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Lie module analysis of hydrodynamic-type systems

Łukasz Chomienia, Alfred Michel Grundland

TL;DR

The paper addresses the analytic description of nonlinear, non-elastic wave interactions in hydrodynamic-type PDEs using a Lie-module and differential-geometric framework. It develops an angle-preserving transformation that maps Lie modules to real Lie algebras, enabling a unified geometric description of wave superpositions through Lie-group ideas; the one-dimensional Euler system provides an explicit realization via the Witt algebra and related semidirect products, with deformations of quasi-rectifiable surfaces described as parallel transport. It also introduces curvature-based criteria for quasi-rectifiability, presents a reduced Euler system through a suitable basis, and generalizes the framework to arbitrary hydrodynamic-type systems with a classification of rescalable Lie modules. Together, these results yield a rigorous geometric and algebraic path to understanding multi-wave interactions and their manifold structures, with concrete Euler-system exemplars such as $L(2,1)\oplus L(1,0)$ illustrating the approach.

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 module 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.

Lie module analysis of hydrodynamic-type systems

TL;DR

The paper addresses the analytic description of nonlinear, non-elastic wave interactions in hydrodynamic-type PDEs using a Lie-module and differential-geometric framework. It develops an angle-preserving transformation that maps Lie modules to real Lie algebras, enabling a unified geometric description of wave superpositions through Lie-group ideas; the one-dimensional Euler system provides an explicit realization via the Witt algebra and related semidirect products, with deformations of quasi-rectifiable surfaces described as parallel transport. It also introduces curvature-based criteria for quasi-rectifiability, presents a reduced Euler system through a suitable basis, and generalizes the framework to arbitrary hydrodynamic-type systems with a classification of rescalable Lie modules. Together, these results yield a rigorous geometric and algebraic path to understanding multi-wave interactions and their manifold structures, with concrete Euler-system exemplars such as illustrating the approach.

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 module 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.

Paper Structure

This paper contains 19 sections, 28 theorems, 246 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wave superpositions
  4. Elastic versus non-elastic wave superpositions
  5. Geometric aspects of quasi-rectifiable families of vector fields
  6. Characterization of quasi-rectifiability in terms of curvature
  7. Quasi-rectifiable families of vector fields in $\mathbb{R}^3$
  8. The one-dimensional compressible Euler system
  9. Representation via infinite-dimensional Lie algebras
  10. Rescaling of the module related to the Euler system
  11. Determining a parametrization of a superposition of waves
  12. Deformations of the quasi-rectifiable surface
  13. Curvatures of the quasi-rectifiable surface
  14. Reduced form of the Euler system
  15. Geometry of general Lie modules associated with waves
  16. ...and 4 more sections

Key Result

Theorem 2.1

(Straightening of vector fields Gru3) Let $X_1,...,X_r$ be a family of vector fields defined on an $n$-dimensional manifold $N$ such that There exists a local coordinate system $\left\{x^1,...,x^n\right\}$ on $N$ such that the first integrals of each vector field $X_i$ are given by the equations for some constants $k_1,...,\hat{k}_i,...,k_n\in\mathbb{R}$, if and only if the commutators for each

Theorems & Definitions (75)

  • Theorem 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Corollary 2.10
  • ...and 65 more