Multi-soliton Collisions and Bäcklund Transformations for the (2+1)-dimensional Modified Nizhnik–Novikov–Vesselov Equations

2015 ◽  
Vol 70 (8) ◽  
pp. 629-635
Author(s):  
Xi-Yang Xie ◽  
Bo Tian ◽  
Yu-Feng Wang ◽  
Wen-Rong Sun ◽  
Ya Sun
Keyword(s):  
Acoustic Waves ◽  
Water Waves ◽  
Soliton Solutions ◽  
Bäcklund Transformations ◽  
Bell Polynomials ◽  
Ion Acoustic Waves ◽  
Backlund Transformations ◽  
Kdv Type Equations ◽  
Binary Bell Polynomials ◽  
Soliton Collisions

AbstractThe Korteweg–de Vries (KdV)-type equations can describe the shallow water waves, stratified internal waves, ion-acoustic waves, plasma physics and lattice dynamics, while the (2+1)-dimensional Nizhnik–Novikov–Vesselov equations are the isotropic extensions of KdV-type equations. In this paper, we investigate the (2+1)-dimensional modified Nizhnik–Novikov–Vesselov equations. By virtue of the binary Bell polynomials, bilinear forms, multi-soliton solutions and Bäcklund transformations are derived. Effects of some parameters on the solitons and monotonic function are graphically illustrated. We can observe the coalescence of the two solitons in their collision region, where their shapes change after the collision.

Multi-Soliton Solutions of the Generalized Sawada–Kotera Equation

2016 ◽  
Vol 71 (4) ◽  
pp. 305-309 ◽  
Author(s):  
Da-Wei Zuo ◽  
Hui-Xia Mo ◽  
Hui-Ping Zhou
Keyword(s):  
Acoustic Waves ◽  
Water Waves ◽  
Soliton Solutions ◽  
Elastic Interaction ◽  
Bell Polynomials ◽  
Nonlinear Phenomena ◽  
Shallow Water Waves ◽  
Ion Acoustic Waves ◽  
Soliton Propagation ◽  
Kdv Type Equations

AbstractKorteweg–de Vries (KdV)-type equations can describe the nonlinear phenomena in shallow water waves, stratified internal waves, and ion-acoustic waves in plasmas. In this article, the two-dimensional generalization of the Sawada–Kotera equation, one of the KdV-type equations, is discussed by virtue of the Bell polynomials and Hirota method. The results show that there exist multi-soliton solutions for such an equation. Relations between the direction of the soliton propagation and coordinate axes are shown. Elastic interaction with the multi-soliton solutions are analysed.

Analysis of the generalized (2+1)-dimensional Nizhnik–Novikov–Veselov equations with variable coefficients in an inhomogeneous medium

2017 ◽  
Vol 31 (22) ◽  
pp. 1750135
Author(s):  
Han-Peng Chai ◽  
Bo Tian ◽  
Hui-Ling Zhen ◽  
Jun Chai ◽  
Yue-Yang Guan
Keyword(s):  
Inhomogeneous Medium ◽  
Acoustic Waves ◽  
Water Waves ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Bilinear Method ◽  
Ion Acoustic Waves ◽  
Dispersion Terms ◽  
Kdv Type Equations ◽  
Higher Order Dispersion

Korteweg-de Vries (KdV)-type equations are seen to describe the shallow-water waves, lattice structures and ion-acoustic waves in plasmas. Hereby, we consider an extension of the KdV-type equations called the generalized (2[Formula: see text]+[Formula: see text]1)-dimensional Nizhnik–Novikov–Veselov equations with variable coefficients in an inhomogeneous medium. Via the Hirota bilinear method and symbolic computation, we derive the bilinear forms, N-soliton solutions and Bäcklund transformation. Effects of the first- and higher-order dispersion terms are investigated. Soliton evolution and interaction are graphically presented and analyzed: Both the propagation velocity and direction of the soliton change when the dispersion terms are time-dependent; The interactions between/among the solitons are elastic, independent of the forms of the coefficients in the equations.

