Solitary waves in the nonlinear Schrödinger equation with spatially modulated Bessel nonlinearity

Solitary waves in the nonlinear Schrödinger equationwith spatially modulated Bessel nonlinearity

Wei-Ping Zhong,1,* Milivoj R. Belić,2,3 and Tingwen Huang2

1Department of Electronic and Information Engineering, Shunde Polytechnic, Guangdong Province,Shunde 528300, China

2Texas A&M University at Qatar, P.O. Box 23874, Doha, Qatar3Institute of Physics, University of Belgrade, P.O. Box 68, Belgrade 11001, Serbia

*Corresponding author: [email protected]

Received January 17, 2013; revised March 15, 2013; accepted March 15, 2013;posted March 19, 2013 (Doc. ID 183696); published April 19, 2013

Using multivariate self-similarity transformation, we construct explicit spatial bright and dark solitary wavesolutions of the generalized nonlinear Schrödinger equation with spatially Bessel-modulated nonlinearity andan external potential. Special kinds of explicit solutions, such as periodically breathing bright and darksolitary waves, are discussed in detail. The stability of these solutions is verified by means of direct numericalsimulation. © 2013 Optical Society of America

OCIS codes: (190.0190) Nonlinear optics; (190.6135) Spatial solitons.http://dx.doi.org/10.1364/JOSAB.30.001276

1. INTRODUCTIONAs is well known, optical solitons are regarded as natural car-riers of information and as important alternatives to the nextgeneration of ultrafast optical telecommunication systems [1].Solitons are formed by the exact balance of dispersion, dif-fraction, and nonlinearity. A particularly important modelfor the generation of solitons is described by the (1� 1)-dimensional nonlinear Schrödinger (NLS) equation with themodulated Kerr nonlinearity and an external potential. In thiscase, the equation is usually known the Gross–Pitaevskii equa-tion [2,3] and the external potential may be time dependent[4]. This equation has drawn a great deal of interest due toits direct applications in the description of Bose–Einstein con-densates (BECs) [2,3], photonic waveguides [5], and otherstationary and nonstationary media [6].

The generalized NLS equation is one of the most importantuniversal nonlinear models that naturally arises in many fieldsof physics, such as nonlinear optics, hydrodynamics, plasmaphysics, condensed matter physics, BECs, and so on [2,3,5,7].The exact periodic wave and solitary wave solutions of theNLS equation have been investigated by many researchers[8–10]. Stable exact soliton solutions of the NLS equation wereobtained by various methods [11–18], including the self-similar transformation [19–26]. Subsequently, the generalizedNLS equation with distributed coefficients was analyzed, con-taining exact sech-shaped and tanh-shaped stable solitonpulses [6,9,10]. Solitary wave formation and propagation indifferent distributed systems has been studied mostly in thecases where the nonlinear refractive index varies along thepropagation distance; as an analog to the dispersion manage-ment, these cases were usually referred to as the nonlinearitymanagement [6]. However, the effects of transversely varyingnonlinear refractive index have only been recently studiedtheoretically for the case of homogeneous [27] as well asspatially modulated [23,24,28] linear refractive indices.

Cases where both linear and nonlinear refractive indiceswere spatially modulated have also been studied in waveguidearrays consisting of interlaced linear and nonlinear wave-guides, and analytical soliton solutions have been found[29,30]. The emphasis of the theoretical research has beenmainly focused on achieving localization and investigatingthe dynamics of localized waves [26,29,30]. While finding sta-ble localized solutions is a common goal in all diverse physicalsystems exhibiting bright and dark solitons, it seems that notenough emphasis has been placed so far on the determinationof localized states with the periodic spatial structure and theircontrollability.

The goal of this paper is to study explicit solutions of theone-dimensional generalized NLS equation with spatiallymodulated diffraction and nonlinearity coefficients. We usethe multivariate self-similarity transformation to transformthis model into the standard NLS equation and find exact sol-utions. Although our method is suitable for arbitrary modula-tions of the coefficients, in this work we investigate in moredetail the generalized NLS equation in which the nonlinearityis spatially modulated by Bessel functions, as well as thedynamics of the corresponding nonlinear solitary waves.

