Automorphisms of Stokes multipliers in higher-order WKBJ theory

TL;DR

This paper develops a framework of automorphisms to analyze the Stokes phenomenon in higher-order WKBJ theory, revealing complex interactions of Stokes lines and automorphisms in systems with multiple WKBJ components.

Contribution

It introduces a novel automorphism framework for Stokes constants in higher-order WKBJ analysis, applied to a complex example from catastrophe theory, and explores automorphism behavior across intersecting Stokes lines.

Findings

Automorphisms capture the Stokes phenomenon in higher-order WKBJ systems.

Automorphism values can change across higher-order Stokes lines in systems with four or more components.

No additional behaviors are observed for systems with five or more WKBJ components.

Abstract

We consider the Stokes phenomenon and higher-order Stokes phenomenon (HOSP) of formal asymptotic transseries arising in the WKBJ analysis of linear differential equations and integral problems. We introduce a framework of automorphisms that act on the Stokes constants of the divergent expansion, explained via late-late-term expansions and parametric Alien calculus, to capture this phenomenon. Our method is applied to a paradigmatic example: we obtain the full Stokes line structure and automorphisms for the Swallowtail problem from catastrophe theory, which contains four WKBJ components. We demonstrate that, in a system with four or more WKBJ components, the automorphism associated with the HOSP can itself change value across another higher-order Stokes line, which occurs when different higher-order Stokes lines intersect. We then argue that no additional special behaviour emerges for transseries with five or more WKBJ components.