Analytical and Multishaped Solitary Wave Solutions for Extended Reduced Ostrovsky Equation

Ben-Gong Zhang

doi:10.1155/2013/670847

Abstract

We present the analytical and multishaped solitary wave solutions for extended reduced Ostrovsky equation (EX-ROE). The exact solitary (traveling) wave solutions are expressed by three types of functions which are hyperbolic function solution, trigonometric function solution, and rational solution. These results generalized the previous results. Multishape solitary wave solutions such as loop-shaped, cusp-shaped, and hump-shaped can be obtained as well when the special values of the parameters are taken. The -expansion method presents a wide applicability for handling nonlinear partial differential equations.

Highlights

IntroductionWhere c0 is the velocity of dispersiveness linear waves, α is the nonlinear coefficient, and β and γ are dispersion coefficients, is a model for weakly nonlinear surface and internal waves in a rotating ocean
The well-known Ostrovsky equation [1](ut + c0ux + αuux + βuxxx)x = γu, (1)where c0 is the velocity of dispersiveness linear waves, α is the nonlinear coefficient, and β and γ are dispersion coefficients, is a model for weakly nonlinear surface and internal waves in a rotating ocean.For long waves, for which high-frequency dispersion is negligible, β = 0, and (1) becomes the so-called reducedOstrovsky equation (ROE) [2](ut + c0ux + αuux)x = γu. (2)Parkes [3] has studied (2) and found its periodic and solitary traveling wave solutions.by applying the following transformation [4]: u 󳨀→ u, α t
We present the analytical and multishaped solitary wave solutions for extended reduced Ostrovsky equation (EX-reducedOstrovsky equation (ROE))

Summary

Introduction

Where c0 is the velocity of dispersiveness linear waves, α is the nonlinear coefficient, and β and γ are dispersion coefficients, is a model for weakly nonlinear surface and internal waves in a rotating ocean. There are many methods being proposed to study the traveling wave solutions of nonlinear partial differential equations which are derived from physics, for example, [16,17,18,19,20,21,22,23,24,25,26,27]. The main idea of (G󸀠/G)-expansion method is to use an integrable ODE to expand a solution to a nonlinear partial differential equation (PDE) as a polynomial or rational function of the solution of the ODE. Such an idea was presented in [36,37,38]. We hope we can find much more interesting properties and new phenomenon of this equation

Exact Traveling Wave Solutions of the Extended Reduced Ostrovsky Equation

Multishaped Solitary Wave Solutions

Conclusion