Cite this Article
Kang Zhou-Zheng, Xia Tie-Cheng, Ma Xi. Multi-soliton solutions for the coupled modified nonlinear Schrödinger equations via Riemann–Hilbert approach. Chinese Physics B, 2018, 27(7): 070201  
Permissions
Multi-soliton solutions for the coupled modified nonlinear Schrödinger equations via Riemann–Hilbert approach
Kang Zhou-Zheng1, 2, Xia Tie-Cheng1, †, Ma Xi1
Department of Mathematics, Shanghai University, Shanghai 200444, China
College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, China

 

† Corresponding author. E-mail: xiatc@shu.edu.cn

Project supported by the National Natural Science Foundation of China (Grant Nos. 61072147 and 11271008).

Abstract

The coupled modified nonlinear Schrödinger equations are under investigation in this work. Starting from analyzing the spectral problem of the Lax pair, a Riemann–Hilbert problem for the coupled modified nonlinear Schrödinger equations is formulated. And then, through solving the obtained Riemann–Hilbert problem under the conditions of irregularity and reflectionless case, N-soliton solutions for the equations are presented. Furthermore, the localized structures and dynamic behaviors of the one-soliton solution are shown graphically.

PACS: 02.30.Jr;02.30.Ik;05.45.Yv
Keyword:coupled modified nonlinear Schrödinger equations;Riemann-Hilbert approach;multi-soliton solutions
1. Introduction

As is well known, the coupled nonlinear Schrödinger type equations[1–3] arise as models to describe different nonlinear waves in an ocean or nonlinear optical fibers, one of which is the coupled modified nonlinear Schrödinger (CMNS) equations

This system describes the pulse propagation in picosecond or femtosecond regime of the birefringent optical fibers.[4–6] Here u and v represent the slowly varying envelopes for two polarizations in the electric fields, the real parameters δ and γ denote the measure of cubic nonlinear strength and derivative cubic nonlinearity, respectively. In particular, setting γ = 0, the CMNS system (1) is reduced to the Manakov equations. Taking γ = 0 as well as δ = 2, the system (1) is called the celebrated Manakov equations.[7] In the case of δ = 0, the CMNS system (1) is changed into the coupled derivative nonlinear Schrödinger equations.[8] There have been plenty of researches on the CMNS system (1). For example, Zhang et al.[4] applied the Hirota’s bilinear method to obtain the bright one- and two-soliton solutions of the CMNS system (1). The complete or partial energy switching in this coupled system was shown as well by the collision dynamics of the bright vector solitons. In Ref. [5], the integrability was proved by finding a new type of Lax pair and infinitely many conservation laws. In addition, the dark and anti-dark vector soliton solutions were obtained by the Hirota’s bilinear method.

Due to the profound theoretical and practical significance, the study of multi-soliton solutions of nonlinear partial differential equations has always been a hot topic. To date, a variety of approaches have been well established to investigate the multi-soliton solutions of nonlinear partial differential equations, such as the inverse scattering method,[9] the Hirota’s bilinear method,[10] the Darboux transformation method,[11–13] the Riemann–Hilbert approach,[14] the KP hierarchy reduction method,[15–17] the generalized unified method,[18,19] and so forth. The Riemann–Hilbert approach as a very powerful approach in dealing with nonlinear partial differential equations with initial-boundary value conditions has attracted more and more attention in recent years. By applying the Riemann–Hilbert method, a number of nonlinear partial differential equations have been studied systematically, including the coupled derivative Schrödinger equation,[20] the derivative nonlinear Schrödinger-type equation,[21] the general coupled nonlinear Schrödinger equations,[22] the generalized Camassa–Holm equation[23] with cubic and quadratic nonlinearity, the generalized Sasa–Satsuma equation,[24] the coupled Sasa–Satsuma equation,[25] the two-component Gerdjikov–Ivanov equation,[26] the coupled modified Korteweg–de Vries equation,[27] the coupled higher-order nonlinear Schrödinger equations, and so on.

