Double Casoratian solutions to the nonlocal semi-discrete modified Korteweg-de Vries equation

2020 ◽  
Vol 34 (05) ◽  
pp. 2050021 ◽  
Author(s):  
Wei Feng ◽  
Song-Lin Zhao ◽  
Ying-Ying Sun
Keyword(s):  
Asymptotic Analysis ◽  
Exact Solutions ◽  
Vries Equation ◽  
Soliton Solutions ◽  
The Real ◽  
Reduced Equations ◽  
De Vries ◽  
Casorati Determinant ◽  
Coupled Equation

Two nonlocal versions of the semi-discrete modified Korteweg-de Vries equation are derived by different nonlocal reductions from a coupled equation set in the Ablowitz–Ladik hierarchy. Different kinds of exact solutions in terms of double Casoratians to the reduced equations are obtained by imposing constraint conditions on the double Casorati determinant solutions of the coupled equation set. Dynamics of the soliton solutions for the real and complex nonlocal semi-discrete modified Korteweg-de Vries equations are analyzed and illustrated by asymptotic analysis.

scholarly journals Rational solutions for three semi-discrete modified Korteweg–de Vries-type equations

2019 ◽  
Vol 33 (32) ◽  
pp. 1950399 ◽  
Author(s):  
Ying-Ying Sun ◽  
Song-Lin Zhao
Keyword(s):  
Asymptotic Analysis ◽  
Exact Solutions ◽  
Vries Equation ◽  
Rational Solutions ◽  
Lattice Potential ◽  
De Vries ◽  
Casorati Determinant

In this paper, we consider three semi-discrete modified Korteweg–de Vries type equations which are the nonlinear lumped self-dual network equation, the semi-discrete lattice potential modified Korteweg–de Vries equation and a semi-discrete modified Korteweg–de Vries equation. We derive several kinds of exact solutions, in particular rational solutions, in terms of the Casorati determinant for these three equations, respectively. For some rational solutions, we present the related asymptotic analysis to understand their dynamics better.

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.

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.

A two dimensional distributed soliton solution of the Korteweg-de Vries equation

1979 ◽  
Vol 366 (1725) ◽  
pp. 185-204 ◽  
Keyword(s):  
Asymptotic Analysis ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Resonant Interaction ◽  
Two Dimensions ◽  
Discrete Soliton ◽  
Variation Of Parameters ◽  
De Vries ◽  
Interaction Of Solitons ◽  
Discrete Solitons

It is suggested that recent discrete soliton solutions of the Korteweg-de Vries equation in two dimensions can be generalized to solutions which have a continuous variation of parameters. In the particular case of the resonant interaction of solitons, as discussed by Miles, an analytic solution is obtained in which only one parameter is varied. The structure of this solution is examined in detail. Asymptotic solutions for large positive and negative time are construc­ted as well as solutions at large distance for finite time. The asymptotic analysis is found to be similar to that developed by Lighthill for Burgers’ equation. A resonant triad of discrete solitons as described by Miles is shown to develop into a curved soliton. Numerical computations confirm the asymptotic analysis.

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.

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.

scholarly journals THE LOCALIZED RESONANCE EXCITATIONS ARE DESCRIBED BY EXACT SOLUTIONS OF THE MODIFIED KORTEWEG-DE VRIES EQUATION

2020 ◽  
Author(s):  
G. Khusainova ◽  
◽  
D. Khusainov ◽  
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Bound State ◽  
Hirota Method ◽  
Resonance Interaction ◽  
Exponential Solution ◽  
De Vries ◽  
Exact Soliton Solutions

The exact soliton solutions of modified Korteweg-de Vries equation are obtained by procedure based on Hirota method. It has shown that thesе solutions described the bound state of soliton-antisoliton pairs which are formed in result resonance interaction of two solitons. Keywords: exact solution, rational-exponential solution, Hirota method

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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