Nonlinear Superposition of Simple Waves in Nonhomogeneous Systems
In this paper we develop a method of constructing the solutions of nonelliptic systems of nonhomogeneous quasilinear partial differential equations of the first order. This method comes from a concept of Riemann’s invariants for homogeneous systems, but it can be generalized for a larger region of solutions than those described by Riemann’s invariants. The base for the construction of solutions are called simple integral elements resulting from the algebraization of the initial system of equations. The notion of the simple state corresponding to the elementary solution of the nonhomogeneous system is introduced, and then some more complicated forms of solutions are proposed and the conditions for their existence are found. The last problem, similarly to the case of the homogeneous systems, is reduced to examining Pfaff’s forms with the use of Cartan’s theory of systems in involution. We propose the physical interpretation of the obtained classes of solutions as a description of the nonlinear superposition of simple waves propagating on a simple state. This method was illustrated by applying it to the Korteweg de Vries equation and to nonhomogeneous equations of magnetohydrodynamics.