Exact solutions and automorphic systems of the geopotential forecast equation

TL;DR

This paper explores the use of group foliation, automorphic, and resolving systems to find invariant solutions of the geopotential forecast equation, comparing different solution methods and their automorphic forms.

Contribution

It advances the understanding of automorphic systems related to the geopotential forecast equation and discusses their role in deriving exact solutions.

Findings

Analysis of automorphic systems for the geopotential forecast equation

Particular solutions of the resolving system are considered

Comparison of solutions from different methods

Abstract

The study of the recently constructed group foliation for the geopotential forecast equation is continued. The group foliation consists of two systems, namely the automorphic and resolving systems, the analysis of which facilitates the derivation of invariant solutions for the original equation. As obtaining a general solution to the resolving system (even to its reductions on subgroups) is problematic, its various particular solutions are considered. Consequently, the question arises concerning the specific forms of automorphic systems that correspond to exact solutions obtained through alternative methods. This is of interest for both comparing solutions derived through different approaches and for the integration of specific automorphic systems. The problem is discussed in a number of examples.