([formula omitted])-dimensional ZK–Burgers equation with the generalized beta effect and its exact solitary solution

Abstract

The (1 ＋1)-dimensional mathematical model had been extensively derived to describe Rossby solitary waves in a line in the past few decades. But as is well known, the (1 ＋1)-dimensional model cannot reflect the generation and evolution of Rossby solitary waves in a plane. In this paper, a (2 ＋1)-dimensional nonlinear Zakharov–Kuznetsov–Burgers equation is derived to describe the evolution of Rossby wave amplitude by using methods of multiple scales and perturbation expansions from the quasi-geostrophic potential vorticity equations with the generalized beta effect. The effects of the generalized beta and dissipation are presented by the Zakharov–Kuznetsov–Burgers equation. We also obtain the new solitary solution of the Zakharov–Kuznetsov equation when the dissipation is absent with the help of the Bernoulli equation, which is different from the common classical solitary solution. Based on the solution, the features of the variable coefficient are discussed by geometric figures Meanwhile, the approximate solitary solution of Zakharov–Kuznetsov–Burgers equation is given by using the homotopy perturbation method. And the amplitude of solitary waves changing with time is depicted by figures. Undoubtedly, these solitary solutions will extend previous results and better help to explain the feature of Rossby solitary waves.