On the Hamiltonian and Lagrangian structures of time-dependent reductions of evolutionary PDEs

Details On the Hamiltonian and Lagrangian structures of time-dependent reductions of evolutionary PDEs On the Hamiltonian and Lagrangian structures of time-dependent reductions of evolutionary PDEs

1999-02-05

arxiv.org/abs/solv-int/9902006v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

46 pages, Tex, to be published in "Differential Geometry and Applications"

Scientific paper

In this paper we study the reductions of evolutionary PDEs on the manifold of the stationary points of time-dependent symmetries. In particular we describe how the finite dimensional Hamiltonian structure of the reduced system is obtained from the Hamiltonian structure of the initial PDE and we construct the time-dependent Hamiltonian function. We also present a very general Lagrangian formulation of the procedure of reduction. As an application we consider the case of the Painleve' equations PI, PII, PIII, PVI and also certain higher order systems appeared in the theory of Frobenius manifolds.

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