Seminarium Gamma

This talk presents an analysis of the conditions for the existence of elastic versus non-elastic wave superpositions governed by the Euler system in (1+1)-dimensions. A review of recently obtained results is presented including the introduction of the notion of quasi-rectifiability of vector fields and its application to both elastic and non-elastic wave superpositions. It is shown that the smallest real Lie algebra containing vector fields associated with the waves admitted by the Euler system is isomorphic to an infinite-dimensional Lie algebra which is the semi-direct sum of an Abelian ideal and the three-dimensional real Lie algebra $K^+$ or $K^-$. The maximal Lie module corresponding to the Euler system can be transformed by an angle preserving transformation to a real algebra isomorphic to the Lie algebras $K^\pm$. The algebras $K^+$ and $K^-$ are quasi-rectifiable and describe the behavior of wave superpositions. Based on these facts, we are able to find a parametrization of the region of non-elastic wave superpositions which allows for the construction of the reduced form of the Euler system.