Generalized Solutions of Nonlinear Equations

Abstract

Equations of mathematical physics and, in particular, nonlinear partial differential equations of the first order appear as a result of certain idealizations. This allows one to achieve elegance of mathematical models, a possibility to use them in order to predict, in an adequately quantitative way, important aspects of various real world phenomena. But any idealizations come at a cost. Factors unaccounted for by these idealizations gradually, and sometimes abruptly, begin to dominate, while the initial models cease to be able to describe what actually happens.