Nonlinear Partial Differential Equations and Invariant Differential Systems

摘要

Partial differential equations arise in all fields of science and engineering, yet, when these equations are nonlinear, systematic methods for their solution are usually not known. Sometimes further integrability or consistency conditions are implied. Sometimes a changes of coordinates can help greatly - even to uncovering hidden linearity. Sometimes subfamilies of solutions (e.g., similarity solutions) can be found which still have sufficient generally to be useful for the engineering or scientific problem at hand. Sometimes special solutions such as solitons exist which cannot have been guessed at perturbative methods, and which turn out to be fundamental and (in a nonlinear way ) superimposable. All these approaches, and many others, all seemingly disparate and ad hoc, can be systematically treated by a so-called geometric approach to partial differential equations initiated almost 50 years ago by Elie Cartan in France. This geometric approach has been little used until recently, as it has still seemed abstract to applied mathematicians. The goal of this research was to formulate and apply geometric methods, and to further their use as a tool of applied mathematics and (in possible discrete versions) computing.