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 Investigation of Interaction Solutions for Modified Korteweg-de Vries Equation by Consistent Riccati Expansion Method

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Jin-Fu Liang ◽  
Xun Wang
Keyword(s):  
Vries Equation ◽  
Soliton Solutions ◽  
Wave Interaction ◽  
Expansion Method ◽  
Periodic Wave ◽  
Mkdv Equation ◽  
Jacobi Elliptic Function ◽  
Interaction Solutions ◽  
De Vries ◽  
Consistent Riccati Expansion

A consistent Riccati expansion (CRE) method is proposed for obtaining interaction solutions to the modified Korteweg-de Vries (mKdV) equation. Using the CRE method, it is shown that interaction solutions such as the soliton-tangent (or soliton-cotangent) wave cannot be constructed for the mKdV equation. More importantly, exact soliton-cnoidal periodic wave interaction solutions are presented. While soliton-cnoidal interaction solutions were found to degenerate to special resonant soliton solutions for the values of modulus (n) closer to one (upper bound of modulus) in the Jacobi elliptic function, a normal kink-shaped soliton was observed for values of n closer to zero (lower bound).

scholarly journals Exact Solutions for the Wick-Type Stochastic Schamel-Korteweg-de Vries Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Xueqin Wang ◽  
Yadong Shang ◽  
Huahui Di
Keyword(s):  
Exact Solutions ◽  
Symbolic Computation ◽  
Vries Equation ◽  
Expansion Method ◽  
Travelling Wave ◽  
Variable Coefficients ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Hermite Transformation

We consider the Wick-type stochastic Schamel-Korteweg-de Vries equation with variable coefficients in this paper. With the aid of symbolic computation and Hermite transformation, by employing the (G′/G,1/G)-expansion method, we derive the new exact travelling wave solutions, which include hyperbolic and trigonometric solutions for the considered 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.

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.

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.

Some Remarks on the Riccati Equation Expansion Method for Variable Separation of Nonlinear Models

2015 ◽  
Vol 70 (10) ◽  
pp. 835-842 ◽  
Author(s):  
Yu-Peng Zhang ◽  
Chao-Qing Dai
Keyword(s):  
Exact Solutions ◽  
Riccati Equation ◽  
Vries Equation ◽  
Nonlinear Models ◽  
Expansion Method ◽  
Variable Separation ◽  
Component Systems ◽  
De Vries ◽  
Physical Engineering

AbstractBased on the Riccati equation expansion method, 11 kinds of variable separation solutions with different forms of (2+1)-dimensional modified Korteweg–de Vries equation are obtained. The following two remarks on the Riccati equation expansion method for variable separation are made: (i) a remark on the equivalence of different solutions constructed by the Riccati equation expansion method. From analysis, we find that these seemly independent solutions with different forms actually depend on each other, and they can transform from one to another via some relations. We should avoid arbitrarily asserting so-called “new” solutions; (ii) a remark on the construction of localised excitation based on variable separation solutions. For two or multi-component systems, we must be careful with excitation structures constructed by all components for the same model lest the appearance of some un-physical structures. We hope that these results are helpful to deeply study exact solutions of nonlinear models in physical, engineering and biophysical contexts.

scholarly journals Hyperbolic Tangent Ansatz Method to Space Time Fractional Modified KdV, Modified EW and Benney–Luke Equations

2018 ◽  
Author(s):  
Ozlem Ersoy Hepson
Keyword(s):  
Exact Solutions ◽  
Vries Equation ◽  
Space Time ◽  
Wave Transformation ◽  
One Dimension ◽  
Governing Equations ◽  
Hyperbolic Tangent ◽  
Ansatz Method ◽  
De Vries

The space time fractional Korteweg-de Vries equation, modified Equal Width equation and Benney-Luke Equations are solved by using simple hyperbolic tangent Ansatz method. A simple compatible wave transformation in one dimension is employed to reduce the governing equations to integer ordered ODEs. Then, the ansatz approximation is used to derive exact solutions. Some illustrative examples are presented for some particular choices of parameters and derivative orders.

scholarly journals Applications of the Novel (G′/G)-Expansion Method for a Time Fractional Simplified Modified Camassa-Holm (MCH) Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Muhammad Shakeel ◽  
Qazi Mahmood Ul-Hassan ◽  
Jamshad Ahmad
Keyword(s):  
Exact Solutions ◽  
Fractional Derivatives ◽  
Traveling Wave Solutions ◽  
Rational Functions ◽  
Expansion Method ◽  
The Novel ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
Liouville Derivative ◽  
Free Parameters

We use the fractional derivatives in modified Riemann-Liouville derivative sense to construct exact solutions of time fractional simplified modified Camassa-Holm (MCH) equation. A generalized fractional complex transform is properly used to convert this equation to ordinary differential equation and, as a result, many exact analytical solutions are obtained with more free parameters. When these free parameters are taken as particular values, the traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions. Moreover, the numerical presentations of some of the solutions have been demonstrated with the aid of commercial software Maple. The recital of the method is trustworthy and useful and gives more new general exact solutions.

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

scholarly journals Traveling wave solutions for the extended modified KORTEWEG-DE VRIES equation

2021 ◽  
Vol 82 (4) ◽  
pp. 197-203
Author(s):  
G. N. Shaikhova ◽  
◽  
B. K. Rakhimzhanov ◽  
Keyword(s):  
Traveling Wave ◽  
Vries Equation ◽  
Nonlinear Partial Differential Equations ◽  
Integrable Model ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Mathematical Tool ◽  
Wave Solutions ◽  
De Vries ◽  
Fifth Order

In this paper, we study an extended modified Korteweg-de Vries equation, which contains the relevant higher-order nonlinear terms and fifth-order dispersion. This equation is the extension of the modified Korteweg-de Vries equation and described by the Ablowitz-Kaup-Newell-Segur hierarchy. The standard Korteweg-de Vries equation is the pioneer integrable model in solitary waves theory, which gives rise to multiple soliton solutions. The Korteweg-de Vries equation arises naturally from shallow water, plasma physics, and other fields of science. To obtain exact solutions the sine-cosine method is applied. It is shown that the sine-cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics. Traveling wave solutions are determined for extended modified Korteweg-de Vries equation. The study shows that the sine–cosine method is quite efficient and practically well suited for use in calculating traveling wave solutions for extended modified Korteweg-de Vries equation.

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