Time\u2013space fractional (2+1) dimensional nonlinear Schr\xf6dinger equation for envelope gravity waves in baroclinic atmosphere and conservation laws as well as exact solutions

Chen Fu,Hong Wei Yang,Chang Na Lu

doi:10.1186/s13662-018-1512-3

Abstract

In this article, nonlinear propagation of envelope gravity waves is studied in baroclinic atmosphere. The classical (2+1) dimensional nonlinear Schrödinger (NLS) equation can be derived by using the multiple-scale, perturbation method. Further, via the semi-inverse method, the Euler–Lagrange equation and Agrawal’s method, the time–space fractional (2+1) dimensional nonlinear Schrödinger (FNLS) equation is obtained to describe the envelope gravity waves. Furthermore, the conservation laws of time–space FNLS equation are discussed on the basis of Lie group analysis method. Finally, the exact solutions to the equation are given by employing the exp(-phi(xi)) method. The results demonstrate that the nonlinear effect caused by the fractional order leads to the change of the propagation characteristics of envelope gravity waves, the construction of fractional model has far-reaching significance for the research of nonlinear propagation of envelope gravity waves in actual atmospheric and ocean movement.

Highlights

1 Introduction It is well known that envelope gravity waves play an important role in atmospheric dynamics [1,2,3,4,5], the troposphere is excited by convection, topography and other excitation processes to transfer energy and momentum from the source to the middle and upper atmosphere [6,7,8,9]
4 Conservation laws of time–space fractional (2 + 1) dimensional nonlinear Schrödinger (NLS) equation In the section, we present a time–space FPDE with three independent variables, G x, y, t, A, Dαt A, D2xβ A, D2yγ A, . . . = 0, α > 0, β > 0, γ > 0
5 Exact solutions of time–space fractional (2 + 1) dimensional NLS equation In this part, the exp(–φ(ξ )) method will be applied to obtain the exact solutions to Eq (45), as follows

Summary

Introduction

It is well known that envelope gravity waves play an important role in atmospheric dynamics [1,2,3,4,5], the troposphere is excited by convection, topography and other excitation processes to transfer energy and momentum from the source (e.g., mountains, thermal forcing) to the middle and upper atmosphere [6,7,8,9]. We discuss the influence of fractional order for the propagation of envelope gravity waves by constructing the time–space fractional (2 + 1) NLS equation. Suppose Eq (10) has the following solution in the form of separate variables: p0 = p∗0(y, z)A(T1, T2, X1, X2, Y ) exp i(kx – ωt) , (12)

Results

Conclusion