Lie modules analysis of hydrodynamic-type systems

Abstract: The objective of this paper is to study nonlinear superpositions of Riemann wave solutions admitted by quasilinear hyperbolic first-order systems of partial differential equations. Particular attention is devoted to the analysis of non-elastic wave superpositions that cannot be decomposed into pairwise independent interactions of waves (quasi-rectifiability). In the case of the compressible Euler system, we describe the structure of the infinite-dimensional Lie algebra of vector fields associated with waves. We prove that a certain class of Lie modules associated with a hydrodynamic-type systems can be uniquely transformed into a real Lie algebra in an angle-preserving manner. For the Euler system, we then demonstrate the connection between the transformed finite-dimensional Lie algebra and the infinite-dimensional algebra associated with waves. This enables a detailed investigation of the geometry of wave superpositions using tools from differential geometry and Lie group theory. In particular, we study the geometry of the manifold of wave superpositions in terms of deformations of submanifolds corresponding to Lie subalgebras associated with waves. Additionally, we present new methods and criteria for determining the quasi-rectifiability of Riemann k-waves.