Combined Jacobi Elliptic Function Solutions Research Articles

FRACTIONAL COMPLEX TRANSFORMS, REDUCED EQUATIONS AND EXACT SOLUTIONS OF THE FRACTIONAL KRAENKEL–MANNA–MERLE SYSTEM

Exact solutions of the fractional Kraenkel–Manna–Merle system in saturated ferromagnetic materials have been studied. Using the fractional complex transforms, the fractional Kraenkel–Manna–Merle system is reduced to ordinary differential equations, (1 + 1)-dimensional partial differential equations and (2 + 1)-dimensional partial differential equations. Based on the obtained ordinary differential equations and taking advantage of the available solutions of Jacobi elliptic equation and Riccati equation, soliton solutions, combined soliton solutions, combined Jacobi elliptic function solutions, triangular periodic solutions and rational function solutions, for the KMM system are obtained. For the obtained (1 + 1)-dimensional partial differential equations, we get the classification of Lie symmetries. Starting from a Lie symmetry, we get a symmetry reduction equation. Solving the symmetry reduction equation by the power series method, power series solutions for the KMM system are obtained. For the obtained (2 + 1)-dimensional partial differential equations, we derive their bilinear form and two-soliton solution. The bilinear form can also be used to study the lump solutions, rogue wave solutions and breathing wave solutions.

Exact Solutions forN-Coupled Nonlinear Schrödinger Equations With Variable Coefficients

AbstractIn this paper, based on similarity transformation and auxiliary equation method, we construct many exact solutions ofN-coupled nonlinear Schrödinger equations with variable coefficients, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions and combined Jacobi elliptic function solutions. These solutions may give insight into many considerable physical processes.

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

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.

Travelling wave solutions to some important equations of mathematical physics

Exact travelling wave solutions of some nonlinear evolution equations of mathematical physics are obtained by using the mapping method and the extended F -expansion method. It is well known that different types of exact solutions of a given auxiliary ordinary differential equation produce new types of exact solutions to nonlinear evolution equations. Many new exact travelling wave solutions of Zakharov-Kuzentsov (ZK), modified Kadomtsev-Petviashvilli (KP) with square root nonlinearity and modified fifth-order Korteweg-de Vries (KdV) equations are constructed by using mapping method. We also apply the extended F -expansion method to the long-short-wave interaction system and the coupled modified KdV equations. The solutions obtained in this paper include single and combined Jacobi elliptic function solutions, rational solutions and hyperbolic function solutions. In the limiting case, the solitary wave solutions of such equations and systems are also studied.

Non-TravellingWave Solutions of the (2+1)-Dimensional Dispersive Long Wave System

With the aid of symbolic computation and the extended F-expansion method, we construct more general types of exact non-travelling wave solutions of the (2+1)-dimensional dispersive long wave system. These solutions include single and combined Jacobi elliptic function solutions, rational solutions, hyperbolic function solutions, and trigonometric function solutions.

Travelling wave solutions for the generalized Zakharov–Kuznetsov equation with higher-order nonlinear terms

With the aid of symbolic computation and two auxiliary ordinary differential equations, the new generalized algebraic method is extended to the generalized Zakharov–Kuznetsov (GZK) equation with higher-order nonlinear terms for constructing a series of new and more general travelling wave solutions. Because of the higher-order nonlinear terms, the equation can not be directly dealt with by the method and require some kinds of techniques. By means of two proper transformations, we transform the GZK equation to an ordinary differential equation that is easy to solve and find a rich variety of new exact travelling wave solutions for the GZK equation, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions, combined Jacobi elliptic function solutions and rational function solutions. The method used here can be also extended to other nonlinear partial differential equations with higher-order nonlinear terms.

Construction of non-travelling wave solutions for the generalized variable-coefficient Gardner equation

With the aid of two first-order nonlinear ordinary differential equations, the new generalized algebraic method is used to study the generalized variable-coefficient Gardner equation. The main idea of the method is to take full advantage of solutions to the first-order ordinary differential equations. More new exact non-travelling solutions, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions and combined Jacobi elliptic function solutions, for the Gardner equation are obtained. The method used here can be also extended to many other nonlinear partial differential equations.

A generalized F-expansion method and new exact solutions of Konopelchenko–Dubrovsky equations

In this paper, a generalized F-expansion method is proposed to seek more general exact solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose the (2 + 1)-dimensional Konopelchenko–Dubrovsky equations to illustrate the validity and advantages of the method. As a result, many new and more general exact solutions are obtained including single and combined Jacobi elliptic function solutions, soliton-like solutions, trigonometric function solutions. The method can be applied to other nonlinear partial differential equations in mathematical physics.

Exact solutions for the coupled Klein–Gordon–Schrödinger equations using the extended F-expansion method

The extended F-expansion method or mapping method is used to construct exact solutions for the coupled Klein-Gordon Schrodinger equations (K-G-S equations) by the aid of the symbolic computation system Mathematica. More solutions in the Jacobi elliptic function form are obtained, including the single Jacobi elliptic function solutions, combined Jacobi elliptic function solutions, rational solutions, triangular solutions, soliton solutions and combined soliton solutions.

Jacobi Elliptic Function Solutions of Three Coupled Nonlinear Physical Equations

Abstract The Jacobi elliptic function solutions of coupled nonlinear partial differential equations, including the coupled modified KdV (mKdV) equations, long-short-wave interaction system and the Davey- Stewartson (DS) equations, are obtained by using the mixed dn-sn method. The solutions obtained in this paper include the single and the combined Jacobi elliptic function solutions. In the limiting case, the solitary wave solutions of the systems are also given. - PACS: 02.30.Jr; 03.40.Kf; 03.65.Fd