On the Exact Soliton Solutions of Some Nonlinear PDE by Tan-Cot Method

2018 ◽  
Vol 5 (1) ◽  
pp. 31-36
Author(s):  
Md Monirul Islam ◽  
Muztuba Ahbab ◽  
Md Robiul Islam ◽  
Md Humayun Kabir
Keyword(s):  
Solitary Wave ◽  
Acoustic Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Rectangular Channel ◽  
Soliton Solutions ◽  
Boussinesq Equations ◽  
Travelling Wave ◽  
Ion Acoustic Waves ◽  
Analytical System

For many solitary wave applications, various approximate models have been proposed. Certainly, the most famous solitary wave equations are the K-dV, BBM and Boussinesq equations. The K-dV equation was originally derived to describe shallow water waves in a rectangular channel. Surprisingly, the equation also models ion-acoustic waves and magneto-hydrodynamic waves in plasmas, waves in elastic rods, equatorial planetary waves, acoustic waves on a crystal lattice, and more. If we describe all of the above situation, we must be needed a solution function of their governing equations. The Tan-cot method is applied to obtain exact travelling wave solutions to the generalized Korteweg-de Vries (gK-dV) equation and generalized Benjamin-Bona- Mahony (BBM) equation which are important equations to evaluate wide variety of physical applications. In this paper we described the soliton behavior of gK-dV and BBM equations by analytical system especially using Tan-cot method and shown in graphically. GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 5(1), Dec 2018 P 31-36

Solitons, Bäcklund transformations, Lax pair and conservation laws for the nonautonomous mKdV–sinh-Gordon equation with time-dependent coefficients

2016 ◽  
Vol 30 (03) ◽  
pp. 1650008 ◽  
Author(s):  
Lei Liu ◽  
Bo Tian ◽  
Wen-Rong Sun ◽  
Yu-Feng Wang ◽  
Yun-Po Wang
Keyword(s):  
Conservation Laws ◽  
Gordon Equation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Time Dependent ◽  
Bäcklund Transformations ◽  
Bell Polynomials ◽  
Bilinear Forms ◽  
Bell Polynomial ◽  
Backlund Transformations

The transition phenomenon of few-cycle-pulse optical solitons from a pure modified Korteweg–de Vries (mKdV) to a pure sine-Gordon regime can be described by the nonautonomous mKdV–sinh-Gordon equation with time-dependent coefficients. Based on the Bell polynomials, Hirota method and symbolic computation, bilinear forms and soliton solutions for this equation are obtained. Bäcklund transformations (BTs) in both the binary Bell polynomial and bilinear forms are obtained. By virtue of the BTs and Ablowitz–Kaup–Newell–Segur system, Lax pair and infinitely many conservation laws for this equation are derived as well.

Bäcklund transformations and soliton solutions for a (2 + 1)-dimensional Korteweg–de Vries-type equation in water waves

2015 ◽  
Vol 81 (4) ◽  
pp. 1815-1821 ◽  
Author(s):  
Yun-Po Wang ◽  
Bo Tian ◽  
Ming Wang ◽  
Yu-Feng Wang ◽  
Ya Sun ◽  
...  
Keyword(s):  
Water Waves ◽  
Soliton Solutions ◽  
Type Equation ◽  
Bäcklund Transformations ◽  
Backlund Transformations ◽  
De Vries

Solitons interaction and integrability for a (2+1)-dimensional variable-coefficient Broer–Kaup system in water waves

2018 ◽  
Vol 32 (08) ◽  
pp. 1750268 ◽  
Author(s):  
Xue-Hui Zhao ◽  
Bo Tian ◽  
Yong-Jiang Guo ◽  
Hui-Min Li
Keyword(s):  
Water Waves ◽  
Variable Coefficient ◽  
Soliton Solutions ◽  
Resonance Phenomenon ◽  
Bell Polynomials ◽  
Bilinear Forms ◽  
Linear Superposition ◽  
Elastic Interactions ◽  
The One ◽  
Dimensional Variable

Under investigation in this paper is a (2+1)-dimensional variable-coefficient Broer–Kaup system in water waves. Via the symbolic computation, Bell polynomials and Hirota method, the Bäcklund transformation, Lax pair, bilinear forms, one- and two-soliton solutions are derived. Propagation and interaction for the solitons are illustrated: Amplitudes and shapes of the one soliton keep invariant during the propagation, which implies that the transport of the energy is stable for the (2+1)-dimensional water waves; and inelastic interactions between the two solitons are discussed. Elastic interactions between the two parabolic-, cubic- and periodic-type solitons are displayed, where the solitonic amplitudes and shapes remain unchanged except for certain phase shifts. However, inelastically, amplitudes of the two solitons have a linear superposition after each interaction which is called as a soliton resonance phenomenon.

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