Darboux formulae for linear hyperbolic equations in discrete case

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.