The interest in NLS equations involving Bessel functions ei-ther as a nonlinearity or as an external potential, or both, haspicked up after the appearance of nondiffracting Bessel beams[31]. The initial appearance of Bessel beams in 1987 [32,33]attracted some controversy, because they represented nondif-fracting beams of infinite energy in free space and as such in-vitedquestionsofphysical reality. Theneedofhaving to involvea finite aperture and finite-size lenses in their productionmeantthat some diffraction must be present. By now, this initial con-troversy has largely settled and nondiffractingBessel andotheroptical beams are a vibrant part of linear optics [34]. However,no such controversy appeared in nonlinear optics, where non-diffracting localized beams—solitons—commonly exist.

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http://dx.doi.org/10.1364/JOSAB.30.001276

Still, research has branched into two directions: in one, theinfluence of nonlinearity, e.g., of Kerr-type, on the nondiffract-ing Bessel beams was investigated [35]; in the other, the non-linearity (or the external potential) was modulated by Besselfunctions and the form of nondiffracting solutions studied[36]. We have presented a study [37] in which both the exter-nal lattice potential and the solutions of the three-dimensionalNLS equation with Kerr nonlinearity involved Bessel func-tions; unfortunately, that study was only approximatelyanalytical. Here we extend the analysis to involve spatiallymodulated Bessel nonlinearity in a rigorously analytical man-ner. Our results show that in such a model there exist severaltypes of exact bright and dark solitary wave solutions, includ-ing breathers and oscillating solitons. Even when breathing,such beams are nondiffracting.

The paper is organized as follows. The model and itssolitary wave solutions are introduced by the multivariateself-similarity transformation in Section 2, and we use the sim-ilarity method to construct explicit solutions of the NLS equa-tion with the transverse spatial modulation of the coefficients.The cases of periodical solitary waves are investigated whenthe solution parameters are chosen appropriately; the discus-sion of solutions is presented in Section 3. Numerical simula-tions and comparison with the analytical results is performedin Section 4. A short summary of results and conclusions aregiven in Section 5.

2. MODEL AND SOLITARY WAVESOLUTIONIn the one-dimensional case, the generalized NLS equationwith variable diffraction (or dispersion) and spatially-modulated nonlinearity is given by the following dimension-less equation [6,22–28]:

i∂u∂z

� 12β�z; x� ∂

2u∂x2

� χ�z; x�juj2u� R�z; x�u � 0; (1)

where u�z; x� is the complex envelope amplitude of the opticalfield, z and x are the normalized longitudinal and transversecoordinates, respectively. The function β�z; x� representsthe diffraction coefficient and χ�z; x� the nonlinearity coeffi-cient. In this paper, we consider the spatial modulation ofthe nonlinearity in the form χ�x� � b0 � J2

n�x�, where b0�>0�is a constant, Jn�x� is the n-order Bessel function of the firstkind, and n is an integer or half-integer. R�z; x� describes areal external potential, to be specified. Such a choice, as ex-plained above, is interesting for the general analysis of non-diffracting Bessel beams.

To obtain exact analytical solutions of Eq. (1), we introducea self-similar transformation of the solution [15–24]:

u�z; x� � A�z; x�V�z; X�eiB�z;x�; (2)

where the multivariate self-similar variable X�z; x�, the ampli-tude AV , and the phase B are all functions of z and x. Writingthe amplitude of u as a product of two auxiliary functions al-lows for more freedom in the treatment of Eq. (1). The goal isto transform Eq. (1) into the standard NLS equation with con-stant coefficients but with transformed variables. SubstitutingEq. (2) into Eq. (1) casts this equation into the followingnonlinear equation:

i∂V∂z

� β

2

�∂X∂x

�2 ∂2V∂X2 � χA2jV j2V � 0: (3)

Requiring that

β

�∂X∂x

�2� 1; (4A)

χA2 � σ; (4B)

transforms Eq. (2) into the standard NLS equation

i∂V∂z

� 12∂2V∂X2 � σjV j2V � 0; (5)

where σ � �1. Now one can use the fundamental soliton sol-utions of Eq. (5), namely, the bright fundamental (single-peak)soliton in the focusing nonlinear medium for σ � 1,

V�z; X� � sech�X�eiz2 (6A)

and the dark fundamental soliton in the defocusing nonlinearmedium for σ � −1,

V�z; X� � tanh�X�e−iz (6B)

to construct novel soliton solutions of Eq. (1). To this end, onemust treat Eqs. (3) and (4) in some detail.

