptoticsolutions invariant under the renormalization grouprdquo Progressof Theoretical Physics vol 116 no 4 pp 605ndash613 2006

[32] M Iwasa and K Nozaki ldquoRenormalization group in differencesystemsrdquo Journal of Physics A Mathematical and Theoreticalvol 41 no 8 Article ID 085204 2008

[33] H Chiba andM Iwasa ldquoLie equations for asymptotic solutionsof perturbation problems of ordinary differential equationsrdquoJournal of Mathematical Physics vol 50 no 4 2009

[34] Y Y Yamaguchi and Y Nambu ldquoRenormalization group equa-tions and integrability in hamiltonian systemsrdquo Progress ofTheoretical Physics vol 100 no 1 pp 199ndash204 1998

[35] N D Goldenfeld Lectures on Phase Transitions and the Renor-malization Group Addison-Wesley 1992

[36] J Bricmont A Kupiainen and G Lin ldquoRenormalization groupand asymptotics of solutions of nonlinear parabolic equationsrdquoCommunications on Pure and Applied Mathematics vol 47 no6 pp 893ndash922 1994

[37] N V Antonov and J Honkonen ldquoField-theoretic renormaliza-tion group for a nonlinear diffusion equationrdquo Physical ReviewE vol 66 no 4 Article ID 046105 7 pages 2002

[38] S Yoshida andT Fukui ldquoExact renormalization group approachto a nonlinear diffusion equationrdquo Physical Review E vol 72 no4 Article ID 046136 2005

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 Journal of Applied Mathematics

(45) read in the lowest order of 120576

(1199062

2minus 1199062

1) (120578119905119905minus 120578119909119909) minus (119906

2

2minus 1199062

1)2

120578119906119906minus 1199062

1119906212057811990611199061

minus 1199062

1119906212057811990621199062

+ 21199061119906 (1199062

2minus 1199062

1) 1205781199061199061

minus 21199062119906 (1199062

2minus 1199062

1) 1205781199061199062

minus 211990611199062119906212057811990611199062

+ 21199061(1199062

2minus 1199062

1) 120578119905119906+ 21199062

11199061205781199051199061

+ 2119906111990621199061205781199051199062

minus 21199062(1199062

2minus 1199062

1) 120578119909119906minus 2119906111990621199061205781199091199061

minus 21199062

21199061205781199091199062

minus 1199061(1199062

2minus 1199062

1) 1205781199061

minus 1199062(1199062

2minus 1199062

1) 1205781199062

minus 119906 (1199062

2minus 1199062

1) 120578119906+ (1199062

2minus 1199062

1) 120578 minus 119906

3(1199062

2minus 1199062

1)

= 0

(47)

Noting the homogeneity of the equation we find a solution120578 (119905 119909 119906 119906

1 1199062)

= minus3

8(1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2) (119886

119905

1199061

minus (1 minus 119886)119909

1199062

)

minus1

32119906 (1199062minus 1199062

1minus 1199062

2)

(48)

where 119886 is an arbitrary constant For this infinitesimalgenerator of the approximate Lie symmetry group the group-invariant solution satisfies

119883 [119906 minus 119906 (119905 119909)]|119906=119906(119905119909) = 0 (49)which reads

120597119906 (119905 119909)

120597120576= 120578 (119905 119909 119906 (119905 119909) 119906

1(119905 119909) 119906

2(119905 119909)) (50)

Asymptotic dynamic behavior can be obtained by consider-ing the cases 119905 ≫ 1 and 119909 ≫ 1 and by neglecting the termswhich are not proportional to 119909 or 119905 By introducing 120591 = 120576119905and 120585 = 120576119909 we obtain

120597119906

120597120591= minus

3 (1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2)

81199061

119886

120597119906

120597120585=

3 (1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2)

81199062

(1 minus 119886)

(51)

By eliminating 119886 the equations are reduced to

1199061

120597119906

120597120591minus 1199062

120597119906

120597120585= minus3

8(1199062+ 1199062

1minus 1199062

2) (1199062

1minus 1199062

2) (52)

This reduced equation describes the asymptotic behavior ofthe original systemThis fact can be confirmed by comparingwith the result derived before By introducing a transforma-tion of the variables

|119860|2=1

4(1199062+ 1199062

1minus 1199062

2)

120596 = 1199061(1199062

1minus 1199062

2)minus12

119896 = 1199062(1199062

1minus 1199062

2)minus12

(53)

the reduced equation (52) is transformed into

[120597

120597120591+119896

120596

120597

120597120585]119860 = minus

3

2120596i |119860|2 119860 (54)

