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.

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 On a Vector Modified Yajima–Oikawa Long-Wave–Short-Wave Equation

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 958
Author(s):  
Xianguo Geng ◽  
Ruomeng Li
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Rogue Wave ◽  
Riccati Equations ◽  
Short Wave ◽  
Lax Pair ◽  
Darboux Transformations ◽  
Wave Solutions ◽  
Zero Curvature ◽  
Long Wave

A vector modified Yajima–Oikawa long-wave–short-wave equation is proposed using the zero-curvature presentation. On the basis of the Riccati equations associated with the Lax pair, a method is developed to construct multi-fold classical and generalized Darboux transformations for the vector modified Yajima–Oikawa long-wave–short-wave equation. As applications of the multi-fold classical Darboux transformations and generalized Darboux transformations, various exact solutions for the vector modified long-wave–short-wave equation are obtained, including soliton, breather, and rogue wave solutions.

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.

An integrable equation governing short waves in a long-wave model

2007 ◽  
Vol 463 (2084) ◽  
pp. 1939-1954 ◽  
Author(s):  
M Faquir ◽  
M.A Manna ◽  
A Neveu
Keyword(s):  
Surface Tension ◽  
Gordon Equation ◽  
Wave Model ◽  
Short Wave ◽  
Lax Pair ◽  
Integrable Equation ◽  
Conserved Quantities ◽  
Long Wave ◽  
Tension Term ◽  
Wave Limit

The dynamics of a nonlinear and dispersive long surface capillary-gravity wave model equation is studied analytically in its short-wave limit. We exhibit its Lax pair and some non-trivial conserved quantities. Through a change of functions, an unexpected connection between this classical surface water-wave model and the sine-Gordon (or sinh-Gordon) equation is established. Numerical and analytical studies show that in spite of integrability their solutions can develop singularities and multivaluedness in finite time. These features can be traced to the fact that the surface tension term in the energy involves second-order derivatives. It would be interesting to see in an experiment whether such singularities actually appear, for which surface tension would be specifically responsible.

Interaction of the variable-coefficient long-wave–short-wave resonance interaction equations

2019 ◽  
Vol 33 (01) ◽  
pp. 1850426
Author(s):  
Hui-Xian Jia ◽  
Da-Wei Zuo
Keyword(s):  
Gravity Waves ◽  
Acoustic Waves ◽  
Variable Coefficient ◽  
Soliton Solutions ◽  
Short Wave ◽  
Resonance Interaction ◽  
Bilinear Method ◽  
Ion Acoustic Waves ◽  
Long Wave ◽  
Wave Resonance

Long-wave–short-wave resonance interaction (LSRI) equations have been studied in the plasmas, gravity waves, nonlinear electron-plasma and ion-acoustic waves. By virtue of the bilinear method, two soliton solutions of the variable-coefficient LSRI equations are attained. Interaction of the solitons are studied when the coefficients are taken as the generalized Gauss functions. New types of the soliton interaction are exhibited. Position and width of the disturbances can be controlled.

Solitons, Bäcklund Transformation, Lax Pair, and Infinitely Many Conservation Law for a (2+1)-Dimensional Generalised Variable-Coefficient Shallow Water Wave Equation

2016 ◽  
Vol 71 (1) ◽  
pp. 69-79 ◽  
Author(s):  
Zhong-Zhou Lan ◽  
Yi-Tian Gao ◽  
Jin-Wei Yang ◽  
Chuan-Qi Su ◽  
Da-Wei Zuo
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Conservation Law ◽  
Water Wave ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Bell Polynomials ◽  
Backlund Transformation ◽  
Shallow Water Wave

AbstractUnder investigation in this article is a (2+1)-dimensional generalised variable-coefficient shallow water wave equation, which describes the interaction of the Riemann wave propagating along the y axis with a long-wave propagating along the x axis in a fluid, where x and y are the scaled space coordinates. Bilinear forms, Bäcklund transformation, Lax pair, and infinitely many conservation law are derived based on the binary Bell polynomials. Multi-soliton solutions are constructed via the Hirota method. Propagation and interaction of the solitons are illustrated graphically: (i) variable coefficients affect the shape of the multi-soliton interaction in the scaled space and time coordinates. (ii) Positions of the solitons depend on the sign of wave numbers after each interaction. (iii) Interaction of the solitons is elastic, i.e. the amplitude, velocity, and shape of each soliton remain invariant after each interaction except for a phase shift.

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