scholarly journals The Study of the Solution to a Generalized KdV-mKdV Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Xiumei Lv ◽  
Tengwei Shao ◽  
Jiacheng Chen
Keyword(s):  
Symbolic Computation ◽  
Elliptic Function ◽  
Mkdv Equation ◽  
High Order ◽  
Auxiliary Equation ◽  
Jacobi Elliptic Function ◽  
Mathematical Technique ◽  
Triangle Function

A mathematical technique based on an auxiliary equation and the symbolic computation system Matlab is employed to investigate a generalized KdV-mKdV equation which possesses high-order nonlinear terms. Some new solutions including the Jacobi elliptic function solutions, the degenerated soliton-like solutions, and the triangle function solutions to the equation are obtained.

scholarly journals Exact Solutions for a Generalized KdV-MKdV Equation with Variable Coefficients

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Bo Tang ◽  
Xuemin Wang ◽  
Yingzhe Fan ◽  
Junfeng Qu
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Mkdv Equation ◽  
Variable Coefficients ◽  
Auxiliary Equation ◽  
Jacobi Elliptic Function ◽  
Mathematical Tool ◽  
Auxiliary Equation Method

By using solutions of an ordinary differential equation, an auxiliary equation method is described to seek exact solutions of variable-coefficient KdV-MKdV equation. As a result, more new exact nontravelling solutions, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions, and combined Jacobi elliptic function solutions, for the KdV-MKdV equation are obtained. It is shown that the considered method provides a very effective, convenient, and powerful mathematical tool for solving many other nonlinear partial differential equations with variable coefficients in mathematical physics.

scholarly journals Solitons for the modified $(2 + 1)$-dimensional Konopelchenko–Dubrovsky equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xiumei Lyu ◽  
Wei Gu
Keyword(s):  
Elliptic Function ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
High Order ◽  
Auxiliary Equation ◽  
Jacobi Elliptic Function ◽  
Equation Approach ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Triangular Function

Abstract In the paper, we consider the modified $(2 + 1)$ ( 2 + 1 ) -dimensional Konopelchenko–Dubrovsky equations which possess high order nonlinear terms. Under the aid of Maple, we derive the exact traveling wave solutions of the mKDs by the auxiliary equation approach. Under some special conditions, Jacobi elliptic function solutions, degenerated triangular function solutions, and solitons for the mKD equations are constructed.

Jacobi Elliptic Function Rational Expansion Method with Symbolic Computation to Construct New Doubly-periodic Solutions of Nonlinear Evolution Equations

2004 ◽  
Vol 59 (9) ◽  
pp. 529-536 ◽  
Author(s):  
Yong Chen ◽  
Qi Wang ◽  
Biao Lic
Keyword(s):  
Periodic Solutions ◽  
Symbolic Computation ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Jacobi Elliptic Function ◽  
Rational Form

A new Jacobi elliptic function rational expansion method is presented by means of a new general ansatz and is very powerful, with aid of symbolic computation, to uniformly construct more new exact doubly-periodic solutions in terms of rational form Jacobi elliptic function of nonlinear evolution equations (NLEEs). We choose a (2+1)-dimensional dispersive long wave equation to illustrate the method. As a result, we obtain the solutions found by most existing Jacobi elliptic function expansion methods and find other new and more general solutions at the same time. When the modulus of the Jacobi elliptic functions m→1 or 0, the corresponding solitary wave solutions and trigonometric function (singly periodic) solutions are also found.

scholarly journals Jacobi elliptic function solutions to the coupled KdV-mKdV equation

2007 ◽  
Vol 56 (10) ◽  
pp. 5585
Author(s):  
Pan Jun-Ting ◽  
Gong Lun-Xun
Keyword(s):  
Elliptic Function ◽  
Mkdv Equation ◽  
Jacobi Elliptic Function

NEW JACOBI ELLIPTIC FUNCTION SOLUTIONS OF (2 +1)-DIMENSIONAL COMPLEX NONLINEAR INTEGRABLE SYSTEM

2003 ◽  
Vol 14 (03) ◽  
pp. 277-284 ◽  
Author(s):  
ZHENYA YAN
Keyword(s):  
Periodic Solutions ◽  
Integrable System ◽  
Symbolic Computation ◽  
Elliptic Function ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Jacobi Elliptic Function ◽  
Jacobian Elliptic Functions ◽  
Jacobi Elliptic Function Expansion ◽  
Nonlinear Integrable System

Recently sixteen types of doubly-periodic solutions were obtained for the new (2 +1)-dimensional complex nonlinear integrable system. In this paper with the aid of computerized symbolic computation, our extended Jacobi elliptic function expansion method is extended to this system. As a result, another eight families of Jacobian elliptic functions solutions are also found.

scholarly journals Infinitely Many Elliptic Solutions to a Simple Equation and Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Long Wei ◽  
Yang Wang
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Nonlinear Differential Equations ◽  
Nonlinear Problems ◽  
Simple Equation ◽  
Equation Method ◽  
Auxiliary Equation ◽  
Jacobi Elliptic Function ◽  
Auxiliary Equation Method ◽  
Elliptic Solutions

Based on auxiliary equation method and Bäcklund transformations, we present an idea to find infinitely many Weierstrass and Jacobi elliptic function solutions to some nonlinear problems. First, we give some nonlinear iterated formulae of solutions and some elliptic function solutions to a simple auxiliary equation, which results in infinitely many Weierstrass and Jacobi elliptic function solutions of the simple equation. Then applying auxiliary equation method to some nonlinear problems and combining the results with exact solutions of the auxiliary equation, we obtain infinitely many elliptic function solutions to the corresponding nonlinear problems. The employed approach is powerful and can be also applied to solve other nonlinear differential equations.

Quasi-Periodic, Periodic Waves, and Soliton Solutions for the Combined KdV-mKdV Equation

2009 ◽  
Vol 64 (9-10) ◽  
pp. 639-645 ◽  
Author(s):  
Emad A.-B. Abdel-Salam
Keyword(s):  
Elliptic Function ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Travelling Wave ◽  
Mkdv Equation ◽  
Jacobi Elliptic Function ◽  
New Exact Solutions ◽  
De Vries ◽  
Jacobi Elliptic Function Method

By introducing the generalized Jacobi elliptic function, a new improved Jacobi elliptic function method is used to construct the exact travelling wave solutions of the nonlinear partial differential equations in a unified way. With the help of the improved Jacobi elliptic function method and symbolic computation, some new exact solutions of the combined Korteweg-de Vries-modified Korteweg-de Vries (KdV-mKdV) equation are obtained. Based on the derived solution, we investigate the evolution of doubly periodic and solitons in the background waves. Also, their structures are further discussed graphically.

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