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N-soliton solutions for the nonlocal two-wave interaction system via the Riemann–Hilbert method

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Chinese Physics B 2018年第12期27卷 157-163页

作者： Si-Qi Xu xian-guo geng School of Mathematics and Statistics Zhengzhou UniversityZhengzhou 450001China College of Civil Engineering Xinyang Normal UniversityXinyang 464000China

In this paper, a nonlocal two-wave interaction system from the Manakov hierarchy is investigated via the Riemann–Hilbert approach. Based on the spectral analysis of the Lax pair, a Riemann–Hilbert problem for the nonlocal two-wave interaction system is constructed. By discussing the solutions of this Riemann–Hilbert problem in both the regular and nonregular cases, we explicitly present the N-soliton solution formula of the nonlocal two-wave interaction system. Moreover,the dynamical behaviour of the single-soliton solution is shown graphically.

关键词： nonlocal two-wave interaction system Riemann–Hilbert problem N-soliton solutions

An extension of integrable equations related to AKNS and WKI spectral problems and their reductions

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Chinese Physics B 2018年第4期27卷 134-137页

作者： xian-guo geng Yun-Yun Zhai School of Mathematics and Statistics Zhengzhou University Zhengzhou 450001 China

A novel hierarchy of integrable nonlinear evolution equations related to the combined Ablowitz–Kaup–Newell–Segur（AKNS） and Wadati–Konno–Ichikawa（WKI） spectral problems is proposed,from which the Lax pair for a corresponding negative flow and its infinite many conservation laws are ***,a reduction of this hierarchy is discussed,by which a generalized sinh-Gordon equation is derived on the basis of its negative flow.

关键词： integrable extension nonlinear evolution equations infinite conservation laws

Spectral Analysis and Long-time Asymptotics for the Coherently-coupled Nonlinear Schrödinger System

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Acta Mathematica Sinica,English Series 2022年第11期38卷 2090-2114页

作者： Ming Ming CHEN xian guo geng Ke Dong WANG School of Mathematics and Statistics Zhengzhou UniversityZhengzhou 450001P.R.China

On the basis of the spectral analysis of the 4×4 matrix Lax pair,the initial value problem of the coherently-coupled nonlinear Schrödinger system is transformed into a 4×4 matrix Riemann–Hilbert *** using the nonlinear steepest decent method,the long-time asymptotics of the solution of the initial value problem for the coherently-coupled nonlinear Schrödinger system is obtained through deforming the Riemann–Hilbert problem into a solvable model one.

关键词： Coherently-coupled nonlinear Schrödinger system nonlinear steepest descent method long-time asymptotics

Two integrable generalizations of WKI and FL equations: Positive and negative flows, and conservation laws

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Chinese Physics B 2020年第5期29卷 70-73页

作者： xian-guo geng Fei-Ying guo Yun-Yun Zhai School of Mathematics and Statistics Zhengzhou UniversityZhengzhou 450001China

With the aid of Lenard recursion equations, an integrable hierarchy of nonlinear evolution equations associated with a 2 × 2 matrix spectral problem is proposed, in which the first nontrivial member in the positive flows can be reduced to a new generalization of the Wadati–Konno–Ichikawa(WKI) equation. Further, a new generalization of the Fokas–Lenells(FL) equation is derived from the negative flows. Resorting to these two Lax pairs and Riccati-type equations, the infinite conservation laws of these two corresponding equations are obtained.

关键词： integrable generalizations positive flow and negative flow infinite conservation laws

Darboux Transformation for Tzitzeica Equation

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Communications in Theoretical Physics 2006年第4期45卷 577-580页

作者： ZHU Jun-Yi geng xian-guo Department of Mathematics Zhengzhou University. Zhengzhou 450052 China

We present a Darboux transformation for Tzitzeica equation associated with 3 × 3 matrix spectral *** explicit solution of Tzitzeica equation is obtained,

关键词： Darboux transformation nonlinear evolution equation soliton solutions

A New Integrable Symplectic Map of Neumann Type

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Communications in Theoretical Physics 2007年第4期47卷 577-581页

作者： ZHU Jun-Yi geng xian-guo Department of Mathematics Zhengzhou University Zhengzhou 450052 China

By resorting to the nonlinearization approach, a Neumann constraint associated with a discrete 3 × 3 matrix eigenvalue problem is considered. A new symplectic map of the Neumann type is obtained through nonlinearization of the discrete eigenvalue problem and its adjoint one. The generating function of integrals of motion is presented, by which the symplectic reap＇is further proved to be completely integrable in the Liouville sense.

关键词： Neumann constraint symplectic map Liouville integrablility

r-Matrix Structure for a Restricted Flow with Bargmann Constraint

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Communications in Theoretical Physics 2005年第3X期44卷 393-395页

作者： CHEN Jin-Bing geng xian-guo Department of Mathematics Zhengzhou University Zhengzhou 450052 China

This paper deals with the integrability of a finite-dimensional Hamiltonian system linked with the generalized coupled KdV hierarchy. For this purpose the associated Lax representation is presented after an elementary calculation. It is shown that the Lax representation enjoys a dynamical r-matrix formula instead of a classical one in the Poisson bracket on R2N. Consequently the resulting system is proved to be completely integrable in view of its r-matrix structure.

关键词： Poisson bracket Lax representation r-matrix Liouville integrability

New Wronskian Representation of Solution for a Variable-Coefficient Kadomtsev-Petviashvili Equation

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Chinese Physics Letters 2013年第6期30卷 46-49页

作者： WU Jian-Ping geng xian-guo Department of Mathematics Zhengzhou University100 Kexue RoadZhengzhou 450001

Utilizing the Wronskian technique,a new Wronskian representation is proposed for a variable-coefficient Kadomtsev–Petviashvili(vcKP)***,some particular forms of Wronskian determinant solutions,includingN-soliton solu... 详细信息

关键词： equation. soliton rational

Grammian Determinant Solution and Pfaffianization for a (3+1)-Dimensional Soliton Equation

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Communications in Theoretical Physics 2009年第11期52卷 791-794页

作者： WU Jian-Ping geng xian-guo2 Institute of Electronic Technology Information Engineering University Zhengzhou 450004 China Department of Mathematics Zhengzhou University Zhengzhou 450052 China

Based on the Pfaffian derivative formulae,a Grammian determinant solution for a(3+1)-dimensionalsoliton equation is ***,the Pfaffianization procedure is applied for the equation to generate a newcoupled *** last,a Gra... 详细信息

关键词： (3+1)-dimensional soliton equation Grammian determinant solution Pfaffianization Gram-type Pfaffian solution