Nonlinear wave superpositions and quasi-rectifiable Lie modules

Abstract

This paper presents a study of nonlinear superpositions of Riemann wave solutions admitted by quasilinear hyperbolic first-order systems of partial differential equations. In particular, we focus on the Euler system and non-elastic wave superpositions that cannot be decomposed into pairwise independent interactions of waves. A crucial tool for this analysis is the property of quasi-rectifiability of the families of vector fields determined by this system. It imposes certain conditions to be satisfied by the commutators of these vector fields. They enable us to find a parametrization of the region of superpositions of Riemann waves which leads to a simplification of the initial system of equations. In order to identify non-elastic superpositions we prove that a class of Lie modules associated with them can be uniquely transformed into a real Lie algebra through an angle-preserving transformation. We are then able to select a particular basis of vector fields associated with a given module which ensures the property of quasi-rectifiability. That, in turn, allows us to construct the reduced form of the Euler system for which a non-elastic superposition of two Riemann waves is then derived analytically. A study of the geometry of the manifold of non-elastic wave superpositions in terms of deformations of submanifolds corresponding to the Lie algebras is performed. Finally, we adapt the described approach to the general form of a hydrodynamic-type system i.e., to arbitrary Lie modules of vector fields associated with such a system, providing the criteria for their quasi-rectifiability. A geometric interpretation of non-elastic wave superpositions in this system is given.