Infinitely Many Elliptic Solutions to a Simple Equation and Applications

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.

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New extended auxiliary equation method for finding many new Jacobi elliptic function solutions of three nonlinear Schrödinger equations

Waves in Random and Complex Media ◽

10.1080/17455030.2016.1259690 ◽

2016 ◽

Vol 27 (3) ◽

pp. 420-439 ◽

Cited By ~ 23

Author(s):

Elsayed M. E. Zayed ◽

Abdul-Ghani Al-Nowehy

Keyword(s):

Elliptic Function ◽

Nonlinear Schrödinger Equations ◽

Equation Method ◽

Schrodinger Equations ◽

Auxiliary Equation ◽

Jacobi Elliptic Function ◽

Schrödinger Equations ◽

Auxiliary Equation Method ◽

Nonlinear Schrödinger ◽

Nonlinear Schrodinger Equations

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

Mathematical Problems in Engineering ◽

10.1155/2016/5274243 ◽

2016 ◽

Vol 2016 ◽

pp. 1-10 ◽

Cited By ~ 1

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.

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Exact Solutions with Variable Coefficient Function Forms for Conformable Fractional Partial Differential Equations by an Auxiliary Equation Method

Advances in Mathematical Physics ◽

10.1155/2018/4596506 ◽

2018 ◽

Vol 2018 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

Fanwei Meng ◽

Qinghua Feng

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Fractional Derivative ◽

Exact Solutions ◽

Variable Coefficient ◽

Equation Method ◽

Auxiliary Equation ◽

Coefficient Function ◽

Fractional Partial Differential Equations ◽

Auxiliary Equation Method

In this paper, an auxiliary equation method is introduced for seeking exact solutions expressed in variable coefficient function forms for fractional partial differential equations, where the concerned fractional derivative is defined by the conformable fractional derivative. By the use of certain fractional transformation, the fractional derivative in the equations can be converted into integer order case with respect to a new variable. As for applications, we apply this method to the time fractional two-dimensional Boussinesq equation and the space-time fractional (2+1)-dimensional breaking soliton equation. As a result, some exact solutions including variable coefficient function solutions as well as solitary wave solutions for the two equations are found.

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Exact solutions of some coupled nonlinear diffusion-reaction equations using auxiliary equation method

Pramana ◽

10.1007/s12043-012-0341-2 ◽

2012 ◽

Vol 79 (3) ◽

pp. 393-402

Author(s):

Ranjit Kumar

Keyword(s):

Exact Solutions ◽

Nonlinear Diffusion ◽

Equation Method ◽

Diffusion Reaction ◽

Auxiliary Equation ◽

Auxiliary Equation Method ◽

Reaction Equations

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Extended auxiliary equation method and its applications for finding the exact solutions for a class of nonlinear Schrödinger-type equations

Applied Mathematics and Computation ◽

10.1016/j.amc.2016.04.014 ◽

2016 ◽

Vol 289 ◽

pp. 111-131 ◽

Cited By ~ 7

Author(s):

E.M.E. Zayed ◽

K.A.E. Alurrfi

Keyword(s):

Exact Solutions ◽

Equation Method ◽

Auxiliary Equation ◽

Auxiliary Equation Method ◽

Nonlinear Schrödinger

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Exact solutions for the high-order dispersive cubic-quintic nonlinear Schrödinger equation by the extended hyperbolic auxiliary equation method

Chaos Solitons & Fractals ◽

10.1016/j.chaos.2006.05.001 ◽

2007 ◽

Vol 34 (5) ◽

pp. 1608-1612 ◽

Cited By ~ 28

Author(s):

Shun-dong Zhu

Keyword(s):

Schrödinger Equation ◽

Exact Solutions ◽

Nonlinear Schrödinger Equation ◽

Schrodinger Equation ◽

Nonlinear Schrodinger Equation ◽

High Order ◽

Equation Method ◽

Auxiliary Equation ◽

Auxiliary Equation Method ◽

Quintic Nonlinear Schrödinger Equation

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A Generalized Auxiliary Equation Method for the Quadratic Nonlinear KG Equation

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.394.571 ◽

2013 ◽

Vol 394 ◽

pp. 571-576

Author(s):

Sheng Zhang ◽

Bo Xu ◽

Ao Xue Peng

Keyword(s):

Partial Differential Equations ◽

Exact Solutions ◽

Nonlinear Partial Differential Equations ◽

Equation Method ◽

Auxiliary Equation ◽

Mathematical Tool ◽

Auxiliary Equation Method ◽

Klein Gordon ◽

Generalized Auxiliary Equation Method ◽

Powerful Mathematical Tool

A generalized auxiliary equation method with symbolic computation is used to construct more general exact solutions of the quadratic nonlinear Klein-Gordon (KG) equation. As a result, new and more general solutions are obtained. It is shown that the generalized auxiliary equation method provides a more powerful mathematical tool for solving nonlinear partial differential equations arising in the fields of nonlinear sciences.

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