The <i>N</i>‐coupled higher‐order nonlinear Schrödinger equation: Riemann‐Hilbert problem and multi‐soliton solutions

Yang, Jun; Tian, Shou‐Fu; Peng, Wei‐Qi; Zhang, Tiantian

doi:10.1002/mma.6055

Cited by 38 publications

(15 citation statements)

References 40 publications

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“…In this section, we consider the N-coupled NLSE, 40,44,45,[49][50][51][52][53] with emphasis on the three-coupled NLSE. Similar to the previous case of two-coupled NLSE, we show that a linear combination of the three components of a given seed solution is also a composite solution.…”

Section: Composite Solutions To the N-coupled Nlsementioning

confidence: 99%

Superposition principle and composite solutions to coupled nonlinear Schrödinger equations

Sakkaf

Khawaja²

2020

Math Methods in App Sciences

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We show that the superposition principle applies to coupled nonlinear Schrödinger equations with cubic nonlinearity where exact solutions may be obtained as a linear combination of other exact solutions. This is possible due to the cancelation of cross terms in the nonlinear coupling. First, we show that a composite solution, which is a linear combination of the two components of a seed solution, is another solution to the same coupled nonlinear Schrödinger equation. Then, we show that a linear combination of two composite solutions is also a solution to the same equation. With emphasis on the case of Manakov system of two-coupled nonlinear Schrödinger equations, the superposition is shown to be equivalent to a rotation operator in a two-dimensional function space with components of the seed solution being its coordinates. Repeated application of the rotation operator, starting with a specific seed solution, generates a series of composite solutions, which may be represented by a generalized solution that defines a family of composite solutions. Applying the rotation operator to almost all known exact seed solutions of the Manakov system, we obtain for each seed solution the corresponding family of composite solutions. Composite solutions turn out, in general, to possess interesting features that do not exist in the seed solution. Using symmetry reductions, we show that the method applies also to systems of N-coupled nonlinear Schrödinger equations. Specific examples for the three-coupled nonlinear Schrödinger equation are given.

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Section: Composite Solutions To the N-coupled Nlsementioning

confidence: 99%

Superposition principle and composite solutions to coupled nonlinear Schrödinger equations

Sakkaf

Khawaja²

2020

Math Methods in App Sciences

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“…These solutions can further help to understand natural phenomena and laws. In recent years, more and more mathematical physicists have begun to pay attention to Riemann-Hilbert (RH) approach [9,10], which is a new powerful method for solving integrable linear and nonlinear evolution equations [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. The main idea of this method is to establish a corresponding matrix RH problem on the Lax pair of integrable equations.…”

Section: Introductionmentioning

confidence: 99%

Dynamic behaviors of soliton solutions for a three-coupled Lakshmanan–Porsezian–Daniel model

Lin

Zhang

2022

Nonlinear Dyn

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In this paper, we use the Riemann-Hilbert (RH) approach to examine the integrable three-coupled Lakshmanan-Porsezian-Daniel (LPD) model, which describe the dynamics of alpha helical protein with the interspine coupling at the fourth-order dispersion term. Through the spectral analysis of Lax pair, we construct the higher order matrix RH problem for the three-coupled LPD model, when the jump matrix of this particular RH problem is a 4 × 4 unit matrix, the exact N-soliton solutions of the three-coupled LPD model can be exhibited. As special examples, we also investigate the nonlinear dynamical behaviors of the single-soliton, two-soliton, three-soliton and breather soliton solutions. Finally, an integrable generalized N-component LPD model with its linear spectral problem is discussed.

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“…On the other hand, some of the higher-order spectral problems have to be transformed into RH problem. This approach, developed by Zakharov et al [34], was successively applied to various integrable systems with a single component [3][4][5][8][9][10][11][12][13][16][17][18][19][20][21][24][25][26][27][28][29][30]32,33,[35][36][37][38]. However, to the best of authors' knowledge, only a few studies deal with multi-component problems.…”

Section: Introductionmentioning

confidence: 99%

The Riemann-Hilbert Approach and $N$-Soliton Solutions of a Four-Component Nonlinear Schrödinger Equation

Zhou¹

2021

EAJAM

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A four-component nonlinear Schrödinger equation associated with a 5×5 Lax pair is investigated. A spectral problem is analysed and the Jost functions are used in order to derive a Riemann-Hilbert problem connected with the equation under consideration. N-soliton solutions of the equation are obtained by solving the Riemann-Hilbert problem without reflection. For N = 1 and N = 2, the local structure and dynamic behavior of some special solutions is analysed by invoking their graphic representations.

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The N‐coupled higher‐order nonlinear Schrödinger equation: Riemann‐Hilbert problem and multi‐soliton solutions

Cited by 38 publications

References 40 publications

Superposition principle and composite solutions to coupled nonlinear Schrödinger equations

Superposition principle and composite solutions to coupled nonlinear Schrödinger equations

Dynamic behaviors of soliton solutions for a three-coupled Lakshmanan–Porsezian–Daniel model

The Riemann-Hilbert Approach and $N$-Soliton Solutions of a Four-Component Nonlinear Schrödinger Equation

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