$X$- and $Y$-invariants of Linear Partial Differential Operators in the plane (In Russian)

TL;DR

This paper introduces $X$- and $Y$-invariants for hyperbolic PDE operators, transforming Darboux theorems into invariant-based equations, facilitating symbolic solutions of PDEs.

Contribution

It presents a novel concept of invariants based on Laplace invariants, with explicit formulas for their transformation under Darboux transformations.

Findings

Defined $X$- and $Y$-invariants in terms of Laplace invariants

Derived explicit formulas for invariants' transformations

Enhanced symbolic solution methods for hyperbolic PDEs

Abstract

We consider a classical problem of Computer Algebra: symbolic solution of PDEs. We transform the famous Darboux theorems on differential transformations of hyperbolic operator into the space of invariants. We introduce a new idea -- $X$- and $Y$-invariants of such operator as solutions of some equations written in terms of the Laplace invariants of this operator. Explicit formula for the changes in the sets of the $X$- and $Y$-invariants under the Darboux transformations are obtained.