Lagrangian multiform structure of discrete and semi-discrete KP systems
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913459250135040 |
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| author | Nijhoff, Frank W |
| author_facet | Nijhoff, Frank W |
| contents | A variational structure for the potential AKP system is established using the novel formalism of a Lagrangian multiforms. The structure comprises not only the fully discrete equation on the 3D lattice, but also its semi-discrete variants including several differential-difference equations asssociated with, and compatible with, the partial difference equation. To this end, an overview is given of the various (discrete and semi-discrete) variants of the KP system, and their associated Lax representations, including a novel `generating PDE' for the KP hierarchy. The exterior derivative of the Lagrangian 3-form for the lattice potential KP equation is shown to exhibit a double-zero structure, which implies the corresponding generalised Euler-Lagrange equations. Alongside the 3-form structures, we develop a variational formulation of the corresponding Lax systems via the square eigenfunction representation arising from the relevant direct linearization scheme. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_13423 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lagrangian multiform structure of discrete and semi-discrete KP systems Nijhoff, Frank W Exactly Solvable and Integrable Systems Mathematical Physics A variational structure for the potential AKP system is established using the novel formalism of a Lagrangian multiforms. The structure comprises not only the fully discrete equation on the 3D lattice, but also its semi-discrete variants including several differential-difference equations asssociated with, and compatible with, the partial difference equation. To this end, an overview is given of the various (discrete and semi-discrete) variants of the KP system, and their associated Lax representations, including a novel `generating PDE' for the KP hierarchy. The exterior derivative of the Lagrangian 3-form for the lattice potential KP equation is shown to exhibit a double-zero structure, which implies the corresponding generalised Euler-Lagrange equations. Alongside the 3-form structures, we develop a variational formulation of the corresponding Lax systems via the square eigenfunction representation arising from the relevant direct linearization scheme. |
| title | Lagrangian multiform structure of discrete and semi-discrete KP systems |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics |
| url | https://arxiv.org/abs/2406.13423 |