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Lagrangian multiforms and dispersionless integrable systems

Evgeny V. Ferapontov, Mats Vermeeren

Abstract

We demonstrate that interesting examples of Lagrangian multiforms appear naturally in the theory of multidimensional dispersionless integrable systems as (a) higher-order conservation laws of linearly degenerate PDEs in 3D, and (b) in the context of Gibbons-Tsarev equations governing hydrodynamic reductions of heavenly type equations in 4D.

Lagrangian multiforms and dispersionless integrable systems

Abstract

We demonstrate that interesting examples of Lagrangian multiforms appear naturally in the theory of multidimensional dispersionless integrable systems as (a) higher-order conservation laws of linearly degenerate PDEs in 3D, and (b) in the context of Gibbons-Tsarev equations governing hydrodynamic reductions of heavenly type equations in 4D.

Paper Structure

This paper contains 20 sections, 3 theorems, 101 equations.

Table of Contents

  1. Lagrangian multiform formalism
  2. Multiform Euler-Lagrange equations
  3. Double zero property
  4. Lagrangian multiforms and conservation laws of linearly degenerate PDEs
  5. Veronese web hierarchy
  6. Translationally non-invariant Veronese web hierarchy
  7. Mikhalev equation
  8. Case 1.
  9. Case 2.
  10. Case 3.
  11. Lagrangian multiforms and Gibbons-Tsarev equations
  12. Second heavenly equation
  13. Lagrangian multiform for system \ref{['h7']}.
  14. First heavenly equation
  15. 6D version of the second heavenly equation
  16. ...and 5 more sections

Key Result

Theorem 1

If the differential $\mathrm{d} \mathcal{L}$ factorises as in equation factorisation, then the full system of multiform Euler-Lagrange equations mfEL follows from the system of equations $A_{ijk}=0, B_{ijk}=0.$

Theorems & Definitions (9)

  • Remark
  • Remark
  • Theorem 1
  • proof
  • Definition : ferapontov2004integrabilityferapontov2004hydrodynamic
  • Lemma 2
  • proof
  • Lemma 3
  • proof