Multidimensional Nonlinear Equations

Abstract

Until now, we have analyzed the properties of one-dimensional fields as a function of time t and only one spatial coordinate. Such fields usually appear in an idealized, simplified description of real processes happening in the three-dimensional space. On the other hand, many ideas and methods of solving partial differential nonlinear equations for one-dimensional fields are in a natural way carried over to the richer with geometric and mechanical developments multidimensional case. Therefore, for completeness, let us discuss multidimensional equations related to those one-dimensional partial differential nonlinear equations of the first order which have been studied in the previous chapter. Complementary information on dynamics and statistics of three-dimensional fields of the hydrodynamic type can be found in [1–13].