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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.
Letters in Mathematical Physics
|November 10, 2025
Summary
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.
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.
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