First-Order Nonlinear Equations and Their Applications

Abstract

First-order, nonlinear, partial differential equations arise in various areas of physical sciences which include geometrical optics, fluid dynamics, and analytical dynamics. An important example of such equations is the Hamilton–Jacobi equation used to describe dynamical systems. Another famous example of the first-order nonlinear equations is the eikonal equation which arises in nonlinear optics and also describes the propagation of wave fronts and discontinuities for acoustic wave equations, Maxwell’s equations, and equations of elastic wave propagation. Evidently, first-order, nonlinear equations play an important role in the development of these diverse areas.

Physics can’t exist without mathematics which provides it with the only language in which it can speak. Thus, services are continuously exchanged between pure mathematical analysis and physics. It is really remarkable that among works of analysis most useful for physics were those cultivated for their own beauty. In exchange, physics, exposing new problems, is as useful for mathematics as it is a model for an artist.

Henri Poincaré

Our present analytical methods seem unsuitable for the solution of the important problems arising in connection with nonlinear partial differential equations and, in fact, with virtually all types of nonlinear problems in pure mathematics. The truth of this statement is particularly striking in the field of fluid dynamics. Only the most elementary problems have been solved analytically in this field…

John Von Neumann