Exact solutions and automorphic systems of the geopotential forecast equation
E. I. Kaptsov
TL;DR
The paper tackles the problem of linking exact solutions of the geopotential forecast equation (GFE) to automorphic systems within the group foliation framework, focusing on the case $\beta \neq 0$. It analyzes the admitted Lie algebra, derives the automorphic and resolving systems in terms of differential invariants, and identifies how invariant solutions induce particular automorphic forms. Through three examples (polynomial, harmonic, and constrained solutions), it demonstrates both explicit automorphic systems and implicit forms that require coordinate changes or reduced resolving systems, illustrating the inverse problem of reconstructing automorphic systems from exact solutions and their integration. The results illuminate the structure of invariant solutions and offer practical pathways to connect alternative solution methods with group-foliation-based automorphic systems, complementing prior work in the field.
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.