Darboux formulae for linear hyperbolic equations in discrete case

TL;DR

This paper develops discrete analogs of Darboux formulae for linear hyperbolic difference operators, extending classical continuous results to the discrete setting and contributing to the theory of discrete conjugate nets.

Contribution

It introduces and proves discrete Darboux formulae for hyperbolic difference operators with finite Laplace series, bridging continuous and discrete hyperbolic PDE theories.

Findings

Discrete Darboux formulae established for hyperbolic difference operators

Extension of classical Darboux formulae to discrete and semi-discrete cases

Enhanced understanding of discrete conjugate nets and their properties

Abstract

In the second half of the 19th century Darboux obtained determinant formulae that provide the general solution for a linear hyperbolic second order PDE with finite Laplace series. These formulae played an important role in his study of the theory of surfaces and, in particular, in the theory of conjugate nets. During the last three decades discrete analogs of conjugate nets (Q-nets) were actively studied. Laplace series can be defined also for hyperbolic difference operators. We prove discrete analogs of Darboux formulae for discrete and semi-discrete hyperbolic operators with finite Laplace series.