Modified Kawahara Equation Research Articles

Periodic and solitary wave solutions of Kawahara and modified Kawahara equations by using Sine–Cosine method

In this paper, we establish exact solutions for nonlinear evolution equations. The sine–cosine method is used to construct periodic and solitary wave solutions of the Kawahara and modified Kawahara equations. These solutions may be important of significance for the explanation of some practical physical problems.

New solitary wave solutions to the modified Kawahara equation

In this work we use the sine–cosine method, the tanh method, the extended tanh method, and ansatze of hyperbolic functions for analytic treatment for the modified Kawahara equation. New solitons solutions and periodic solutions are formally derived. The change of the parameters, that will drastically change the characteristics of the equation, is examined. The employed approaches are reliable and manageable.

A analytic and numerical solution to a modified Kawahara equation and a convergence analysis of the method

In this paper, we present an Adomian’s decomposition method (shortly ADM) for numerical approximation traveling wave solutions of the modified Kawahara equation. The numerical solutions are compared with the known analytical solutions. We also prove the convergence of Adomian decomposition method (ADM) applied to the modified Kawahara equation.

A analytic and numerical solution to a modified Kawahara equation and a convergence analysis of the method

In this paper, we present an Adomian’s decomposition method (shortly ADM) for numerical approximation traveling-wave solutions of the modified Kawahara equation. The numerical solutions are compared with the known analytical solutions. We also prove the convergence of Adomian decomposition method (ADM) applied to the modified Kawahara equation.

Exact Traveling Wave Solutions for the Modified Kawahara Equation

The mapping method is used with the aid of the symbolic computation system Mathematica for constructing exact solutions of the modified Kawahara equation. By this method the modified Kawahara equation is investigated and new exact traveling wave solutions are obtained. The solutions obtained in this paper include Jacobi elliptic solutions, combined Jacobi elliptic solutions, solitary wave solutions, periodic wave solutions, trigonometric solutions and rational solutions.

Doubly periodic solutions of the modified Kawahara equation

Some doubly periodic (Jacobi elliptic function) solutions of the modified Kawahara equation are presented in closed form. Our approach is to introduce a new auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct doubly periodic solutions of the modified Kawahara equation. When the module m → 1, these solutions degenerate to the exact solitary wave solutions of the equation. Then we reveal the relation of some exact solutions for the modified Kawahara equation obtained by other authors.

New exact travelling wave solutions for the Kawahara and modified Kawahara equations

By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact travelling wave solutions for nonlinear evolution equations. By this method the Kawahara and the modified Kawahara equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation.

The JEFE method and periodic solutions of two kinds of nonlinear wave equations

The Jacobi elliptic function expansion (JEFE) method is applied to construct the exact periodic solutions to two kinds of nonlinear wave equations, such as BBM equation, fifth-order dispersive equation, Kawahara equation, modified Kawahara equation, second-order BO equation and symmetrical-regular long wave equation. The corresponding shock wave solutions or solitary wave solutions are obtained as special cases of the periodic solutions. It is shown that this method is very powerful for some nonlinear wave equations, and its applying domain is given.

Stability of periodic waves governed by the modified Kawahara equation

The stability of spatially periodic traveling waves, generated by a long-wavelength oscillatory instability, is investigated. The modified Kawahara equation, derived formerly in a number of physical problems, is used. The stability region and the main instability modes are found.