Integrable variable-coefficient derivative Spin-1 Gross–Pitaevskii equations and their explicit solutions

2021 ◽  
Author(s):  
Ting Su ◽  
Junhong Yao ◽  
Yanan Huang
Keyword(s):  
Variable Coefficient ◽  
Separation Of Variables ◽  
Lax Pair ◽  
Explicit Solutions ◽  
Dressing Method

Based on the generalized dressing method, we propose a new integrable variable coefficient Spin-1 Gross–Pitaevskii equations and derive their Lax pair. Using separation of variables, we have derived explicit solutions of the equations. In order to analyze the characteristic of derived solution, the graphical wave of the solutions is plotted with the aid of Matlab.

scholarly journals A New Integrable Variable-Coefficient 2+1-Dimensional Long Wave-Short Wave Equation and the Generalized Dressing Method

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Ting Su ◽  
Hui Hui Dai
Keyword(s):  
Wave Equation ◽  
Variable Coefficient ◽  
Separation Of Variables ◽  
Short Wave ◽  
Lax Pair ◽  
Explicit Solutions ◽  
Dressing Method ◽  
Long Wave

Based on the generalized dressing method, we propose a new integrable variable-coefficient 2+1-dimensional long wave-short wave equation and derive its Lax pair. Using separation of variables, we have derived the explicit solutions of the equation. With the aid of Matlab, the curves of the solutions are drawn.

scholarly journals ON THE APPLICATION OF A GENERALIZED VERSION OF THE DRESSING METHOD TO THE INTEGRATION OF VARIABLE COEFFICIENT N-COUPLED NONLINEAR SCHRÖDINGER EQUATION

2012 ◽  
Vol 19 (04) ◽  
pp. 1250028
Author(s):  
TING SU ◽  
HUIHUI DAI ◽  
XIAN GUO GENG
Keyword(s):  
Optical Fibers ◽  
Variable Coefficient ◽  
High Order ◽  
Explicit Solutions ◽  
Compatibility Conditions ◽  
Dressing Method ◽  
Integrable Equations ◽  
Nonlinear Schrödinger ◽  
Coupled Nonlinear Schrödinger Equation ◽  
Hirota Equations

N-coupled nonlinear Schrödinger (NLS) equations have been proposed to describe N-pulse simultaneous propagation in optical fibers. When the fiber is nonuniform, N-coupled variable-coefficient NLS equations can arise. In this paper, a family of N-coupled integrable variable-coefficient NLS equations are studied by using a generalized version of the dressing method. We first extend the dressing method to the versions with (N + 1) × (N + 1) operators and (2N + 1) × (2N + 1) operators. Then, we obtain three types of N-coupled variable-coefficient equations (N-coupled NLS equations, N-coupled Hirota equations and N-coupled high-order NLS equations). Then, the compatibility conditions are given, which insure that these equations are integrable. Finally, the explicit solutions of the new integrable equations are obtained.

Dark solitons, Lax pair and infinitely-many conservation laws for a generalized (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation in the inhomogeneous Heisenberg ferromagnetic spin chain

2017 ◽  
Vol 31 (03) ◽  
pp. 1750013 ◽  
Author(s):  
Xue-Hui Zhao ◽  
Bo Tian ◽  
De-Yin Liu ◽  
Xiao-Yu Wu ◽  
Jun Chai ◽  
...  
Keyword(s):  
Conservation Laws ◽  
Spin Chain ◽  
Variable Coefficient ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Nonlinear Schrodinger Equation ◽  
Dark Solitons ◽  
Heisenberg Ferromagnetic Spin Chain ◽  
Dimensional Variable

Under investigation in this paper is a generalized (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation in an inhomogeneous Heisenberg ferromagnetic spin chain. Lax pair and infinitely-many conservation laws are derived, indicating the existence of the multi-soliton solutions for such an equation. Via the Hirota method with an auxiliary function, bilinear forms, dark one-, two- and three-soliton solutions are derived. Propagation and interactions for the dark solitons are illustrated graphically: Velocity of the solitons is linearly related to the coefficients of the second- and fourth-order dispersion terms, while amplitude of the solitons does not depend on them. Interactions between the two solitons are shown to be elastic, while those among the three solitons are pairwise elastic.

A BRIEF SURVEY ON CONSTRUCTING HOMOCLINIC STRUCTURES OF SOLITON EQUATIONS

2006 ◽  
Vol 16 (10) ◽  
pp. 2799-2813 ◽  
Author(s):  
RANCHAO WU ◽  
JIANHUA SUN
Keyword(s):  
Nonlinear Partial Differential Equations ◽  
Plane Waves ◽  
Homoclinic Orbits ◽  
Lax Pair ◽  
Dressing Method ◽  
Soliton Equations ◽  
Coupled Equations ◽  
Mathematical Proofs ◽  
Extended Application ◽  
Uniform Plane

To give rigorous mathematical proofs of chaotic behaviors in a given system, it is necessary to identify the homoclinic structures in the system. In this tutorial review, methods for constructing explicit solutions for nonlinear partial differential equations are presented, with more emphasis placed on those utilizing complete integrability associated with soliton equations. As an extended application, homoclinic orbits to spatial uniform plane waves of coupled modified nonlinear Schrödinger equations are obtained via the dressing method. During the procedure, it is necessary to introduce the Lax pair for these coupled equations, as well as its Floquet spectral analysis and corresponding Bloch functions.

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