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On Riemann wave superpositions obtained from the Euler system

Łukasz Chomienia, Alfred Michel Grundland

TL;DR

This work develops an algebraic-geometric framework for analyzing Riemann wave superpositions in the ($1+1$)-dimensional Euler system. It introduces quasi-rectifiability of vector fields and shows that the minimal real Lie algebra containing the wave fields is infinite-dimensional, realized as a semidirect sum with Abelian ideals and Virasoro/Witt-like components, enabling a unified view of elastic and non-elastic interactions. Elastic two-wave superpositions, encoded by quasi-rectifiable pairs such as $(oldsymbol{ u}_+,oldsymbol{ u}_-)$, admit rescalings that commute and yield explicit double-wave solutions with a reduced Euler system; non-elastic interactions involve the entropic mode and are analyzed through a parametrization of the interaction region and a corresponding reduced system (κ=3). The paper further shows that the wave-surfaces carry local Lie-group structures, with affine connections linking the geometry to wave evolution via parallel transport, offering a geometric route to systematic reduction and interpretation of wave interactions in compressible flows.

Abstract

The paper contains 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. The maximal Lie module corresponding to the Euler system can be transformed, by an angle preserving transformation, to this algebra which is quasi-rectifiable and describes 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.

On Riemann wave superpositions obtained from the Euler system

TL;DR

This work develops an algebraic-geometric framework for analyzing Riemann wave superpositions in the ()-dimensional Euler system. It introduces quasi-rectifiability of vector fields and shows that the minimal real Lie algebra containing the wave fields is infinite-dimensional, realized as a semidirect sum with Abelian ideals and Virasoro/Witt-like components, enabling a unified view of elastic and non-elastic interactions. Elastic two-wave superpositions, encoded by quasi-rectifiable pairs such as , admit rescalings that commute and yield explicit double-wave solutions with a reduced Euler system; non-elastic interactions involve the entropic mode and are analyzed through a parametrization of the interaction region and a corresponding reduced system (κ=3). The paper further shows that the wave-surfaces carry local Lie-group structures, with affine connections linking the geometry to wave evolution via parallel transport, offering a geometric route to systematic reduction and interpretation of wave interactions in compressible flows.

Abstract

The paper contains 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. The maximal Lie module corresponding to the Euler system can be transformed, by an angle preserving transformation, to this algebra which is quasi-rectifiable and describes 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.

Paper Structure

This paper contains 7 sections, 18 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Elastic and non-elastic wave superpositions
  3. Elastic wave superposition in the Euler system
  4. Infinite-dimensional Lie algebra
  5. Parametrization of the region of wave superpositions
  6. Reduced form of the Euler system
  7. Surfaces with Lie group structure

Key Result

Theorem 1.1

Suppose that $(\lambda,\gamma)$ is a set of $C^1$ functions satisfying the algebraic equation algebra-ic and that $f:\mathbb{R} \to \mathbb{R}^q$ is an integral curve $\Gamma$ of the vector field $\gamma^{\alpha}(u)\frac{\partial}{\partial u^{\alpha}}$ on $\mathbb{R}^q$ with parameter $r$, i.e. Then, the relations (where $\phi$ is an arbitrary function of one variable $\lambda_i(r)x^i$) constitu

Theorems & Definitions (21)

  • Theorem 1.1: Riemann wave solution $[1,2]$
  • Theorem 1.2: Straightening of vector fields $[4]$
  • Definition 1.3: Quasi-rectifiability property $[4]$
  • Theorem 1.4: The rescaling theorem ${[4]}$
  • Theorem 1.5: Integrating factor ${[4]}$
  • Corollary 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 11 more