Symmetries of the space of solutions to special double confluent Heun equation of negative integer order and its applications

TL;DR

This paper investigates the symmetries and automorphisms of solutions to a special double confluent Heun equation of negative integer order, deriving explicit representations and functional equations, with applications to nonlinear differential equations modeling Josephson junctions.

Contribution

It introduces new ${ m L}$-operators and their composition rules, providing explicit matrix representations and functional equations for solutions, advancing the understanding of symmetries in sDCHE.

Findings

Derived explicit matrix representations of ${ m L}$-operators and monodromy.

Established composition rules leading to functional equations for solutions.

Applied results to nonlinear differential equations in superconductivity modeling.

Abstract

Three linear operators (${\mathcal L}$-operators) determining automorphisms of the space of solutions to a special double confluent Heun equation (sDCHE) of negative integer order are considered. Their composition rules involving in a natural way the monodromy transformation are given. Introducing eigenfunctions $E_{\{+\}}, E_{\{-\}}$ of one of the ${\mathcal L}$-operators (${\mathcal L}_C$) satisfying sDCHE, the four polylocal quadratic functionals playing role of the first integrals of sDCHE are derived. Their use allows to construct the explicit matrix representations of ${\mathcal L}$-operators and the monodromy operator with respect to the basis constituted by $E_{\{\pm\}}$. The composition rules of ${\mathcal L}$-operators lead to functional equations for the eigenfunctions $E_{\{\pm\}}$ which can be interpreted as analytic continuations of solutions to sDCHE from the half-plane $\Re z>0$ to their whole domain by means of algebraic transformations. Application of the above results to the theory of the first order non-linear differential equation utilized, in particular, for the modeling of overdamped Josephson junctions in superconductors and closely related to sDCHE is presented. The automorphisms of the set of its solutions induced by ${\mathcal L}$-operators and by the monodromy operator (certain shift of the solution domain in the latter case) are found which are realized by certain algebraic operations. Among them, one transformation is involutive and another one can be regarded as the square root of the transformation induced by the monodromy transformation.