April  1998, 4(2): 359-378. doi: 10.3934/dcds.1998.4.359

Periodic systems for the higher-dimensional Laplace transformation

1. 

Department of Mathematics & Statistics, McGill University, Montréal, Québec H3A 2K6, Canada

2. 

Departamento de Matemática, Universidade de Brasília, Brasília, DF 70910, Brazil

Received  December 1996 Revised  May 1997 Published  February 1998

We consider overdetermined systems of linear partial differential equations of the form

$ y_{,k\l}+a_{k\l}^ky_{,k}+a_{k\l}^\ly_{,\l}+c_{k\l}y=0\ , \quad 1\le k\ne\l\le n\ , $

where the coefficients are smooth functions satisfying certain integrability conditions. Generalizing the classical theory of second order linear hyperbolic partial differential equation in the plane, we consider higher-dimensional Laplace invariants of a system of the above class. These invariants are characterized as functions which must satisfy a set of differential equations. We establish a normal form for any system of the above class in terms of these invariants. Moreover, we solve the periodicity problem for the higher-dimensional Laplace transformation applied to such systems, generalizing a classical theorem of Darboux which shows that for $n=2$ a 1-periodic equation is equivalent to the Klein-Gordon equation.

Citation: N. Kamran, K. Tenenblat. Periodic systems for the higher-dimensional Laplace transformation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 359-378. doi: 10.3934/dcds.1998.4.359
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