The aim of the current study is to seek multi-soliton solutions for the coupled modified nonlinear Schrödinger equations (1). The rest of the paper is arranged as follows. In Section 2, we use the spectral problem of the Lax pair related to the CMNS system (1) to formulate a Riemann–Hilbert problem. In Section 3, we discuss the construction of multi-soliton solutions by using the formalism of the Riemann–Hilbert problem. Finally, some discussions and conclusions are given.

2. The Riemann–Hilbert problem

This section is devoted to formulating a Riemann–Hilbert problem for the CMNS system (1). The Lax pair of the CMNS system (1) reads

Here ψ = (ψ1,ψ2,ψ3)T, ς ∈ ℂ is a spectral parameter, Λ = diag(−1,1,1),
where the asterisk stands for the complex conjugation, and

For the sake of simplicity, we only discuss the particular case of δ = 2 and γ = 1. Therefore, the Lax pair (2) can be written as

where
By introducing a new matrix spectral function μ = μ(x,t;ς),
the Lax pair (3) is then converted into
where [Λ,μ] = Λμ − μΛ is the commutator.

Now, considering the direct scattering process, we construct two matrix Jost solutions for the spectral problem (4a)

meeting the asymptotic conditions
where each [μ±]l (l = 1,2,3) denotes the l-th column of the matrices μ± respectively, I is the 3×3 identity matrix, and μ± are uniquely determined by the Volterra integral equations
Through the direct analysis of Eq. (6), it can be seen that [μ+]1, [μ−]2, and [μ−]3 are analytic for and continuous for , while [μ−]1, [μ+]2, and [μ+]3 are analytic for and continuous for , where

Due to the Abel’s identity and trU1 = 0, the determinants of μ± are constants for all x. Based on the asymptotic conditions (5), we obtain

As to the original spectral problem (3), there exist two fundamental matrix solutions μ− E and μ+ E, which are linearly associated with a 3 × 3 scattering matrix S(ς) = (skj)3 × 3,
where E = e−i(ς2+1)Λx. This gives rise to
Furthermore, we find from the property of μ− that s11 accepts analytic extensions to and skj (k,j = 2,3) extend to analytically.

A Riemann–Hilbert problem desired for the CMNS system (1) involves two matrix functions: one is analytic in and the other is analytic in . Define the first matrix function

which is an analytic function of ς in . And then, we can study the very large-ς asymptotic behavior of H1, which has the asymptotic expansion
Inserting Eq. (9) into Eq. (4a) and comparing the coefficients of ς directly bring about
from which we have , namely,

In order to set up a Riemann–Hilbert problem for the CMNS system (1), it is essential for us to construct a matrix function H2 that is analytic for ς in . Actually, we only need to consider the inverse matrices of μ±

in which each (l = 1, 2, 3) denotes the l-th row of the matrices , respectively. Moreover, meet the equation
which is called the adjoint scattering equation of Eq. (4a). From Eq. (7), it is easy to see that
here R(ς) = (rkj)3×3 = S−1(ς). The matrix function H2 which is analytic in is introduced in terms of
In a similar process as H1, we can obtain the very large-ς asymptotic behavior of H2,

Substitution of μ− = ([μ−]1,[μ−]2,[μ−]3) along with μ+ = ([μ+]1,[μ+]2,[μ+]3) into Eq. (7) gives

from which the expression of [μ−]1 reads
Hence, H1 is of the form

On the other hand, through carrying as well as into Eq. (10), we derive

from which we can express as
As a result, H2 takes the form

Having discussed the construction of two matrix functions H1 and H2 which are analytic in and , respectively, we are in a position to present a Riemann–Hilbert problem for the CMNS system (1). Here we denote that the limit of H1 is H+ when approaches ℝ ∪ iℝ and the limit of H2 is H− when approaches ℝ ∪ iℝ, based on which a Riemann–Hilbert problem can be set up as follows:

whose canonical normalization conditions are
and r11s11+r12s21+r13s31 = 1.

