ZK-Burgers equation for three-dimensional Rossby solitary waves and its solutions as well as chirp effect

Hong Wei Yang,Bao Shu Yin,De Zhou Yang,Zhen Hua Xu,Huan He Dong,Xing Ru Feng

doi:10.1186/s13662-016-0901-8

Abstract

Two-dimensional Rossby solitary waves propagating in a line have attracted much attention in the past decade, whereas there is few research on three-dimensional Rossby solitary waves. But as is well known, three-dimensional Rossby solitary waves are more suitable for real ocean and atmosphere conditions. In this paper, using multiscale and perturbation expansion method, a new Zakharov-Kuznetsov (ZK)-Burgers equation is derived to describe three-dimensional Rossby solitary waves that propagate in a plane. By analyzing the equation we obtain the conservation laws of three-dimensional Rossby solitary waves. Based on the sine-cosine method, we give the classical solitary wave solutions of the ZK equation; on the other hand, by the Hirota method we also obtain the rational solutions, which are similar to the solutions of the Benjamin-Ono (BO) equation, the solutions of which can describe the algebraic solitary waves. The rational solutions of the ZK equations are worth of attention. Finally, with the help of the classical solitary wave solutions, similar to the fiber soliton communication, we discuss the dissipation and chirp effect of three-dimensional Rossby solitary waves.

Highlights

When nonlinearity and dispersion are exactly matched in a dispersive wave system, a kind of the stable waves of permanent form, or the solitary waves, may arise
The research on solitary waves has become an advanced subject in the fields of mechanics, physics, applied mathematics, and atmospheric and oceanic sciences and attracted more and more attention [ – ]
After tedious calculation, we find that the energy of three-dimensional Rossby solitary waves decreases with the increasing of time T and dissipative coefficient μ

Summary

Introduction

When nonlinearity and dispersion are exactly matched in a dispersive wave system, a kind of the stable waves of permanent form, or the solitary waves, may arise. The rational solutions of ZK equation are obtained firstly in this paper Based on these analytical solutions, the dissipation and chirp effect is discussed. C > , 4.2 Rational solutions In the same way, neglecting dissipation effect and assuming that α = and μ = , Eq ( ) can be rewritten as AT + a AAX + a AXXX + a AXYY =. We introduce the chirp idea of optical soliton communication and analyze the dispersion and nonlinear effect during the propagation process of three-dimensional Rossby solitary waves. Based on the analytical solution of the ZK equation, we take the initial wave form of threedimensional Rossby solitary waves as.

C ΔT sech

Conclusion