Multi-soliton solutions of a variable coefficient Schrödinger equation derived from vorticity equation

A variable coefficient Schrödinger equation which is derived by using the multi-scale expansion and coordinate expansion transformation method from nonlinear inviscid barotropic nondivergent vorticity equation in a β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-plane is discussed in this paper. It is different from the previous model that the background basic flow is assumed as a function of time t. Then, the bilinear forms are obtained based on a transformation. By using the Hirota bilinear method, exact analytical solutions to the Schrödinger equation are achieved. These solutions include single-soliton, two-soliton and three-soliton solutions, whose interactions have been presented in the form of 3-d solid figure or density graphic to describe the dynamic characteristics, and might help to describe Rossby waves more suitable. Furthermore, the effects on solitons of coefficients except which relate to the dispersion relation of the model are discussed.