3. Multi-soliton solutions

Having established the Riemann–Hilbert problem for the system (1), we will now move on to solve the Riemann–Hilbert problem (13) that is assumed to be irregular. The irregularity means that both det H1 and det H2 possess certain zeros in their analytic domains. By the definitions of H1 and H2 as well as the scattering relationship (7), we have

From these two equalities we find the facts that the zeros of det H1 and det H2 are the same as s11 and r11, respectively.

Let us now turn to discuss the characteristics of the zeros. As regards the potential matrix U1 possessing the symmetry condition

where the superscript “†” represents the Hermitian of a matrix, we obtain
According to Eq. (7), we have
which gives the following relationships
It is apparent from equality (17) that each zero ±ςk of s11 generates each zero of r11 correspondingly. From the relationships (14) and (17), we have

In addition, there also exists the symmetry relationship for the potential matrix U1

where Λ = diag(−1,1,1). By use of Eq. (19), we have
which reveals the relationships
The equality (23) shows that each zero ςk of s11 is accompanied by another zero −ςk, i.e., s11(±ςk) = 0. Here we point out that equations (20) and (23) yield the property
At this point, we posit that det H1 has 2N simple zeros {ςj} (1 ≤ j ≤ 2N) in , where
and det H2 has 2N simple zeros (1 ≤ j ≤ 2N) in , where

In our discussion, the continuous scattering data s21, s31 and the discrete scattering data ςj, , νj, are very essential. To determine νj and , the following two equations are needed:

where νj and denote nonzero column vectors and row vectors, respectively. Without loss of generality, taking advantage of Eqs. (18), (24), (25) as well as (26), we find

Now we shall obtain the spatial evolutions of the vectors νj. For this purpose, performing the x-derivative with respect to Eq. (25) directly results in

in which νj,0 is independent of x.

Here we examine the Riemann–Hilbert problem (13) with reflectionless case, that is s21 = s31 = 0. We introduce a 2N × 2N matrix M whose entries are

If we assume that the inverse matrix of M exists, then the solution for the Riemann–Hilbert problem (13) can be described as

With the aid of Eq. (30), we are going to restructure the potentials u and v. As a matter of fact, we need to consider the asymptotic expansion of H1,

Carrying expression (31) into Eq. (4a) yields
Consequently, a direct calculation shows the potential functions
where and are the (1,2)-entry and (1,3)-entry of matrix , respectively. Whereas, the can be obtained from Eq. (30) as

So far we merely focus on solving the CMNS system (1) at a fixed time. To seek solutions of the system (1) at any time, it is required to reveal the temporal evolutions of the scattering data. By means of Eqs. (4b) and (7) as well as the sufficiently fast decay of u,v on the boundary, we obtain

which leads to the temporal evolutions
Taking account of Eq. (4b), we gain
Thus, νj and are generated based on Eqs. (27)–(29) as
Here νj,0 is a complex column vector that is independent of x and t. In order to obtain the multi-soliton solutions to the CMNS system (1) conveniently, we make the assumption νj,0 = (αj,βj, γj)T, 1 ≤ j ≤ N.

After summarizing the above results, the expression of N-soliton solutions for the CMNS system (1) can be immediately obtained as follows:

where
and

At the end of this section, two simplest cases arise by taking N = 1 and N = 2 in the N-soliton solution formula (34). Suppose that N = 1 and ς1 = a1 + ib1, a1b1 < 0,

the one-soliton solution for the CMNS system (1) follows at once
in which ξ1 = −i((a1 + ib1)2 + 1)x + 2i((a1 + ib1)2 + 1)2t, and
The solution is graphically shown in Figs. 1–3.

