Some New Exact Solutions of the Modified KdV Equation Using Lie Point Symmetry Method

2017 ◽  
Vol 3 (S1) ◽  
pp. 1163-1171 ◽  
Author(s):  
A. H. Abdel Kader ◽  
M. S. Abdel Latif ◽  
H. M. Nour
Keyword(s):  
Exact Solutions ◽  
Kdv Equation ◽  
Point Symmetry ◽  
New Exact Solutions ◽  
Modified Kdv Equation ◽  
Lie Point Symmetry ◽  
Symmetry Method

scholarly journals AKNS Formalism and Exact Solutions of KdV and Modified KdV Equations with Variable-Coefficients

2016 ◽  
Vol 6 ◽  
pp. 32-41 ◽  
Author(s):  
Supratim Das ◽  
Dibyendu Ghosh
Keyword(s):  
Exact Solutions ◽  
Kdv Equation ◽  
Elliptic Functions ◽  
Variable Coefficients ◽  
Jacobian Elliptic Functions ◽  
Generalized Kdv Equation ◽  
New Exact Solutions ◽  
Kdv Equations ◽  
Modified Kdv Equation ◽  
Modified Kdv Equations

We apply the AKNS hierarchy to derive the generalized KdV equation andthe generalized modified KdV equation with variable-coefficients. We system-atically derive new exact solutions for them. The solutions turn out to beexpressible in terms of doubly-periodic Jacobian elliptic functions.

A NOTE ON THE EXACT SOLUTION OF THE SPECIAL MODIFIED KdV EQUATION

2008 ◽  
Vol 22 (04) ◽  
pp. 289-293
Author(s):  
HONGLEI WANG ◽  
CHUNHUAN XIANG
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
General Equation ◽  
Vries Equation ◽  
Kdv Equation ◽  
Variable Coefficients ◽  
New Exact Solutions ◽  
Modified Kdv Equation ◽  
De Vries ◽  
Physical Fields

The modified KdV (Korteweg–de Vries) equation with two different variable coefficients can be employed in many different physical fields with time changing. In the present work, by using the truncated expansion, some new exact solutions of the equation are obtained. The general equation may change into lots of other forms KdV equation if we select different parameters.

Symmetry analysis and some new exact solutions of some nonlinear KdV-like equations

2018 ◽  
Vol 11 (03) ◽  
pp. 1850040 ◽  
Author(s):  
A. H. Abdel Kader ◽  
M. S. Abdel Latif ◽  
F. El Bialy ◽  
A. Elsaid
Keyword(s):  
Exact Solutions ◽  
Soliton Solutions ◽  
Dark Soliton ◽  
Symmetry Analysis ◽  
Point Symmetry ◽  
New Exact Solutions ◽  
Lie Point Symmetry

In this paper, we obtained some new exact solutions of some nonlinear KdV-like equations using Lie point symmetry and [Formula: see text]-symmetry methods. The obtained solutions are in the form of doubly periodic, bright and dark soliton solutions.

scholarly journals Lie-Point Symmetries and Backward Stochastic Differential Equations

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1153
Author(s):  
Na Zhang ◽  
Guangyan Jia
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Backward Stochastic Differential Equation ◽  
Backward Stochastic Differential Equations ◽  
Point Symmetry ◽  
Lie Point Symmetries ◽  
Lie Point Symmetry ◽  
Symmetry Method

In this paper, we introduce the Lie-point symmetry method into backward stochastic differential equation and forward–backward stochastic differential equations, and get the corresponding deterministic equations.

scholarly journals New Generalized Hyperbolic Functions to Find New Exact Solutions of the Nonlinear Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Yusuf Pandir ◽  
Halime Ulusoy
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Wave Equations ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equations ◽  
Hyperbolic Function ◽  
Hyperbolic Functions ◽  
Partial Differential ◽  
New Exact Solutions

We firstly give some new functions called generalized hyperbolic functions. By the using of the generalized hyperbolic functions, new kinds of transformations are defined to discover the exact approximate solutions of nonlinear partial differential equations. Based on the generalized hyperbolic function transformation of the generalized KdV equation and the coupled equal width wave equations (CEWE), we find new exact solutions of two equations and analyze the properties of them by taking different parameter values of the generalized hyperbolic functions. We think that these solutions are very important to explain some physical phenomena.

scholarly journals Soliton Molecules, Rational Positon Solution and Rogue Waves for the Extended Complex Modified KdV Equation

2021 ◽  
Author(s):  
Lin Huang ◽  
Nannan Lv
Keyword(s):  
Periodic Solution ◽  
Numerical Analysis ◽  
Exact Solutions ◽  
Soliton Solution ◽  
Vries Equation ◽  
Kdv Equation ◽  
Rational Solution ◽  
Rogue Waves ◽  
Modified Kdv Equation ◽  
Extended Complex

Abstract We consider the integrable extended complex modified Korteweg–de Vries equation, which is generalized modified KdV equation. The first part of the article considers the construction of solutions via the Darboux transformation. We obtain some exact solutions, such as soliton solution, soliton molecules, positon solution, rational positon solution, rational solution, periodic solution and rogue waves solution. The second part of the article analyzes the dynamics of rogue waves. By means of the numerical analysis, under the standard decomposition, we divide the rogue waves into three patterns: fundamental patterns, triangular patterns and ring patterns. For the fundamental patterns, we define the length and width of the rogue waves and discuss the effect of different parameters on rogue waves.

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