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

TL;DR

This paper characterizes nonlinear PDEs that preserve a finite-dimensional space of functions, enabling their reduction to ODE systems and extending concepts from quasi-exactly solvable quantum models.

Contribution

It provides a complete characterization of invariant PDEs and introduces new methods for identifying annihilating operators, extending classical Wronskian techniques.

Findings

Invariant subspaces lead to PDE reduction to ODE systems

Extension of Wronskian determinant condition for nonlinear cases

New approaches to characterize annihilating differential operators

Abstract

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.