scholarly journals N-Soliton Solutions of the Coupled Kundu Equations Based on the Riemann-Hilbert Method

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Mengshuang Tao ◽  
Huanhe Dong
Keyword(s):  
Spectral Problem ◽  
Hilbert Problem ◽  
Soliton Solutions ◽  
Species Competition ◽  
Identity Matrix ◽  
Coupled Equations ◽  
Jump Matrix ◽  
As Species ◽  
Riemann Hilbert Problem

The Kundu equation, which can be used to describe many phenomena in physics and mechanics, has crucial theoretical meaning and research value. In previous studies, the single Kundu equation has been investigated by the Riemann-Hilbert method, but few researchers have focused on the coupled Kundu equations. To our knowledge, many phenomena in nature can be only described by coupled equations, such as species competition and signal interactions. In this paper, we discuss N-soliton solutions of the coupled Kundu equations according to the Riemann-Hilbert method. Starting from the spectral problem, the coupled Kundu equations are generated, and the Riemann-Hilbert problem is presented. When the jump matrix of the Riemann-Hilbert problem is the identity matrix, the N-soliton solutions of the coupled Kundu equations can be expressed explicitly.

scholarly journals N-Soliton Solutions for the NLS-Like Equation and Perturbation Theory Based on the Riemann–Hilbert Problem

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 826 ◽  
Author(s):  
Yuxin Lin ◽  
Huanhe Dong ◽  
Yong Fang
Keyword(s):  
Perturbation Theory ◽  
Soliton Solution ◽  
Hilbert Problem ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Nls Equation ◽  
Jump Matrix ◽  
Potential Matrix ◽  
Relationship Of ◽  
Riemann Hilbert Problem

In this paper, a kind of nonlinear Schrödinger (NLS) equation, called an NLS-like equation, is Riemann–Hilbert investigated. We construct a 2 × 2 Lax pair associated with the NLS equation and combine the spectral analysis to formulate the Riemann–Hilbert (R–H) problem. Then, we mainly use the symmetry relationship of potential matrix Q to analyze the zeros of det P + and det P − ; the N-soliton solutions of the NLS-like equation are expressed explicitly by a particular R–H problem with an unit jump matrix. In addition, the single-soliton solution and collisions of two solitons are analyzed, and the dynamic behaviors of the single-soliton solution and two-soliton solutions are shown graphically. Furthermore, on the basis of the R–H problem, the evolution equation of the R–H data with the perturbation is derived.

ASYMPTOTIC EXPANSIONS FOR RIEMANN–HILBERT PROBLEMS

2008 ◽  
Vol 06 (03) ◽  
pp. 269-298 ◽  
Author(s):  
W.-Y. QIU ◽  
R. WONG
Keyword(s):  
Asymptotic Expansion ◽  
Complex Analysis ◽  
Elementary Proof ◽  
Hilbert Problem ◽  
Jump Condition ◽  
Similar Type ◽  
Matrix Solution ◽  
Jump Matrix ◽  
The Matrix ◽  
Riemann Hilbert Problem

Let Γ be a piecewise smooth contour in ℂ, which could be unbounded and may have points of self-intersection. Let V(z, N) be a 2 × 2 matrix-valued function defined on Γ, which depends on a parameter N. Consider a Riemann–Hilbert problem for a matrix-valued analytic function R(z, N) that satisfies a jump condition on the contour Γ with the jump matrix V(z, N). Assume that V(z, N) has an asymptotic expansion, as N → ∞, on Γ. An elementary proof is given for the existence of a similar type of asymptotic expansion for the matrix solution R(z, N), as n → ∞, for z ∈ ℂ\Γ. Our method makes use of only complex analysis.