This is the nonlinear Schrodinger equation which corre-sponds to the reduced equation derived with the renormal-ization group method [9]

5 Concluding Remarks

In summary this paper presents an investigation of perturba-tion problems which involves the application of a particulartype of Lie group method to partial as well as ordinarydifferential equations As a consequence we have obtainedthe dynamics which describes the asymptotic behavior ofthe original system The main characteristic of the presentmethod is the addition of some constraint equations whensearching for a Lie symmetry group an approach whichallows us to find a partial Lie symmetry groupThe constraintequations are determined from the nonperturbative solutionon which we try to investigate the effect of the perturbationThe reason why it is necessary to specify a nonperturbativesolution beforehand for the partial differential equations isthat unlike for ordinary differential equations the non-perturbative system has an infinite number of independentsolutions In terms of technique those constraint equationsare constructed in such a way that the arbitrary constantsor functions included in the nonperturbative solution areeliminated as we have done in deriving (4) (18) and (41) Inthis study we have investigated the effect of the perturbationof its lowest order however extending our work to higher-order approximations would require us to use higher-orderconstraint equations instead

Examples towhich themethod is applied here are singularperturbation problems in which secular terms appear in thenaıve expansion of the solution [3 6 8 10 11 34] Termsproportional to the independent variables emerge in theinfinitesimal generator of the Lie symmetry group as seen in(10) (30) and (48) corresponding to the emergence of thesecular terms in the naıve expansion As inferred from thismathematical similarity the presentmethod is also applicableto other systems which exhibit the same type of singularperturbation problems In the singular perturbationmethodsproposed thus far it is only after the construction of the naıveexpansion that it becomes clear whether the method shouldbe used or not However the present method can be appliedwithout making the expansion enabling the appropriatereduced equations to be derived

Future work should involve a clarification of the relationsbetween the present method and other types of singularperturbation methods presented thus far such as the WKBor boundary layer problem methods In addition there areexisting methods that have enabled the successful deriva-tion of the asymptotic behavior for various other partialdifferential equations [13ndash18 20ndash30 35ndash38] Therefore itshould be clarified whether or not the present method canbe appropriately extended and applied to the problems thosemethods have resolved

Journal of Applied Mathematics 7

Appendix

Searching for Lie Symmetry withthe Normal Procedure

Let us search for the Lie symmetry group of (1) withoutconstraint equations The criterion for the invariance of theequation is given by (7) namely

119883lowast[2119906 + minus 120576]

10038161003816100381610038162119906+minus120576=0= 0 (A1)

which reads

21199062120578119905119905minus 119906 + (4119906

2120578119906119905+ 2119906120578

119905)

+ 2(21199062120578119906119906+ 119906120578 minus 120578) + 120576 (120578 + 119906120578

119906) = 0

(A2)

In the lowest order of 120576 120578 has to satisfy the followingdifferential equations

21199062120578119905119905minus 119906 = 0

41199062120578119906119905+ 2119906120578

119905= 0

21199062120578119906119906+ 119906120578 minus 120578 = 0

(A3)

However we can easily find out that there is no solution thatsatisfies all the above equations which implies that there is noLie symmetry group

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by JSPS KAKENHI Grant no26800208

References

[1] N N Bogoliubov and C M Place Asymptotic Methods in theTheory of Non-Linear Oscillations Gordon and Break 1961

[2] GHori ldquoTheory of general perturbationwith unspecified cano-nical variablerdquo Publications of the Astronomical Society of Japanvol 18 p 287 1966

[3] T Taniuchi ldquoReductive perturbation method and far fields ofwave equationsrdquo Progress ofTheoretical Physics Supplement vol55 pp 1ndash35 1974

[4] J Guckenheimer andPHolmesNonlinearOscillationsDynam-ical systems and Bifurcations of Vector Fields Springer NewYork NY USA 1983

[5] A H Nayfeh Method of Normal Forms John Wiley amp SonsNew York NY USA 1993

[6] T Kunihiro ldquoA geometrical formulation of the renormalizationgroup method for global analysisrdquo Progress of TheoreticalPhysics vol 94 no 4 pp 503ndash514 1995

[7] L-Y ChenNGoldenfeld YOono andG Paquette ldquoSelectionstability and renormalizationrdquo Physica A Statistical Mechanicsand its Applications vol 204 no 1ndash4 pp 111ndash133 1994

