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Quasi-rectifiable Lie algebras for partial differential equations

A. M. Grundland, J. de Lucas

TL;DR

The paper addresses identifying and exploiting quasi-rectifiable Lie algebras of vector fields to study hydrodynamic-type PDEs, using a rescaled Fröbenius framework to rectify noncommuting families. It develops a geometric and algebraic toolkit, including dual-form criteria, Frobenius-type results, and a classification up to dimension five, and connects these structures to PDE Lie systems and Hamiltonian dynamics. Key contributions include definitions of quasi-rectifiable vector fields and Lie algebras, practical rectification methods, a detailed 4D–5D classification, and explicit constructions of k-wave solutions for hydrodynamic-type equations. The framework enables systematic derivation of nonlinear superposition rules and Sundman-type transformations, with significant implications for Riemann invariants and multi-wave PDE solutions in fluid and gas dynamics.

Abstract

We introduce families of quasi-rectifiable vector fields and study their geometric and algebraic aspects. Then, we analyse their applications to systems of partial differential equations. Our results explain, in a simpler manner, previous findings about hydrodynamic-type equations. Facts concerning families of quasi-rectifiable vector fields, their relation to Hamiltonian systems, and practical procedures for studying such families are developed. We introduce and analyse quasi-rectifiable Lie algebras, which are motivated by geometric and practical reasons. We classify different types of quasi-rectifiable Lie algebras, e.g. indecomposable ones up to dimension five. New methods for solving systems of hydrodynamic-type equations are established to illustrate our results. In particular, we study hydrodynamic-type systems admitting $k$-wave solutions through quasi-rectifiable Lie algebras of vector fields. We develop techniques for obtaining the submanifolds related to quasi-rectifiable Lie algebras of vector fields and systems of partial differential equations admitting a nonlinear superposition rule: the PDE Lie systems.

Quasi-rectifiable Lie algebras for partial differential equations

TL;DR

The paper addresses identifying and exploiting quasi-rectifiable Lie algebras of vector fields to study hydrodynamic-type PDEs, using a rescaled Fröbenius framework to rectify noncommuting families. It develops a geometric and algebraic toolkit, including dual-form criteria, Frobenius-type results, and a classification up to dimension five, and connects these structures to PDE Lie systems and Hamiltonian dynamics. Key contributions include definitions of quasi-rectifiable vector fields and Lie algebras, practical rectification methods, a detailed 4D–5D classification, and explicit constructions of k-wave solutions for hydrodynamic-type equations. The framework enables systematic derivation of nonlinear superposition rules and Sundman-type transformations, with significant implications for Riemann invariants and multi-wave PDE solutions in fluid and gas dynamics.

Abstract

We introduce families of quasi-rectifiable vector fields and study their geometric and algebraic aspects. Then, we analyse their applications to systems of partial differential equations. Our results explain, in a simpler manner, previous findings about hydrodynamic-type equations. Facts concerning families of quasi-rectifiable vector fields, their relation to Hamiltonian systems, and practical procedures for studying such families are developed. We introduce and analyse quasi-rectifiable Lie algebras, which are motivated by geometric and practical reasons. We classify different types of quasi-rectifiable Lie algebras, e.g. indecomposable ones up to dimension five. New methods for solving systems of hydrodynamic-type equations are established to illustrate our results. In particular, we study hydrodynamic-type systems admitting -wave solutions through quasi-rectifiable Lie algebras of vector fields. We develop techniques for obtaining the submanifolds related to quasi-rectifiable Lie algebras of vector fields and systems of partial differential equations admitting a nonlinear superposition rule: the PDE Lie systems.
Paper Structure (13 sections, 16 theorems, 195 equations, 2 figures, 4 tables)

This paper contains 13 sections, 16 theorems, 195 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Quasi-rectifiable families of vector fields
  3. Methods for constructing quasi-rectifiable families of vector fields
  4. Integrable systems arising from quasi-rectifiable families of vector fields
  5. Analysis of quasi-rectifiable Lie algebras
  6. Definition and general properties
  7. On quasi-rectifiable two- and three-dimensional Lie algebras
  8. On four-, five- and higher-dimensional quasi-rectifiable Lie algebras
  9. Applications to hydrodynamic-type systems
  10. Solutions of (1+1)-dimensional hydrodynamic system
  11. Barotropic fluid flow in $(2k+1)$-dimensions
  12. $k$-wave solutions involving quasi-rectifiable Lie algebras
  13. Appendix: Classification of quasi-rectifiable indecomposable Lie algebras

Key Result

Theorem 2.1

Let $X_1,\ldots ,X_r$ be a family of vector fields on $N$ such that $X_1\wedge\ldots\wedge X_r$ does not vanish on $N$. There exists a coordinate system $\{x^1,\ldots,x^n\}$ on $N$ such that the integral curves of each $X_i$ are given by $x^1=k_1,\ldots,{x}^{i-1}=k_{i-1},{x}^{i+1}=k_{i+1},\ldots,x^n for a family of $r(r-1)$ functions $f_{ij}^i,f_{ij}^j\in C^\infty(N)$ with $1\leq i<j\leq r$.

Figures (2)

  • Figure 1: Hexagon centred in the $\mathbb{R}^2$ plane with three vertices on the horizontal axis. This represents the root diagram for $\mathfrak{sl}_3$. Note that $H_1,H_4$ belong to $G$, while $B$ and $A$ contain $H_2$ and $H_3$, respectively. Moreover, $C$, $D$, $E$ and $F$ have $H_7$, $H_5$, $H_6$ and $H_8$, respectively.
  • Figure 2: Schematic root diagram for $\mathfrak{sl}_4$, which has 6 positive roots $\alpha_1,\alpha_2,\alpha_3,\alpha_1+\alpha_2,\alpha_2+\alpha_3,\alpha_1+\alpha_2+\alpha_3$ (cf. GHLMMRRT20). Each root is represented by its eigenvalues relative to a basis of the Cartan algebra for $\mathfrak{sl}_4$. Note that the plane $z=0$ represents the root diagram for $\mathfrak{sl}_3$.

Theorems & Definitions (33)

  • Theorem 2.1
  • proof
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Proposition 3.3
  • ...and 23 more