Time-Space Fractional Coupled Generalized Zakharov-Kuznetsov Equations Set for Rossby Solitary Waves in Two-Layer Fluids

Lei Fu,Yaodeng Chen,Hongwei Yang

doi:10.3390/math7010041

Abstract

In this paper, the theoretical model of Rossby waves in two-layer fluids is studied. A single quasi-geostrophic vortex equation is used to derive various models of Rossby waves in a one-layer fluid in previous research. In order to explore the propagation and interaction of Rossby waves in two-layer fluids, from the classical quasi-geodesic vortex equations, by employing the multi-scale analysis and turbulence method, we derived a new (2+1)-dimensional coupled equations set, namely the generalized Zakharov-Kuznetsov(gZK) equations set. The gZK equations set is an extension of a single ZK equation; they can describe two kinds of weakly nonlinear waves interaction by multiple coupling terms. Then, for the first time, based on the semi-inverse method and the variational method, a new fractional-order model which is the time-space fractional coupled gZK equations set is derived successfully, which is greatly different from the single fractional equation. Finally, group solutions of the time-space fractional coupled gZK equations set are obtained with the help of the improved ( G ′ / G ) -expansion method.

Highlights

IntroductionThe existence of solitary waves has been known in hydrokinetics for about a century, it was not until recently that the theory was applied to wave phenomena in the atmospheric, ocean [1,2,3,4,5,6]
The existence of solitary waves has been known in hydrokinetics for about a century, it was not until recently that the theory was applied to wave phenomena in the atmospheric, ocean [1,2,3,4,5,6]and large lake dynamics system, such as solitary waves [7], internal gravity waves [8], internal Kelvin waves [9] and so on
For the exploration of the Rossby waves propagation and action between two-layer fluids, the coupled gZK equations set of the objective function is derived from the following two layers of quasi-geostrophic vortex equations set by using the multi-scale analysis and turbulence method q At + J (ψ A, q A ) + βψ Ax = 0, q Bt + J + βψBx = 0, (1)

Summary

Introduction

The existence of solitary waves has been known in hydrokinetics for about a century, it was not until recently that the theory was applied to wave phenomena in the atmospheric, ocean [1,2,3,4,5,6]. Since Long(1964) derived the Korteweg de-Vries(KdV) equation on the positive pressure system, the isolated Rossby wave theory has been gradually developed [10,11,12], but in many complex atmospheric and oceanic systems, waves interact with each other. Mathematics 2019, 7, 41 physics, the gZK equations set as the extension of a single equation can be used to describe the interaction of nonlinear Rossby waves in two-layer fluids [24,25,26]. We choose the modified ( G 0 /G )-expansion method to solve the time-space fractional coupled gZK equations set, and get several different kinds of solutions

Derivation of Coupled gZK Equations Set

The Time-Space Fractional Coupled gZK Equations set

Solutions of Time-Space Fractional Coupled gZK Equations Set

Conclusions