[8] L Y Chen N Goldenfeld and Y Oono ldquoRenormalizationgroup and singular perturbations multiple scales boundarylayers and reductive perturbation theoryrdquo Physical Review Evol 54 no 1 pp 376ndash394 1996

[9] S-I Goto Y Masutomi and K Nozaki ldquoLie-group approachto perturbative renormalization group methodrdquo Progress ofTheoretical Physics vol 102 no 3 pp 471ndash497 1999

[10] R E L DeVille A Harkin M Holzer K Josic and T J KaperldquoAnalysis of a renormalization group method and normal formtheory for perturbed ordinary differential equationsrdquoPhysicaDvol 237 no 8 pp 1029ndash1052 2008

[11] H Chiba ldquo1198621 approximation of vector fields based on therenormalization group methodrdquo SIAM Journal on AppliedDynamical Systems vol 7 no 3 pp 895ndash932 2008

[12] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1986

[13] V A Baıkov R K Gazizov and N H Ibragimov ldquoApproximatesymmetriesrdquoMathematics of the USSR-Sbornik vol 64 no 2 p427 1989

[14] W I Fushchich andWM Shtelen ldquoOn approximate symmetryand approximate solutions of the nonlinear wave equation witha small parameterrdquo Journal of Physics A Mathematical andGeneral vol 22 no 18 pp L887ndashL890 1989

[15] N Eular MW Shulga andW-H Steeb ldquoApproximate symme-tries and approximate solutions for amultidimensional Landau-Ginzburg equationrdquo Journal of Physics A Mathematical andGeneral vol 25 no 18 Article ID L1095 1992

[16] Y Y Bagderina and R K Gazizov ldquoEquivalence of ordinarydifferential equations 11991010158401015840 = 119877(119909 119910)11991010158402 + 2119876(119909 119910)1199101015840 + 119875(119909 119910)rdquoDifferential Equations vol 43 no 5 pp 595ndash604 2007

[17] V N Grebenev andM Oberlack ldquoApproximate Lie symmetriesof theNavier-Stokes equationsrdquo Journal ofNonlinearMathemat-ical Physics vol 14 no 2 pp 157ndash163 2007

[18] R K Gazizov and N K Ibragimov ldquoApproximate symmetriesand solutions of the Kompaneets equationrdquo Journal of AppliedMechanics and Technical Physics vol 55 no 2 pp 220ndash2242014

[19] R K Gazizov and C M Khalique ldquoApproximate transforma-tions for van der Pol-type equationsrdquoMathematical Problems inEngineering vol 2006 Article ID 68753 11 pages 2006

[20] G Gaeta ldquoAsymptotic symmetries and asymptotically symmet-ric solutions of partial differential equationsrdquo Journal of PhysicsA Mathematical and General vol 27 no 2 pp 437ndash451 1994

[21] G Gaeta ldquoAsymptotic symmetries in an optical latticerdquo PhysicalReview A vol 72 no 3 Article ID 033419 2005

[22] G Gaeta D Levi and R Mancinelli ldquoAsymptotic symmetriesof difference equations on a latticerdquo Journal of NonlinearMathe-matical Physics vol 12 supplement 2 pp 137ndash146 2005

[23] G Gaeta and RMancinelli ldquoAsymptotic scaling symmetries fornonlinear PDEsrdquo International Journal of Geometric Methods inModern Physics vol 2 no 6 pp 1081ndash1114 2005

[24] G Gaeta and R Mancinelli ldquoAsymptotic scaling in a modelclass of anomalous reaction-diffusion equationsrdquo Journal ofNonlinear Mathematical Physics vol 12 no 4 pp 550ndash5662005

[25] D Levi and M A Rodrıguez ldquoAsymptotic symmetries andintegrability the KdV caserdquo Europhysics Letters vol 80 no 6Article ID 60005 2007

[26] V F Kovalev V V Pustovalov and D V Shirkov ldquoGroup anal-ysis and renormgroup symmetriesrdquo Journal of MathematicalPhysics vol 39 no 2 pp 1170ndash1188 1998

8 Journal of Applied Mathematics

[27] V F Kovalev ldquoRenormalization group analysis for singularitiesin the wave beam self-focusing problemrdquo Theoretical andMathematical Physics vol 119 no 3 pp 719ndash730 1999

[28] V F Kovalev and D V Shirkov ldquoFunctional self-similarityand renormalization group symmetry inmathematical physicsrdquoTheoretical and Mathematical Physics vol 121 no 1 pp 1315ndash1332 1999

