First-Order Nonlinear PDEs and Conservation Laws

Abstract

Linear partial differential equations discussed in Part III often offer only a very simplified description of physical phenomena. To get a deeper understanding of some of them, it is necessary to move beyond the linear “universe” and consider nonlinear models, which in the case of continuous media, means nonlinear partial differential equations. Even today, their theory is far from complete and is the subject of intense study. On closer inspection, almost all physical phenomena in continuous media—from growing molecular interfaces at atomic scales to the structure of the distribution of matter in the universe at intergalactic scales—are nonlinear. The variety of nonlinear physical phenomena necessitates the use of various mathematical models and techniques to study them. In this part we shall restrict our attention to nonlinear waves of hydrodynamic type in media with weak or no dispersion. Since weak dispersion has little influence on the development of many nonlinear effects, we shall have a chance to observe typical behavior of these systems in strongly nonlinear regimes. The basic features of strongly nonlinear fields and waves are already evident in solutions of first-order nonlinear partial differential equations, and we take them as our starting point.