Open Access
2011 Solutions of Smooth Nonlinear Partial Differential Equations
Jan Harm van der Walt
Abstr. Appl. Anal. 2011: 1-37 (2011). DOI: 10.1155/2011/658936


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 Schrödinger equation in one spatial dimension.


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Jan Harm van der Walt. "Solutions of Smooth Nonlinear Partial Differential Equations." Abstr. Appl. Anal. 2011 1 - 37, 2011.


Published: 2011
First available in Project Euclid: 12 August 2011

zbMATH: 1221.35022
MathSciNet: MR2817264
Digital Object Identifier: 10.1155/2011/658936

Rights: Copyright © 2011 Hindawi

Vol.2011 • 2011
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