[29] D V Shirkov and V F Kovalev ldquoThe Bogoliubov renormaliza-tion group and solution symmetry in mathematical physicsrdquoPhysics Report vol 352 no 4ndash6 pp 219ndash249 2001

[30] V F Kovalev and D V Shirkov ldquoThe renormalization groupsymmetry for solution of integral equationsrdquo Proceedings of theInstitute of Mathematics of NAS of Ukraine vol 50 no 2 pp850ndash861 2004

[31] M Iwasa and K Nozaki ldquoA method to construct asymptoticsolutions invariant under the renormalization grouprdquo Progressof Theoretical Physics vol 116 no 4 pp 605ndash613 2006

[32] M Iwasa and K Nozaki ldquoRenormalization group in differencesystemsrdquo Journal of Physics A Mathematical and Theoreticalvol 41 no 8 Article ID 085204 2008

[33] H Chiba andM Iwasa ldquoLie equations for asymptotic solutionsof perturbation problems of ordinary differential equationsrdquoJournal of Mathematical Physics vol 50 no 4 2009

[34] Y Y Yamaguchi and Y Nambu ldquoRenormalization group equa-tions and integrability in hamiltonian systemsrdquo Progress ofTheoretical Physics vol 100 no 1 pp 199ndash204 1998

[35] N D Goldenfeld Lectures on Phase Transitions and the Renor-malization Group Addison-Wesley 1992

[36] J Bricmont A Kupiainen and G Lin ldquoRenormalization groupand asymptotics of solutions of nonlinear parabolic equationsrdquoCommunications on Pure and Applied Mathematics vol 47 no6 pp 893ndash922 1994

[37] N V Antonov and J Honkonen ldquoField-theoretic renormaliza-tion group for a nonlinear diffusion equationrdquo Physical ReviewE vol 66 no 4 Article ID 046105 7 pages 2002

[38] S Yoshida andT Fukui ldquoExact renormalization group approachto a nonlinear diffusion equationrdquo Physical Review E vol 72 no4 Article ID 046136 2005

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

Journal of Applied Mathematics 7

Appendix

Searching for Lie Symmetry withthe Normal Procedure

Let us search for the Lie symmetry group of (1) withoutconstraint equations The criterion for the invariance of theequation is given by (7) namely

119883lowast[2119906 + minus 120576]

10038161003816100381610038162119906+minus120576=0= 0 (A1)

which reads

21199062120578119905119905minus 119906 + (4119906

2120578119906119905+ 2119906120578

119905)

+ 2(21199062120578119906119906+ 119906120578 minus 120578) + 120576 (120578 + 119906120578

119906) = 0

(A2)

In the lowest order of 120576 120578 has to satisfy the followingdifferential equations

21199062120578119905119905minus 119906 = 0

41199062120578119906119905+ 2119906120578

119905= 0

21199062120578119906119906+ 119906120578 minus 120578 = 0

(A3)

However we can easily find out that there is no solution thatsatisfies all the above equations which implies that there is noLie symmetry group

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by JSPS KAKENHI Grant no26800208

References

[1] N N Bogoliubov and C M Place Asymptotic Methods in theTheory of Non-Linear Oscillations Gordon and Break 1961

[2] GHori ldquoTheory of general perturbationwith unspecified cano-nical variablerdquo Publications of the Astronomical Society of Japanvol 18 p 287 1966

[3] T Taniuchi ldquoReductive perturbation method and far fields ofwave equationsrdquo Progress ofTheoretical Physics Supplement vol55 pp 1ndash35 1974

[4] J Guckenheimer andPHolmesNonlinearOscillationsDynam-ical systems and Bifurcations of Vector Fields Springer NewYork NY USA 1983

[5] A H Nayfeh Method of Normal Forms John Wiley amp SonsNew York NY USA 1993

[6] T Kunihiro ldquoA geometrical formulation of the renormalizationgroup method for global analysisrdquo Progress of TheoreticalPhysics vol 94 no 4 pp 503ndash514 1995

[7] L-Y ChenNGoldenfeld YOono andG Paquette ldquoSelectionstability and renormalizationrdquo Physica A Statistical Mechanicsand its Applications vol 204 no 1ndash4 pp 111ndash133 1994

[8] L Y Chen N Goldenfeld and Y Oono ldquoRenormalizationgroup and singular perturbations multiple scales boundarylayers and reductive perturbation theoryrdquo Physical Review Evol 54 no 1 pp 376ndash394 1996