Considering Eq. (3), when one substitutes Eq. (2) intoEq. (1), separates its real and imaginary parts, and requiresthe coefficients of V and VX to be zero, then one obtains asystem of partial differential equations for X , A, and B:

∂∂x

�A2 ∂X

∂x

�� 0; (7A)

−

∂B∂z

� β

2A∂2A∂x2

−

β

2

�∂B∂x

�2� R � 0; (7B)

∂B∂x

� −

1β

Xz

Xx; (7C)

∂∂z

A2 � β∂∂x

�A2 ∂B

∂x

�� 0; (7D)

where the subscripts indicate the partial derivatives. FromEq. (7A) we find:

A2 � λ2�z�Xx

; (8A)

where λ�z� (>0) is a real integration constant that may dependon z. Substituting Eq. (8A) into Eq. (7D), and eliminating Bx

using Eq. (7C), one obtains

λzλ−

XxXxz

X2x

� βx2β

Xz

Xx� XzXxx

X2x

� 0; (8B)

Zhong et al. Vol. 30, No. 5 / May 2013 / J. Opt. Soc. Am. B 1277

From Eq. (4A) we have �βx∕2β� � −�Xxx∕Xx�; substituting thisinto Eq. (8B), one finds

X�z; x� � λ�z�F�x� � θ�z�; (9)

where θ�z� is a new integration “constant”. Therefore, fromEqs. (4) and (9), one obtains β � �1∕λ2F2

x� and Fx � σ−1λχ.In other words, the nonlinearity coefficient should separatethe variables: χ�z; x� � �σFx�∕λ.

Concerning the phase, it follows from Eq. (7C) thatB�z; x� � −�1∕2�λλzF2

− λωzF � φ�z�, where ω�z� and φ�z�are another pair of integration constants. Now, using Eq. (7B),the external potential R�z; x� can be expressed in terms offunctions λ�z�, F�x�, and φ�z�:

R�z; x� � −

12λλzzF2

−

3F2xx − 2FxFxxx

8λ2F4x

� φz: (10)

This completes the procedure. To summarize, starting withEq. (1), one casts its solution into a convenient form that de-pends on the self-similarity variable X (and the original vari-ables x and z). This variable depends on another auxiliaryvariable F�x� � σ−1λ

Rχ�z; x�dx and two arbitrary integration

constants λ�z� and θ�z�. The form of the auxiliary variable F iscrucial in the procedure: it depends in a straightforward fash-ion on the nonlinearity χ. On the other hand, the amplitudeand the phase of u in turn depend in a simple manner on Fand another two integration “constants” ω�z� and φ�z�. Theprice to pay in this procedure is the connection betweenthe diffraction coefficient β�z; x� and the nonlinearity coeffi-cient χ�z; x�: β�z; x� � 1∕λ4χ2. However, this connection maybe understood as an integrability condition on the method ofobtaining the particular solutions described here. Anotherconsequence is the appearance of the external potentialR�z; x� in a prescribed form, given by Eq. (10). In other words,the method of solution places specific constraints on themedium in which the solutions found can propagate as

solitary waves. The reward is in obtaining explicit analyticalsolutions that involve the modulated nonlinearity—in thiscase Bessel functions—and an external potential that alsoexplicitly depends on the nonlinearity.

With the solutions of the standard NLS equation [Eq. (6)]given above, one can obtain the particular analytical solitarywave solutions of Eq. (1). As can be seen, Eqs. (2) and (6A)lead to the bright solitary solution of Eq. (1):

uB�z; x� ��λ

jFxj

ssech�λF � θ�ei�−1

2λλzF2−λωzF�z

2�φ�; (11A)

while Eq. (6B) leads to the dark solitary wave solution:

uD�z; x� ��λ

jFxj

stanh�λF � θ�ei�−1

2λλzF2−λωzF−z�φ�; (11B)

where, as mentioned, F�x� � σ−1λRχ�z; x�dx. With Eqs. (11A)

and (11B), as long as we choose appropriately the form of thearbitrary integration functions λ�z�, θ�z�, ω�z�, and φ�z�, andthe nonlinearity coefficient χ�z; x�, we obtain analytical soli-tary wave solutions to the generalized NLS equation with var-iable spatially modulated diffraction and nonlinearitycoefficients. It is worth mentioning that the solutions obtainedinvolve the nonlinearity coefficient explicitly—albeit in a com-plicated form—that points to a way in which Bessel functionsmay appear in both the modulation of nonlinearity and in thesoliton solutions.

