Invariant Modules and the Reduction of Nonlinear Partial Differential Equations to Dynamical Systems

Details Invariant Modules and the Reduction of Nonlinear Partial Differential Equations to Dynamical Systems Invariant Modules and the Reduction of Nonlinear Partial Differential Equations to Dynamical Systems

1999-04-15

arxiv.org/abs/solv-int/9904014v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

28 pages. To appear in Advances in Mathematics

Scientific paper

We completely characterize all nonlinear partial differential equations leaving a given finite-dimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a re duction of the associated dynamical partial differential equations to a system of ordinary differential equations, and provide a nonlinear counterpart to quasi-exactly solvable quantum Hamiltonians. These results rely on a useful extension of the classical Wronskian determinant condition for linear independence of functions. In addition, new approaches to the characterization o f the annihilating differential operators for spaces of analytic functions are presented.

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