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Quasi-Lie schemes for PDEs

Jose F. Carinena, Janusz Grabowski, Javier de Lucas

Abstract

The theory of quasi-Lie systems, i.e. systems of first order ordinary differential equations which can be related via a generalised flow to Lie systems, is extended to systems of partial differential equations and its applications to obtaining $t$-dependent superposition rules and integrability conditions are analysed. We develop a procedure of constructing quasi-Lie systems through a generalisation to PDEs of the so-called theory of quasi-Lie schemes. Our techniques are illustrated with the analysis of Wess-Zumino-Novikov-Witten models, generalised Abel differential equations, Baecklund transformations, as well as other differential equations of physical and mathematical relevance.

Quasi-Lie schemes for PDEs

Abstract

The theory of quasi-Lie systems, i.e. systems of first order ordinary differential equations which can be related via a generalised flow to Lie systems, is extended to systems of partial differential equations and its applications to obtaining -dependent superposition rules and integrability conditions are analysed. We develop a procedure of constructing quasi-Lie systems through a generalisation to PDEs of the so-called theory of quasi-Lie schemes. Our techniques are illustrated with the analysis of Wess-Zumino-Novikov-Witten models, generalised Abel differential equations, Baecklund transformations, as well as other differential equations of physical and mathematical relevance.

Paper Structure

This paper contains 9 sections, 17 theorems, 100 equations, 1 table.

Table of Contents

  1. Introduction.
  2. Generalised flows and $t$-dependent polyvector fields
  3. Bundle transformations and integrability conditions
  4. PDE Lie systems: basics and new examples
  5. Families of PDEs, $t$-dependent superpositions rules and foliations.
  6. Generalised PDE Lie--Scheffers Theorem.
  7. Quasi-Lie systems and schemes for PDEs
  8. Quasi-Lie invariants and integrability
  9. Conclusions and Outlook

Key Result

Proposition 2.1

Any two integrable systems of PDEs related to $t$-dependent polyvector fields ${\bf X}_1$ and ${\bf X}_2$ are bundle $\phi$-related for some bundle isomorphism $\phi:\mathbb{R}^s\times N\rightarrow \mathbb{R}^s\times N$.

Theorems & Definitions (47)

  • Proposition 2.1
  • proof
  • Example 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 3.1
  • Remark 1
  • Example 3.1
  • Example 3.2
  • ...and 37 more