Hirota's Bilinear Method and its Generalization

1997 ◽  
Vol 12 (01) ◽  
pp. 43-51 ◽  
Author(s):  
Jarmo Hietarinta
Keyword(s):  
Soliton Solution ◽  
Gauge Invariance ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
Singularity Structure ◽  
Soliton Equations ◽  
Hirota’S Method ◽  
Hirota's Method ◽  
Multisoliton Solutions

We review Hirota's bilinear method for constructing multisoliton solutions, its use in searching for new soliton equations, and its generalization to higher multi-linearity using gauge invariance as the determining property. Hirota's method is relevant even when a soliton solution is not the object of the study, as an example we show how it clarifies the singularity structure of the discrete Painlevé I equation.

scholarly journals Soliton Solutions for a Nonisospectral Semi-Discrete Ablowitz–Kaup–Newell–Segur Equation

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1889 ◽  
Author(s):  
Song-Lin Zhao
Keyword(s):  
Asymptotic Analysis ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Hirota’S Method ◽  
De Vries ◽  
Hirota's Method ◽  
Multisoliton Solutions

In this paper, we study a nonisospectral semi-discrete Ablowitz–Kaup–Newell–Segur equation. Multisoliton solutions for this equation are given by Hirota’s method. Dynamics of some soliton solutions are analyzed and illustrated by asymptotic analysis. Multisoliton solutions and dynamics to a nonisospectral semi-discrete modified Korteweg-de Vries equation are also discussed.

Multisoliton solutions of a (2+1)-dimensional variable-coefficient Toda lattice equation via Hirota’s bilinear method

2014 ◽  
Vol 92 (3) ◽  
pp. 184-190 ◽  
Author(s):  
Sheng Zhang ◽  
Dong Liu
Keyword(s):  
Variable Coefficient ◽  
Toda Lattice ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Hirota’S Bilinear Method ◽  
Lattice Equation ◽  
Bilinear Method ◽  
Dimensional Variable ◽  
Toda Lattice Equation ◽  
Multisoliton Solutions

In this paper, Hirota’s bilinear method is extended to construct multisoliton solutions of a (2+1)-dimensional variable-coefficient Toda lattice equation. As a result, new and more general one-soliton, two-soliton, and three-soliton solutions are obtained, from which the uniform formula of the N-soliton solution is derived. It is shown that Hirota’s bilinear method can be used for constructing multisoliton solutions of some other nonlinear differential-difference equations with variable coefficients.

scholarly journals Bilinearization and new multi-soliton solutions of mKdV hierarchy with time-dependent coefficients

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 69-75 ◽  
Author(s):  
Sheng Zhang ◽  
Luyao Zhang
Keyword(s):  
Soliton Solution ◽  
Bilinear Form ◽  
Soliton Solutions ◽  
Time Dependent ◽  
Time Varying ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
De Vries ◽  
Mkdv Hierarchy

AbstractIn this paper, Hirota’s bilinear method is extended to a new modified Kortweg–de Vries (mKdV) hierarchy with time-dependent coefficients. To begin with, we give a bilinear form of the mKdV hierarchy. Based on the bilinear form, we then obtain one-soliton, two-soliton and three-soliton solutions of the mKdV hierarchy. Finally, a uniform formula for the explicit N-soliton solution of the mKdV hierarchy is summarized. It is graphically shown that the obtained soliton solutions with time-dependent functions possess time-varying velocities in the process of propagation.

scholarly journals The Soliton Solutions and Long-Time Asymptotic Analysis for a General Coupled KdV Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Changhao Zhang ◽  
Guiying Chen
Keyword(s):  
Asymptotic Behavior ◽  
Soliton Solution ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Elastic Collision ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
Coupled Kdv Equation ◽  
Long Time ◽  
The One

A general coupled KdV equation, which describes the interactions of two long waves with different dispersion relation, is considered. By employing the Hirota’s bilinear method, the bilinear form is obtained, and the one-soliton solution and two-soliton solution are constructed. Moreover, the elasticity of the collision between two solitons is proved by analyzing the asymptotic behavior of the two-soliton solution. Some figures are displayed to illustrate the process of elastic collision.

scholarly journals Construction of Higher-Order Smooth Positons and Breather Positons via Hirota's Bilinear Method

2021 ◽  
Author(s):  
Zhao Zhang ◽  
Biao Li ◽  
Junchao Chen ◽  
QI GUO
Keyword(s):  
Soliton Solution ◽  
Darboux Transformation ◽  
Mathematical Expression ◽  
Higher Order ◽  
Second Order ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
Advantages And Disadvantages ◽  
De Vries ◽  
Generalized Darboux Transformation

Abstract Based on the Hirota's bilinear method, a more classic limit technique is perfected to obtain second-order smooth positons. Immediately afterwards, we propose an extremely ingenious limit approach in which higher-order smooth positons and breather positons can be quickly derived from N-soliton solution. Under this ingenious technique, the smooth positons and breather positons of the modified Korteweg-de Vries system are quickly and easily derived. Compared with the generalized Darboux transformation, the approach mentioned in this paper has the following advantages and disadvantages: the advantage is that it is simple and fast; the disadvantage is that this method cannot get a concise general mathematical expression of nth-order smooth positons.

Emergence and Interaction of the Lump-Type Solution with the (3+1)-D Jimbo-Miwa Equation

2017 ◽  
Vol 73 (1) ◽  
pp. 43-49 ◽  
Author(s):  
Wei Tan ◽  
Zheng-de Dai ◽  
Jing-li Xie ◽  
Ling-li Hu
Keyword(s):  
Exact Solutions ◽  
Soliton Solution ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Hirota’S Bilinear Method ◽  
Type Solution ◽  
Bilinear Method ◽  
Limit Behaviour

AbstractA kinky breather-soliton solution and kinky periodic-soliton solution are obtained using Hirota’s bilinear method and homoclinic test approach for the (3+1)-dimensional Jimbo-Miwa equation. Based on these two exact solutions, some lump-type solutions are emerged by limit behaviour. Meanwhile, two kinds of new dynamical phenomena, kinky breather degeneracy and kinky periodic degeneracy, are discussed and presented. Finally, the interaction between a stripe soliton and a lump-type soliton is discussed by the standardisation of the lump-type solution; the fusion and fission phenomena of soliton solutions are investigated and simulated by three-dimensional plots.

Multiple Soliton Solutions for a Variety of Coupled Modified Korteweg–de Vries Equations

2011 ◽  
Vol 66 (10-11) ◽  
pp. 625-631
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Soliton Solutions ◽  
Mkdv Equation ◽  
Resonance Phenomenon ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
Multiple Soliton Solutions ◽  
De Vries ◽  
Coupled Equation ◽  
Singular Soliton ◽  
Multiple Singular Soliton Solutions

We make use of Hirota’s bilinear method with computer symbolic computation to study a variety of coupled modified Korteweg-de Vries (mKdV) equations. Multiple soliton solutions and multiple singular soliton solutions are obtained for each coupled equation. The resonance phenomenon of each coupled mKdV equation is proved not to exist.

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