[9] S-I Goto Y Masutomi and K Nozaki ldquoLie-group approachto perturbative renormalization group methodrdquo Progress ofTheoretical Physics vol 102 no 3 pp 471ndash497 1999

[10] R E L DeVille A Harkin M Holzer K Josic and T J KaperldquoAnalysis of a renormalization group method and normal formtheory for perturbed ordinary differential equationsrdquoPhysicaDvol 237 no 8 pp 1029ndash1052 2008

[11] H Chiba ldquo1198621 approximation of vector fields based on therenormalization group methodrdquo SIAM Journal on AppliedDynamical Systems vol 7 no 3 pp 895ndash932 2008

[12] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1986

[13] V A Baıkov R K Gazizov and N H Ibragimov ldquoApproximatesymmetriesrdquoMathematics of the USSR-Sbornik vol 64 no 2 p427 1989

[14] W I Fushchich andWM Shtelen ldquoOn approximate symmetryand approximate solutions of the nonlinear wave equation witha small parameterrdquo Journal of Physics A Mathematical andGeneral vol 22 no 18 pp L887ndashL890 1989

[15] N Eular MW Shulga andW-H Steeb ldquoApproximate symme-tries and approximate solutions for amultidimensional Landau-Ginzburg equationrdquo Journal of Physics A Mathematical andGeneral vol 25 no 18 Article ID L1095 1992

[16] Y Y Bagderina and R K Gazizov ldquoEquivalence of ordinarydifferential equations 11991010158401015840 = 119877(119909 119910)11991010158402 + 2119876(119909 119910)1199101015840 + 119875(119909 119910)rdquoDifferential Equations vol 43 no 5 pp 595ndash604 2007

[17] V N Grebenev andM Oberlack ldquoApproximate Lie symmetriesof theNavier-Stokes equationsrdquo Journal ofNonlinearMathemat-ical Physics vol 14 no 2 pp 157ndash163 2007

[18] R K Gazizov and N K Ibragimov ldquoApproximate symmetriesand solutions of the Kompaneets equationrdquo Journal of AppliedMechanics and Technical Physics vol 55 no 2 pp 220ndash2242014

[19] R K Gazizov and C M Khalique ldquoApproximate transforma-tions for van der Pol-type equationsrdquoMathematical Problems inEngineering vol 2006 Article ID 68753 11 pages 2006

[20] G Gaeta ldquoAsymptotic symmetries and asymptotically symmet-ric solutions of partial differential equationsrdquo Journal of PhysicsA Mathematical and General vol 27 no 2 pp 437ndash451 1994

[21] G Gaeta ldquoAsymptotic symmetries in an optical latticerdquo PhysicalReview A vol 72 no 3 Article ID 033419 2005

[22] G Gaeta D Levi and R Mancinelli ldquoAsymptotic symmetriesof difference equations on a latticerdquo Journal of NonlinearMathe-matical Physics vol 12 supplement 2 pp 137ndash146 2005

[23] G Gaeta and RMancinelli ldquoAsymptotic scaling symmetries fornonlinear PDEsrdquo International Journal of Geometric Methods inModern Physics vol 2 no 6 pp 1081ndash1114 2005

[24] G Gaeta and R Mancinelli ldquoAsymptotic scaling in a modelclass of anomalous reaction-diffusion equationsrdquo Journal ofNonlinear Mathematical Physics vol 12 no 4 pp 550ndash5662005

[25] D Levi and M A Rodrıguez ldquoAsymptotic symmetries andintegrability the KdV caserdquo Europhysics Letters vol 80 no 6Article ID 60005 2007

[26] V F Kovalev V V Pustovalov and D V Shirkov ldquoGroup anal-ysis and renormgroup symmetriesrdquo Journal of MathematicalPhysics vol 39 no 2 pp 1170ndash1188 1998

8 Journal of Applied Mathematics

[27] V F Kovalev ldquoRenormalization group analysis for singularitiesin the wave beam self-focusing problemrdquo Theoretical andMathematical Physics vol 119 no 3 pp 719ndash730 1999

[28] V F Kovalev and D V Shirkov ldquoFunctional self-similarityand renormalization group symmetry inmathematical physicsrdquoTheoretical and Mathematical Physics vol 121 no 1 pp 1315ndash1332 1999

