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Fixed point theorems in the Arnol'd model about instability of the action-variables in phase-space
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 |
$ 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.
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