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Lagrangian multiforms and dispersionless integrable systems.

Evgeny V Ferapontov1, Mats Vermeeren1

  • 1Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU UK.

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Summary
This summary is machine-generated.

Lagrangian multiforms are shown to be integral to multidimensional dispersionless integrable systems. They appear as conservation laws in 3D partial differential equations and in Gibbons-Tsarev equations for 4D hydrodynamic reductions.

Keywords:
Dispersionless integrabilityGibbons–Tsarev equationsHigher conservation lawsHydrodynamic reductionsLagrangian multiforms

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Area of Science:

  • Mathematical Physics
  • Integrable Systems
  • Differential Geometry

Background:

  • Multidimensional dispersionless integrable systems are a key area of research in mathematical physics.
  • Lagrangian multiforms offer a powerful framework for studying these systems.
  • Understanding conservation laws and hydrodynamic reductions is crucial for analyzing complex PDEs.

Purpose of the Study:

  • To demonstrate the natural emergence of Lagrangian multiforms within multidimensional dispersionless integrable systems.
  • To connect Lagrangian multiforms to specific applications in 3D and 4D systems.
  • To highlight the role of these structures in conservation laws and hydrodynamic reductions.

Main Methods:

  • Analysis of linearly degenerate PDEs in 3D.
  • Investigation of Gibbons-Tsarev equations in the context of 4D heavenly type equations.
  • Application of Lagrangian multiform theory to identify conserved quantities and reduction structures.

Main Results:

  • Identification of interesting examples of Lagrangian multiforms as higher-order conservation laws for 3D linearly degenerate PDEs.
  • Demonstration of Lagrangian multiforms in the context of Gibbons-Tsarev equations for 4D hydrodynamic reductions.
  • Established a natural link between Lagrangian multiforms and key features of multidimensional integrable systems.

Conclusions:

  • Lagrangian multiforms are fundamental structures in the study of multidimensional dispersionless integrable systems.
  • The findings provide new insights into conservation laws and hydrodynamic reductions.
  • This work opens avenues for further exploration of Lagrangian multiforms in related mathematical physics contexts.