Orbits of Darboux groupoid for hyperbolic operators of order three

TL;DR

This paper studies the structure of Darboux transformations for third-order hyperbolic operators, describing their orbits and how these transformations affect differential invariants, thus advancing understanding of integrable systems.

Contribution

It characterizes the orbits of Darboux transformations of type I for third-order hyperbolic operators and explicitly computes their action on differential invariants.

Findings

Darboux transformations form a subgroupoid with well-defined orbits.

Explicit calculation of how transformations lift to differential invariants.

Identification of the algebraic structure of invariants under these transformations.

Abstract

Darboux transformations are viewed as morphisms in a Darboux category. Darboux transformations of type I which we defined previously, make an important subgroupoid consists of Darboux transformations of type I. We describe the orbits of this subgroupoid for hyperbolic operators of order three. We consider the algebras of differential invariants for our operators. In particular, we show that the Darboux transformations of this class can be lifted to transformations of differential invariants (which we calculate explicitly).