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 reduction of the associated dynamical partial differential equations to a system of ordinary differential equations and provides 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 of the annihilating differential operators for spaces of analytic functions are presented.
Bibliographical noteFunding Information:
1Supported in part by NSERC Grant OGP0105490. 2Supported in part by an NSERC Postdoctoral Fellowship. 3Supported in part by NSF Grant DMS 98-03154. http: www.math.umn.edu