Non-linear Dynamics and Exact Solutions for the Variable-Coefficient Modified Korteweg–de Vries Equation

2018 ◽  
Vol 73 (2) ◽  
pp. 143-149 ◽  
Author(s):  
Jiangen Liu ◽  
Yufeng Zhang
Keyword(s):  
Exact Solutions ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Rational Solutions ◽  
Transformation Technique ◽  
Linear Dynamics ◽  
New Exact Solutions ◽  
Non Linear ◽  
De Vries

AbstractThis paper presents some new exact solutions which contain soliton solutions, breather solutions and two types of rational solutions for the variable-coefficient-modified Korteweg–de Vries equation, with the help of the multivariate transformation technique. Furthermore, based on these new soliton solutions, breather solutions and rational solutions, we discuss their non-linear dynamics properties. We also show the graphic illustrations of these solutions which can help us better understand the evolution of solution waves.

THE GENERALIZED WRONSKIAN SOLUTIONS OF THE INTEGRABLE VARIABLE-COEFFICIENT KORTEWEG-DE VRIES EQUATION

2011 ◽  
Vol 25 (32) ◽  
pp. 4615-4626 ◽  
Author(s):  
YI ZHANG ◽  
HAI-QIONG ZHAO ◽  
LING-YA YE ◽  
YI-NENG LV
Keyword(s):  
Fluid Dynamics ◽  
Exact Solutions ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Sufficient Conditions ◽  
Rational Solutions ◽  
Wronskian Determinant ◽  
Linear Partial Differential Equations ◽  
De Vries ◽  
Bose Einstein

A broad set of sufficient conditions consisting of systems of linear partial differential equations are presented which guarantee that the Wronskian determinant is the solutions of the integrable variable-coefficient Korteweg-de Vries model from Bose–Einstein condensates and fluid dynamics. The generalized Wronskian solutions provide us with a comprehensive approach to construct many exact solutions including rational solutions, solitons, negatons, positons, and complexitons.

Analytical study of exact solutions of the nonlinear Korteweg–de Vries equation with space–time fractional derivatives

2018 ◽  
Vol 32 (02) ◽  
pp. 1850012 ◽  
Author(s):  
Jiangen Liu ◽  
Yufeng Zhang
Keyword(s):  
Exact Solutions ◽  
Fractional Derivatives ◽  
Analytical Study ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Space Time ◽  
Strong Basis ◽  
De Vries

This paper gives an analytical study of dynamic behavior of the exact solutions of nonlinear Korteweg–de Vries equation with space–time local fractional derivatives. By using the improved [Formula: see text]-expansion method, the explicit traveling wave solutions including periodic solutions, dark soliton solutions, soliton solutions and soliton-like solutions, are obtained for the first time. They can better help us further understand the physical phenomena and provide a strong basis. Meanwhile, some solutions are presented through 3D-graphs.

Solitons for a forced generalized variable-coefficient Korteweg–de Vries equation for the atmospheric blocking phenomenon

2016 ◽  
Vol 30 (35) ◽  
pp. 1650318 ◽  
Author(s):  
Jun Chai ◽  
Bo Tian ◽  
Xi-Yang Xie ◽  
Han-Peng Chai
Keyword(s):  
Variable Coefficient ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Atmospheric Blocking ◽  
Graphic Analysis ◽  
Bilinear Bäcklund Transformation ◽  
Different Types ◽  
De Vries ◽  
Soliton Amplitude ◽  
The One

Investigation is given to a forced generalized variable-coefficient Korteweg–de Vries equation for the atmospheric blocking phenomenon. Applying the double-logarithmic and rational transformations, respectively, under certain variable-coefficient constraints, we get two different types of bilinear forms: (a) Based on the first type, the bilinear Bäcklund transformation (BT) is derived, the [Formula: see text]-soliton solutions in the Wronskian form are constructed, and the [Formula: see text]- and [Formula: see text]-soliton solutions are proved to satisfy the bilinear BT; (b) Based on the second type, via the Hirota method, the one- and two-soliton solutions are obtained. Those two types of solutions are different. Graphic analysis on the two types shows that the soliton velocity depends on [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], the soliton amplitude is merely related to [Formula: see text], and the background depends on [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are the dissipative, dispersive, nonuniform and line-damping coefficients, respectively, and [Formula: see text] is the external-force term. We present some types of interactions between the two solitons, including the head-on and overtaking interactions, interactions between the velocity- and amplitude-unvarying two solitons, between the velocity-varying while amplitude-unvarying two solitons and between the velocity- and amplitude-varying two solitons, as well as the interactions occurring on the constant and varying backgrounds.

Periodic and rational solutions of variable-coefficient modified Korteweg–de Vries equation

2017 ◽  
Vol 89 (1) ◽  
pp. 617-622 ◽  
Author(s):  
Ritu Pal ◽  
Harleen Kaur ◽  
Thokala Soloman Raju ◽  
C. N. Kumar
Keyword(s):  
Variable Coefficient ◽  
Vries Equation ◽  
Rational Solutions ◽  
De Vries

scholarly journals New exact solutions of fractional Hirota-Satsuma coupled Korteweg-de Vries equations

2015 ◽  
Vol 19 (4) ◽  
pp. 1173-1176 ◽  
Author(s):  
Lian-Xiang Cui ◽  
Li-Mei Yan ◽  
Yan-Qin Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Function Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Fractional Complex Transform ◽  
New Exact Solutions ◽  
Non Linear ◽  
De Vries

An improved extended tg-function method, which combines the fractional complex transform and the extended tanh-function method, is applied to find exact solutions of non-linear fractional partial differential equations. Generalized Hirota-Satsuma coupled Korteweg-de Vries equations are used as an example to elucidate the effectiveness and simplicity of the method.

scholarly journals New Exact Solutions for the (3+1)-Dimensional Generalized BKP Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Jun Su ◽  
Genjiu Xu
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Wronskian Technique ◽  
Rational Solutions ◽  
Wronskian Determinant ◽  
New Exact Solutions ◽  
Positon Solutions ◽  
Bkp Equation

The Wronskian technique is used to investigate a (3+1)-dimensional generalized BKP equation. Based on Hirota’s bilinear form, new exact solutions including rational solutions, soliton solutions, positon solutions, negaton solutions, and their interaction solutions are formally derived. Moreover we analyze the strangely mechanical behavior of the Wronskian determinant solutions. The study of these solutions will enrich the variety of the dynamics of the nonlinear evolution equations.

scholarly journals Inverse scattering transform for a supersymmetric Korteweg-de Vries equation

2019 ◽  
Vol 23 (Suppl. 3) ◽  
pp. 677-684
Author(s):  
Sheng Zhang ◽  
Caihong You
Keyword(s):  
Exact Solutions ◽  
Inverse Scattering ◽  
Evolution Equations ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Grassmann Algebra ◽  
Inverse Scattering Transform ◽  
Linear Evolution ◽  
Scattering Transform ◽  
De Vries

In this paper, the inverse scattering transform is extended to a super Korteweg-de Vries equation with an arbitrary variable coefficient by using Kulish and Zeitlin?s approach. As a result, exact solutions of the super Korteweg-de Vries equation are obtained. In the case of reflectionless potentials, the obtained exact solutions are reduced to soliton solutions. More importantly, based on the obtained results, an approach to extending the scattering transform is proposed for the supersymmetric Korteweg-de Vries equation in the 1-D Grassmann algebra. It is shown the the approach can be applied to some other supersymmetric non-linear evolution equations in fluids.

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