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.

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.

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.

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.

Rogue wave solutions for the higher-order nonlinear Schrödinger equation with variable coefficients by generalized Darboux transformation

2016 ◽  
Vol 30 (10) ◽  
pp. 1650106 ◽  
Author(s):  
Hai-Qiang Zhang ◽  
Jian Chen
Keyword(s):  
Darboux Transformation ◽  
Variable Coefficient ◽  
Rogue Wave ◽  
Variable Coefficients ◽  
Higher Order ◽  
Optical Systems ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Order Variable ◽  
Generalized Darboux Transformation

In this paper, we study a higher-order variable coefficient nonlinear Schrödinger (NLS) equation, which plays an important role in the control of the ultrashort optical pulse propagation in nonlinear optical systems. Then, we construct a generalized Darboux transformation (GDT) for the higher-order variable coefficient NLS equation. The [Formula: see text]th order rogue wave solution is obtained by the iterative rule and it can be expressed by the determinant form. As application, we calculate rogue waves (RWs) from first- to fourth-order in accordance with different kinds of parameters. In particular, the dynamical properties and spatial-temporal structures of RWs are discussed and compared with Hirota equation through some figures.

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.

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