Invariants for Darboux transformations of Arbitrary Order for $D_x D_y +aD_x + bD_y +c$

TL;DR

This paper develops explicit formulas for invariants of differential operators under gauge transformations, aiding the construction of Darboux transformations of arbitrary order.

Contribution

It introduces a method to compute basis invariants for operators of any order, extending the invariant theory for Darboux transformations.

Findings

Derived explicit formulas for basis invariants.

Extended invariant theory to arbitrary order operators.

Facilitated invariant construction of Darboux transformations.

Abstract

We develop the method of regularized moving frames of Fels and Olver to obtain explicit general formulas for the basis invariants that generate all the joint differential invariants, under gauge transformations, for the operators [\o{L}=D_xD_y +a(x,y) D_x + b(x,y) D_y +c(x,y)] and an operator of arbitrary order. The problem appeared in connection with invariant construction of Darboux transformations for $\o{L}$.