Differential transformations of parabolic second-order operators in the plane

TL;DR

This paper extends Darboux's classical transformation results from hyperbolic to parabolic second-order operators in the plane, providing a general method to determine and compose transformations, with implications for integrable equations like Boussinesq.

Contribution

It introduces a general theorem for transformations of parabolic operators, showing how higher-order transformations can be composed of first-order ones and linking inverse transformations to integrable equations.

Findings

Transformations of parabolic operators can be systematically determined.

Higher-order transformations are compositions of first-order transformations.

Inverse transformations impose differential constraints related to integrable equations.

Abstract

Here, Darboux's classical results about transformations with differential substitutions for hyperbolic equations are extended to the case of parabolic equations of the form $L u = \big(D^2_{x} + a(x,y) D_x + b(x,y) D_y + c(x,y)\big)u=0$. We prove a general Theorem that provides a way to determine transformations for parabolic equations shown above. It turnes out that transforming operators $M$ of some higher order can be always represented as a composition of some first-order operators that consecutively define a series of transformations. Existence of inverse transformations implies some differential constrains on the coefficients of the initial operator. We show that these relations can imply famous integrable equations, in particular, the Boussinesq equation.