3. EXAMPLES OF SOLITARY WAVESFrom the Eqs. (11A) and (11B), one can conclude thatboth bright solitary wave solutions in the focusing nonlinearmedium for σ � 1 and dark solitary wave solutions in thedefocusing nonlinear medium for σ � −1 can propagate inthe given nonlinear medium. Recall that the arbitraryfunctions λ�z� and θ�z� are also very important in our

Fig. 1. Comparison of the nonlinearity coefficient χ�x� and the diffraction coefficient β�x�, for different n. (a) Nonlinearity coefficient with theparameters b0 � 0.3, λ � 10. (b) Diffraction coefficient with the parameter b0 � 0.1.


solution procedure, as they modulate both the amplitudeand the phase. The similarity variable X�z; x� of the generalsolution in Eqs. (11) is also affected by λ�z� and θ�z�, as wellas by R�z; x� given in Eq. (10). Functions ω�z� and φ�z� affectonly the phase, and as such do not influence intensitydistributions. We have the freedom in selecting these para-metric functions appropriately, according to some actualphysical requirements, so as to improve the solitary wavecharacteristics.

In this paper, we consider the nonlinearity coefficientinvolving Bessel functions: χ�x� � b0 � J2

n�x� with b0 > 0,namely, Fx � σ−1λ⌊b0 � J2

n�x�⌋, where Jn�x� is the n-orderBessel function of the first kind and n is an integer or half-in-teger. As mentioned above, such a choice is of interest whenthe importance of Bessel functions in the appearance of

diffraction-free optical beams is taken into account. Consider-ing that different choices of the parameter functions λ�z� andθ�z� brings different solutions; we single out two characteris-tic examples:

λ�z� � λ0; θ�z� � a cos�ω0z�; (12A)

where λ0 (≠ 0) is a constant, a ∈ �0; 1�, and ω0 ≠ 0, and

θ�z� � θ0; λ�z� � λ0; (12B)

where θ0 is a constant. Such a choice is realistic, because itavoids the appearance of singularities. The diffraction andnonlinearity coefficients corresponding to the coefficientsχ�x� � b0 � J2

n�x� and β�z; x� � �1∕λ4�b0 � J2n�x��2� are shown

Fig. 2. Intensity distributions of the solitary wave (left column) and the profiles of the external potential (right column) when n � 1∕2; theparameters are as in the text. (a) Bright soliton and (b) Dark soliton.

Fig. 3. Breather solitons (left column) and the external potentials (right column) for n � 1∕2, for the parameters from Eq. (12B). The figure setupis as in Fig. 2.


in Fig. 1. As seen, there is no divergence, but there existmaxima and minima. The amplitudes show oscillationsalong the transverse x direction.

First, we investigate the Bessel function with half-integer n,n � 1∕2, in which the solitary waves possess a single peakwhen the parameter functions are chosen according toEq. (12A). Figure 2 shows the spatial structure of intensity dis-tributions I � juj2 for the solitary wave solution [Eqs. (11A)and (11B)] and the profile of the external potential R�z; x� forthe parameter values λ0 � 1, b0 � 0.6, a � 1, ω0 � 1, andφ � 0. In this case, the parameter σ � 1, which denotes thefocusing nonlinear medium and the bright solitary wave, asseen in Fig. 2(a). In the opposite case, with the parameterσ � −1, the plot displays the dark solitary wave, see Fig. 2(b);this corresponds to the defocusing nonlinear medium. As seenin Fig. 2, the intensity distributions of the solitary wave showbreathing oscillations. Actually, the formation of a breather is

the result of the periodic modulation function cos�ω0z�. Thecorresponding external potential function R�z; x� is displayedin the right column of the same figure; R�x� is negative every-where for the bright solitary wave. Since we have chosen theparameter λ�z� to be constant, from Eq. (10) we find thatR�z; x� is independent of the propagation distance z, if φ�z�is also chosen constant.

As we have demonstrated, the breather solution exists byselecting the parameters according to Eq. (12A) for half-integer Bessel function. Here we consider another examplefor the following set of parameters: λ0 � 1, n � 1∕2,θ�z� � 0, and φ�z� � 0, by choosing the modulation accordingto the condition in Eq. (12B). In Fig. 3, we illustrate the cor-responding simple breather; the top row represents the brightsolitary wave and the bottom row represents the dark solitarywave. The right column of Fig. 3 depicts the correspondingprofiles of the external potential.

Fig. 4. Intensity of the solitary waves (11)—bright soliton (top row) and dark soliton (bottom row)—for different values of n; other parametersb0 � 0.3, λ0 � 1, a � 1, θ � 0, ω0 � 2. The left column is the intensity distribution, the right column the external potential.


