Time–space fractional (2+1) $(2+1)$ dimensional nonlinear Schrödinger equation for envelope gravity waves in baroclinic atmosphere and conservation laws as well as exact solutions

Abstract

Abstract In this article, nonlinear propagation of envelope gravity waves is studied in baroclinic atmosphere. The classical (2+1) $(2+1)$ 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) $(2+1)$ 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(−ϕ(ξ)) $\exp(-\phi(\xi))$ 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.

Keywords

• Time–space fractional ( 2 + 1 ) $(2+1)$ dimensional nonlinear Schrödinger equation
• Envelope gravity waves
• Conservation laws
• exp ( − ϕ ( ξ ) ) $\exp(-\phi(\xi))$ method