Simple waves in quasilinear hyperbolic systems. I. Theory of simple waves and simple states. Examples of applications

Abstract

This paper presents a new method of construction of solutions to nonlinear, nonelliptic systems of partial differential equations and especially nonhomogeneous ones. These equations have been considered from the point of view of integral elements. In particular the connections between the structure of the set of integral elements and the possibility of a construction of special classes of solutions have been studied. These classes consist of what is called simple waves and k waves (for homogeneous systems) and simple states (in the case of nonhomogeneous systems). They provide us with a possibility for a selection of simple integral elements from the set of all integral elements. Analyses have been performed using differential forms and Cartan theory of system in involution. The problem has been reduced to examining Pfaff forms. The Cauchy problem for Pfaff systems has been formulated and solved using the Riemann function. Some remarks concerning the notion of Bäcklund transformations for the case of k waves have been formulated. It is shown that, in contrast to simple wave, the simple state has no gradient catastrophy. The technique presented of constructing the solutions in form of simple states has been illustrated by the examples of Korteweg and de Vries and four-dimensional Klein–Gordon, sine–Gordon, and Liouville equations. It has been shown that the known soliton equations are closely connected with the elliptical functions and especially with the P–Weierstrass functions.