Solutions of smooth nonlinear partial differential equations

Abstract

The method of order completion provides a general and type-independent theory for the existence and basic regularity of the solutions of large classes of systems of nonlinear partial differential equations PDEs . Recently, the application of convergence spaces to this theory resulted in a significant improvement upon the regularity of the solutions and provided new insight into the structure of solutions. In this paper, we show how this method may be adapted so as to allow for the infinite differentiability of generalized functions. Moreover, it is shown that a large class of smooth nonlinear PDEs admit generalized solutions in the space constructed here. As an indication of how the general theory can be applied to particular nonlinear equations, we construct generalized solutions of the parametrically driven, damped nonlinear Schr¨odinger equation in one spatial dimension.