Riemann–Hilbert approach and N-soliton solutions for the Chen–Lee–Liu equation

2019 ◽  
Vol 33 (02) ◽  
pp. 1950002 ◽  
Author(s):  
Ming-Jun Xu ◽  
Tie-Cheng Xia ◽  
Bei-Bei Hu
Keyword(s):  
Soliton Solution ◽  
Hilbert Problem ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Potential Matrix ◽  
Riemann Hilbert Problem

In this paper, we construct the Riemann–Hilbert problem to the Lax pair of Chen–Lee–Liu (CLL) equation. As far as we know, many researchers have studied various equations with Riemann–Hilbert method before, but no one compared the N-soliton solutions calculated by different symmetries of potential matrix. Using different symmetries of potential matrix, we get two N-soliton solution formulae of the CLL equation. The interesting thing is that we find the equivalence of these two N-soliton solutions.

N-soliton solution for a higher-order Chen–Lee–Liu equation with nonzero boundary conditions

2019 ◽  
Vol 34 (04) ◽  
pp. 2050054 ◽  
Author(s):  
Yi Zhao ◽  
Engui Fan
Keyword(s):  
Boundary Conditions ◽  
Soliton Solution ◽  
Hilbert Problem ◽  
Asymptotic Properties ◽  
Soliton Solutions ◽  
Higher Order ◽  
Third Order ◽  
Third Order Dispersion ◽  
Nonzero Boundary ◽  
Riemann Hilbert Problem

In this paper, the Riemann–Hilbert approach is applied to investigate a higher-order Chen–Lee–Liu equation with third-order dispersion and quintic nonlinearity terms. Based on the analytical, symmetric and asymptotic properties of eigenfunctions, a generalized Riemann–Hilbert problem associated with Chen–Lee–Liu equation with nonzero boundary conditions is constructed. Further, the [Formula: see text]-soliton solution is found by solving the generalized Riemann–Hilbert problem. As an illustrative example, two kinds of one-soliton solutions with different forms of parameters are obtained.

On a Riemann–Hilbert problem for the NLS-MB equations

2021 ◽  
pp. 2150420
Author(s):  
Leilei Liu ◽  
Weiguo Zhang ◽  
Jian Xu
Keyword(s):  
Hilbert Problem ◽  
Soliton Solutions ◽  
Coupled System ◽  
Soliton Interaction ◽  
Nls Equation ◽  
Nonzero Boundary ◽  
Complex Symmetric ◽  
Complex Phenomena ◽  
Symmetric Relations ◽  
Riemann Hilbert Problem

In this paper, we study a coupled system of the nonlinear Schrödinger (NLS) equation and the Maxwell–Bloch (MB) equation with nonzero boundary conditions by Riemann–Hilbert (RH) method. We obtain the formulae of the simple-pole and the multi-pole solutions via a matrix Riemann–Hilbert problem (RHP). The explicit form of the soliton solutions for the NLS-MB equations is obtained. The soliton interaction is also given. Furthermore, we show that the multi-pole solutions can be viewed as some proper limits of the soliton solutions with simple poles, and the multi-pole solutions constitute a novel analytical viewpoint in nonlinear complex phenomena. The advantage of this way is that it avoids solving the complex symmetric relations and repeatedly solving residue conditions.

Riemann–Hilbert Problem and Multi-Soliton Solutions of the Integrable Spin-1 Gross–Pitaevskii Equations

2019 ◽  
Vol 74 (2) ◽  
pp. 139-145 ◽  
Author(s):  
Xiu-Bin Wang ◽  
Bo Han
Keyword(s):  
Spectral Analysis ◽  
Nonlinear Optical ◽  
Light Transmission ◽  
Hilbert Problem ◽  
Soliton Solutions ◽  
Nonlinear Phenomena ◽  
Optical Fibres ◽  
Collision Dynamics ◽  
Integrable Spin ◽  
Riemann Hilbert Problem

AbstractUnder investigation in this article is the integrable spin-1 Gross–Pitaevskii (SGP) equations, which can be used to describe light transmission in bimodal nonlinear optical fibres. The spectral analysis with 4 × 4 Lax pairs is performed for the integrable SGP equations, from which a Riemann Hilbert problem is formulated. Furthermore, N-soliton solutions of this integrable SGP equations are expressed in terms of solutions of the Riemann–Hilbert problem by using the Plemelj formulae. Finally, collision dynamics between two solitons is also analyzed. Our results can be used to enrich and explain some related nonlinear phenomena.

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