scholarly journals Recursion Operator and Local and Nonlocal Symmetries of a New Modified KdV Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Qian Suping ◽  
Li Xin
Keyword(s):  
Exact Solutions ◽  
Kdv Equation ◽  
Symmetry Algebra ◽  
Original Model ◽  
Identity Transformation ◽  
Recursion Operator ◽  
Nonlocal Symmetries ◽  
Symmetry Reductions ◽  
Finite Dimensional ◽  
Modified Kdv Equation

The recursion operator of a new modified KdV equation and its inverse are explicitly given. Acting the recursion operator and its inverse on the trivial symmetry 0 related to the identity transformation, the infinitely many local and nonlocal symmetries are obtained. Using a closed finite dimensional symmetry algebra with both local and nonlocal symmetries of the original model, some symmetry reductions and exact solutions are found.

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.

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.

scholarly journals Further Results about Traveling Wave Exact Solutions of the (2+1)-Dimensional Modified KdV Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Yang Yang ◽  
Jian-ming Qi ◽  
Xue-hua Tang ◽  
Yong-yi Gu
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Kdv Equation ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Mkdv Equation ◽  
Nonlinear Evolution Equations ◽  
Complex Method ◽  
Modified Kdv Equation

We used the complex method and the exp(-ϕ(z))-expansion method to find exact solutions of the (2+1)-dimensional mKdV equation. Through the maple software, we acquire some exact solutions. We have faith in that this method exhibited in this paper can be used to solve some nonlinear evolution equations in mathematical physics. Finally, we show some simulated pictures plotted by the maple software to illustrate our results.

scholarly journals Conditional Lie-Bäcklund Symmetry Reductions and Exact Solutions of a Class of Reaction-Diffusion Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Xinyang Wang ◽  
Junquan Song
Keyword(s):  
Dynamical Systems ◽  
Exact Solutions ◽  
Separation Of Variables ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Financial Mathematics ◽  
Symmetry Reductions ◽  
Finite Dimensional ◽  
Wide Range

The method of conditional Lie-Bäcklund symmetry is applied to solve a class of reaction-diffusion equations ut+uxx+Qxux2+Pxu+Rx=0, which have wide range of applications in physics, engineering, chemistry, biology, and financial mathematics theory. The resulting equations are either solved exactly or reduced to some finite-dimensional dynamical systems. The exact solutions obtained in concrete examples possess the extended forms of the separation of variables.

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