[29] D V Shirkov and V F Kovalev ldquoThe Bogoliubov renormaliza-tion group and solution symmetry in mathematical physicsrdquoPhysics Report vol 352 no 4ndash6 pp 219ndash249 2001

[30] V F Kovalev and D V Shirkov ldquoThe renormalization groupsymmetry for solution of integral equationsrdquo Proceedings of theInstitute of Mathematics of NAS of Ukraine vol 50 no 2 pp850ndash861 2004

[31] M Iwasa and K Nozaki ldquoA method to construct asymptoticsolutions invariant under the renormalization grouprdquo Progressof Theoretical Physics vol 116 no 4 pp 605ndash613 2006

[32] M Iwasa and K Nozaki ldquoRenormalization group in differencesystemsrdquo Journal of Physics A Mathematical and Theoreticalvol 41 no 8 Article ID 085204 2008

[33] H Chiba andM Iwasa ldquoLie equations for asymptotic solutionsof perturbation problems of ordinary differential equationsrdquoJournal of Mathematical Physics vol 50 no 4 2009

[34] Y Y Yamaguchi and Y Nambu ldquoRenormalization group equa-tions and integrability in hamiltonian systemsrdquo Progress ofTheoretical Physics vol 100 no 1 pp 199ndash204 1998

[35] N D Goldenfeld Lectures on Phase Transitions and the Renor-malization Group Addison-Wesley 1992

[36] J Bricmont A Kupiainen and G Lin ldquoRenormalization groupand asymptotics of solutions of nonlinear parabolic equationsrdquoCommunications on Pure and Applied Mathematics vol 47 no6 pp 893ndash922 1994

[37] N V Antonov and J Honkonen ldquoField-theoretic renormaliza-tion group for a nonlinear diffusion equationrdquo Physical ReviewE vol 66 no 4 Article ID 046105 7 pages 2002

[38] S Yoshida andT Fukui ldquoExact renormalization group approachto a nonlinear diffusion equationrdquo Physical Review E vol 72 no4 Article ID 046136 2005

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

8 Journal of Applied Mathematics

[27] V F Kovalev ldquoRenormalization group analysis for singularitiesin the wave beam self-focusing problemrdquo Theoretical andMathematical Physics vol 119 no 3 pp 719ndash730 1999

[28] V F Kovalev and D V Shirkov ldquoFunctional self-similarityand renormalization group symmetry inmathematical physicsrdquoTheoretical and Mathematical Physics vol 121 no 1 pp 1315ndash1332 1999

[29] D V Shirkov and V F Kovalev ldquoThe Bogoliubov renormaliza-tion group and solution symmetry in mathematical physicsrdquoPhysics Report vol 352 no 4ndash6 pp 219ndash249 2001

[30] V F Kovalev and D V Shirkov ldquoThe renormalization groupsymmetry for solution of integral equationsrdquo Proceedings of theInstitute of Mathematics of NAS of Ukraine vol 50 no 2 pp850ndash861 2004

[31] M Iwasa and K Nozaki ldquoA method to construct asymptoticsolutions invariant under the renormalization grouprdquo Progressof Theoretical Physics vol 116 no 4 pp 605ndash613 2006

[32] M Iwasa and K Nozaki ldquoRenormalization group in differencesystemsrdquo Journal of Physics A Mathematical and Theoreticalvol 41 no 8 Article ID 085204 2008

[33] H Chiba andM Iwasa ldquoLie equations for asymptotic solutionsof perturbation problems of ordinary differential equationsrdquoJournal of Mathematical Physics vol 50 no 4 2009

[34] Y Y Yamaguchi and Y Nambu ldquoRenormalization group equa-tions and integrability in hamiltonian systemsrdquo Progress ofTheoretical Physics vol 100 no 1 pp 199ndash204 1998

[35] N D Goldenfeld Lectures on Phase Transitions and the Renor-malization Group Addison-Wesley 1992

[36] J Bricmont A Kupiainen and G Lin ldquoRenormalization groupand asymptotics of solutions of nonlinear parabolic equationsrdquoCommunications on Pure and Applied Mathematics vol 47 no6 pp 893ndash922 1994

[37] N V Antonov and J Honkonen ldquoField-theoretic renormaliza-tion group for a nonlinear diffusion equationrdquo Physical ReviewE vol 66 no 4 Article ID 046105 7 pages 2002

[38] S Yoshida andT Fukui ldquoExact renormalization group approachto a nonlinear diffusion equationrdquo Physical Review E vol 72 no4 Article ID 046136 2005

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