On classification and invariants of second order non-parabolic linear partial differential equations in two variables

TL;DR

This paper classifies second order linear partial differential equations in two variables, identifies invariants under transformations, and provides criteria for reducing these equations to constant coefficient form, advancing the understanding of their structure and symmetries.

Contribution

It introduces a classification scheme and invariants for second order LPDEs under specific transformations, and offers a criterion for their reducibility to constant coefficients.

Findings

Classification and invariants for non-parabolic LPDEs are established.

Solutions for subclasses of LPDEs are provided.

A criterion for reducibility to constant coefficients is developed.

Abstract

The paper deals with second order abstract linear partial differential equations (LPDE) over a partial differential field with two commuting differential operators. In terms of usual differential equations the main content can be presented as follows. The classification and invariants problems for second order LPDEs with respect to transformations \[x=x(\xi,\eta),\ y=y(\xi,\eta),\ u=h(x,y)v(\xi,\eta),\] where $x,y$ are independent and $u$ is the dependent variable of the LPDE, are considered. Solutions to these problems are given for different subclasses of non-parabolic LPDE which appear naturally in the equivalence problem of LPDE. A criterion for reducibility of such LPDE to LPDE with constant coefficients is offered as well.