Fig. 1. (color online) One-soliton solution (35) with α1 = 1, β1 = 1, γ1 = 2, a1 = −0.1, b1 = 1: (a) perspective view of modulus of u, (b) perspective view of modulus of v. (c) The soliton along the x axis with different time in panel (a). (d) The soliton along the x axis with different time in panel (b).
Fig. 2. (color online) One-soliton solution (35) with α1 = 1, β1 = 1, γ1 = 2, a1 = −0.1, b1 = 1 : (a) perspective view of real part of u, (b) perspective view of imaginary of u, (c) perspective view of real part of v, (d) perspective view of imaginary of v.
Fig. 3. (color online) One-soliton solution (35) with α1 = 1, β1 = 1, γ1 = 2, a1 = −0.1, b1 = 1. (a) The soliton along the x axis with different time in Fig. 2(a); (b) The soliton along the x axis with different time in Fig. 2(b); (c) The soliton along the x axis with different time in Fig. 2(c); (d) The soliton along the x axis with different time in Fig. 2(d).

By taking N = 2 and assuming

the two-soliton solution for the CMNS system (1) is given by
where M = (mkj)4×4 and its entries are
with

4. Discussion and conclusion

In this work, we have successfully constructed multi-soliton solutions for the coupled modified nonlinear Schrödinger equations based on the Riemann–Hilbert approach. To sum up, the process of the Riemann–Hilbert approach can be generally outlined as follows: first, we deduce a Riemann–Hilbert problem for the CMNS system (1) via analyzing the related spectral problem. Second, by studying the presented Riemann–Hilbert problem with irregularity and reflectionless case, we gain the solution of the Riemann–Hilbert problem explicitly. After restructuring the potentials and considering the temporal evolutions of the scattering data, we finally derive the N-soliton solutions for the coupled modified nonlinear Schrödinger equations. The localized structures and dynamic behaviors of one-soliton solution are displayed vividly by figures.

Reference
[1] Zhao L C Guo B L Ling L M 2016 J. Math. Phys. 57 043508
[2] Wang X Li Y Q Chen Y 2014 Wave Motion 51 1149
[3] Xu T Chen Y Lin J 2017 Chin. Phys. 26 120201
[4] Zhang H Q Tian B LU X Li H Meng X H 2009 Phys. Lett. 373 4315
[5] Li M Tian B Liu W J Jiang Y Sun K 2010 Eur. Phys. J. 59 279
[6] Hu B B Xia T C 2018 Math. Meth. Appl. Sci. 2012 1
[7] Manakov S 1974 Sov. Phys. JETP 38 248
[8] Morris H C Dodd P K 1979 Phys. Scr. 20 505
[9] Gardner C S Greene J M Kruskal M D Miura R M 1976 Phys. Rev. E 19 1095
[10] Zhang S Tian C Qian W Y 2016 Pramana J. Phys. 86 1259
[11] Wang X Chen Y 2014 Commun. Theor. Phys. 61 423
[12] Zhaqilao 2013 Chin. Phys. 22 040201
[13] Ling L M Liu Q P 2010 J. Phys. 43 434023
[14] Wang D S Wang X L 2018 Nonlinear Anal. Real World Appl. 41 334
[15] Han Z Chen Y 2017 Z. Naturforsch. 72 745
[16] Han Z Chen Y 2017 Chin. Phys. Lett. 34 070202
[17] Han Z Chen Y Chen J C 2017 J. Phys. Soc. Jpn. 86 074005
[18] Osman M S Wazwaz A M 2018 Appl. Math. Comput. 321 282
[19] Osman M S 2017 Comput. Math. Appl. 75 1
[20] Guo B L Ling L M 2012 J. Math. Phys. 53 073506
[21] Geng X G Nie H 2017 Math. Meth. Appl. Sci. 1
[22] Wang D S Zhang D J Yang J K 2010 J. Math. Phys. 51 023510
[23] Wang Z Qiao Z J 2016 J. Math. Phys. 57 073505
[24] Geng X G Wu J P 2016 Wave Motion 60 62
[25] Wu J P Geng X G 2017 Commun. Theor. Phys. 67 527
[26] Zhang Y S Cheng Y He J S 2017 J. Nonlinear Math. Phys. 24 210
[27] Wu J P Geng X G 2017 Commun. Nonlinear Sci. Numer. Simulat. 53 83