General description on extended homoclinic orbit solutions of the KdV-type bilinear equations

2020 ◽  
pp. 2150092
Author(s):  
Shu-Zhi Liu ◽  
Da-Jun Zhang
Keyword(s):  
Soliton Solution ◽  
Homoclinic Orbit ◽  
Soliton Solutions ◽  
General Description ◽  
De Vries ◽  
Bilinear Derivatives ◽  
Bilinear Equations ◽  
Special Case ◽  
Bilinear Equation

The Korteweg–de Vries (KdV)-type bilinear equations always allow 2-soliton solutions. In this paper, for a general KdV-type bilinear equation, we interpret how the so-called extended homoclinic orbit solutions arise from a special case of its 2-soliton solution. Two properties of bilinear derivatives are developed to deal with bilinear equation deformations. A non-integrable (3+1)-dimensional bilinear equation is employed as an example.

THE SECOND-ORDER KdV EQUATION AND ITS SOLITON-LIKE SOLUTION

2009 ◽  
Vol 23 (14) ◽  
pp. 1771-1780 ◽  
Author(s):  
CHUN-TE LEE ◽  
JINN-LIANG LIU ◽  
CHUN-CHE LEE ◽  
YAW-HONG KANG
Keyword(s):  
Solitary Waves ◽  
Soliton Solution ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Second Order ◽  
Close Resemblance ◽  
De Vries

This paper presents both the theoretical and numerical explanations for the existence of a two-soliton solution for a second-order Korteweg-de Vries (KdV) equation. Our results show that there exists "quasi-soliton" solutions for the equation in which solitary waves almost retain their identities in a suitable physical regime after they interact, and bear a close resemblance to the pure KdV solitons.

scholarly journals Analytical and Numerical Methods for the CMKdV-II Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Ömer Akin ◽  
Ersin Özuğurlu
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Iterative Method ◽  
Soliton Solution ◽  
Bilinear Form ◽  
Difference Method ◽  
Soliton Solutions ◽  
Hirota’S Bilinear Form ◽  
De Vries

Hirota's bilinear form for the Complex Modified Korteweg-de Vries-II equation (CMKdV-II) is derived. We obtain one- and two-soliton solutions analytically for the CMKdV-II. One-soliton solution of the CMKdV-II equation is obtained by using finite difference method by implementing an iterative method.

Pairs of Bilinear Equations in a Finite Field

1966 ◽  
Vol 18 ◽  
pp. 561-565
Author(s):  
A. Duane Porter
Keyword(s):  
Finite Field ◽  
Quadratic Forms ◽  
Number Of Solutions ◽  
Image Position ◽  
Bilinear Equations ◽  
Special Case ◽  
Bilinear Equation

Let F = GF(g) be the finite field of q = pr elements, p arbitrary. We wish to consider the system of bilinear equations1.1where all coefficients are from F. The number of solutions in F of a single bilinear equation may be obtained from a theorem of John H. Hodges (3, Theorem 3) by properly defining the matrices U, V, A, B. In 1954, L. Carlitz (1) obtained, as a special case of his work on quadratic forms, the number of simultaneous solutions in F of (1.1) when all aj = 1 and p is odd. Carlitz considered the case p = 2 separately.

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 Multiple-Pole Solutions to a Semidiscrete Modified Korteweg-de Vries Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Zhixing Xiao ◽  
Kang Li ◽  
Junyi Zhu
Keyword(s):  
Soliton Solution ◽  
Vries Equation ◽  
Hilbert Problem ◽  
Soliton Solutions ◽  
Higher Order ◽  
Symmetry Condition ◽  
De Vries ◽  
Riemann Hilbert Problem

Multiple-pole soliton solutions to a semidiscrete modified Korteweg-de Vries equation are derived by virtue of the Riemann-Hilbert problem with higher-order zeros. A different symmetry condition is introduced to build the nonregular Riemann-Hilbert problem. The simplest multiple-pole soliton solution is presented. The dynamics of the solitons are studied.

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.

scholarly journals The Evolutionary Properties on Solitary Solutions of Nonlinear Evolution Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Wenxia Chen ◽  
Danping Ding ◽  
Xiaoyan Deng ◽  
Gang Xu
Keyword(s):  
Soliton Solution ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Evolution Process ◽  
Solution Curve ◽  
Solitary Solutions ◽  
Basic Calculus

The evolution process of four class soliton solutions is investigated by basic calculus theory. For any given x, we describe the special curvature evolution following time t for the curve of soliton solution and also study the fluctuation of solution curve.

N-soliton solutions and localized wave interaction solutions of a (3 + 1)-dimensional potential-Yu–Toda–Sasa–Fukuyamaf equation

2021 ◽  
pp. 2150277
Author(s):  
Hongcai Ma ◽  
Qiaoxin Cheng ◽  
Aiping Deng
Keyword(s):  
Dynamic Behavior ◽  
Soliton Solution ◽  
Soliton Solutions ◽  
Wave Interaction ◽  
Bilinear Transformation ◽  
Interaction Solution ◽  
Interaction Solutions ◽  
Ytsf Equation ◽  
Line Soliton

[Formula: see text]-soliton solutions are derived for a (3 + 1)-dimensional potential-Yu–Toda–Sasa–Fukuyama (YTSF) equation by using bilinear transformation. Some local waves such as period soliton, line soliton, lump soliton and their interaction are constructed by selecting specific parameters on the multi-soliton solutions. By selecting special constraints on the two soliton solutions, period and lump soliton solution can be obtained; three solitons can reduce to the interaction solution between period soliton and line soliton or lump soliton and line soliton under special parameters; the interaction solution among period soliton and two line solitons, or the interaction solution for two period solitons or two lump solitons via taking specific constraints from four soliton solutions. Finally, some images of the results are drawn, and their dynamic behavior is analyzed.

Sign in / Sign up

Export Citation Format

Share Document