Next, we study the solutions generated by Eqs. (11) and theexternal potential from Eq. (10) for integer n. In Fig. 4, wedisplay the corresponding spatial distributions of the solitarywave intensity �I � juj2� for different values of n, according tothe solution [Eqs. (11A) and (11B)] and the parameter func-tions (12A). The plots in Fig. 4(a) correspond to n � 0,λ0 � 1, b0 � 0.3, a � 1, and ω0 � 2, while Fig. 4(b) corre-sponds to n � 2.

The respective external potential R�z; x� is shown in theright column of the same figure, for θ � 0. We produce brightand dark solitary waves by adjusting the parameters a and ω0;this suggests the possibility of controlling the amplitude of thesolitary wave oscillation and the curvature of its trajectory.

Overall, it is possible to use parameters in different ways,to control the propagation of solitary waves. We conclude thatthe frequency of oscillations of the solitary wave increaseswith the increase in ω0 and the amplitude of peak oscillationincreases with the increase in the modulation amplitude,according to Eq. (12A).

To obtain new solitary wave structures, we choose func-tions θ�z� and λ�z� according to Eq. (12B) and choose differentvalues of n. A breather presented in Fig. 5(a) is for the brightsolitary wave (11A) with n � 0, where the parameters areλ0 � 1 and b0 � 0.3. The external potential correspondingto the functions θ�z� and λ�z�, given by Eq. (12B), is shownin the right column of the same figure. Similarly, we can

Fig. 5. Distribution plots of the bright and dark breathers; the parameters are as in the text. The setup is as in Fig. 3.

Fig. 6. Comparing the analytical solution with the numerical simulation at z � 80. The left column represents the analytical solution given byEqs. (11), while the right column depicts the numerical simulation of Eq. (1) with w � 10. (a) Bright soliton and (b) Dark soliton.


construct a dark breather for Eq. (11B) with the same param-eters as for the bright solitary wave; this solitary distribution ispresented in Fig. 5(b).

4. NUMERICAL SIMULATIONSSolitary waves associated with the Kerr-type variable nonli-nearity preserve their shape; however their stability is notguaranteed because of the nonintegrable nature of the under-lying generalized NLS equation. To confirm the validity of sol-utions in Eqs. (11) and to test their stability, we perform adirect numerical simulation and compare analytical solutionswith the numerical ones. We perform numerical simulationsby using the split-step beam propagation method [10,38]and adding 10% white noise to the initial condition; we setthe order of the Bessel function to n � 5∕2, to have a moreoscillating case. We consider the input pulse from Eq. (11).Figure 6(a) shows the comparison of the exact bright solutiongiven by Eq. (11A) with the result of numerical simulation. Wealso consider evolution of a dark soliton solution, given byEq. (11B), imposed on a finite super-Gaussian backgroundpulse [39] u0�z � 0; x� � exp�−x4∕w4�, where w is a big num-ber (of the order of 10), as shown in Fig. 6(b). Such a widebackground pulse is still small at the transverse boundary,allowing the use of the split-step beam propagation method.From the figure we see that the dark soliton riding on a finitebackground retains its soliton character even when the finitewhite noise is included.

It is also seen from Fig. 6 that the analytical solution is con-sistent with the numerical simulation. It is evident that thenoise does not cause the collapse or the dispersal of thesolitary wave and that the stability of the solitary waves overprolonged propagation distances is there.

5. CONCLUSIONSIn conclusion, we have utilized the multivariate self-similartransformation to construct explicit solutions of the NLS equa-tion with the spatially Bessel-modulated nonlinearity and ex-ternal potential. The results show that the shape of the solitarywaves can be controlled by the nonlinearity coefficient. It isdemonstrated that the breathing behavior of solitary wavescan be obtained by appropriately selecting the parameterfunctions λ�z� and ω�z�. The similarity method can be ex-tended to higher dimensional NLS equations along the linesdescribed in [40–43] to exhibit multidimensional bright anddark solitary wave structures.

ACKNOWLEDGMENTSThis work was supported by the National Natural ScienceFoundation of China under Grant No. 61275001. The workat the Texas A&M University at Qatar was supported bythe NPRP 09-462-1-074 project of the Qatar National ResearchFund.

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Solitary waves in the nonlinear Schrödinger equation with spatially modulated Bessel nonlinearity