Invariant modules and the reduction of nonlinear partial differential equations to dynamical systems

We completely characterize all nonlinear partial differential equations lea ving a given finite-dimensional vector space uf analytic functions invarian t. Existence of an invariant subspace leads to a reduction of the associate d dynamical partial differential equations to a system of ordinary differen tial equations and provides a nonlinear counterpart to quasi-exactly solvab le quantum Hamiltonians. These results rely on a useful extension of the cl assic;tl Wronskian determinant condition for linear independence of functio ns. In addition. new. approaches to the characterization of the annihilatin g differential operators for spaces of analytic functions are presented. (C ) 2000 Academic Press.