Lagrangian multiform structure for the lattice KP system

Lobb, Sarah; Nijhoff, F.W.; Quispel, G.

doi:10.1088/1751-8113/42/47/472002

Cited by 47 publications

(67 citation statements)

References 17 publications

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“…This line of research was developed in several directions, mainly by two teams: by Nijhoff with collaborators (under the name "theory of Lagrangian multi-forms"), see [2,[11][12][13]21], and by the authors of the present paper with collaborators, who termed the corresponding structures "pluri-Lagrangian", see [7][8][9][10][17][18][19][20]. As argued in [8], the unconventional idea to consider the action on arbitrary two-dimensional surfaces in the multi-dimensional space of independent variables has significant precursors.…”

Section: Introductionmentioning

confidence: 99%

Variational symmetries and pluri-Lagrangian systems in classical mechanics

Petrera¹,

Suris²

2021

JNMP

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We analyze the relation of the notion of a pluri-Lagrangian system, which recently emerged in the theory of integrable systems, to the classical notion of variational symmetry, due to E. Noether. We treat classical mechanical systems and show that, for any Lagrangian system with m commuting variational symmetries, one can construct a pluri-Lagrangian 1-form in the (m + 1)-dimensional time, whose multi-time Euler-Lagrange equations coincide with the original system supplied with m commuting evolutionary flows corresponding to the variational symmetries. We also give a Hamiltonian counterpart of this construction, leading, for any system of commuting Hamiltonian flows, to a pluri-Lagrangian 1-form with coefficients depending on functions in the phase space.

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Section: Introductionmentioning

confidence: 99%

Variational symmetries and pluri-Lagrangian systems in classical mechanics

Petrera¹,

Suris²

2021

JNMP

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“…Another important property of the classical star-triangle relation (1.4), is that this relation implies the invariance of the action functional A(x), under "star-triangle" transformations of the lattice [10,12,50], which is a natural counterpart of Baxter's Z-invariance for lattice models. The classical star-triangle relation (1.4) and the associated constraint equation (1.5) are also related [50] to the 3D consistency relation [15] and more generally to the multiform Lagrangian structures and multidimensional consistency [50,52], further studied in the recent papers [53][54][55][56][57][58].…”

Section: Introductionmentioning

confidence: 99%

Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs

Bazhanov¹,

Kels²,

Сергеев³

2016

J. Phys. A: Math. Theor.

109

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In this paper we give an overview of exactly solved edge-interaction models, where the spins are placed on sites of a planar lattice and interact through edges connecting the sites. We only consider the case of a single spin degree of freedom at each site of the lattice. The Yang-Baxter equation for such models takes a particular simple form called the star-triangle relation. Interestigly all known solutions of this relation can be obtained as particular cases of a single "master solution", which is expressed through the elliptic gamma function and have continuous spins taking values on the circle. We show that in the low-temperature (or quasi-classical) limit these lattice models reproduce classical discrete integrable systems on planar graphs previously obtained and classified by Adler, Bobenko and Suris through the consistency-around-a-cube approach. We also discuss inversion relations, the physicical meaning of Baxter's rapidity-independent parameter in the startriangle relations and the invariance of the action of the classical systems under the star-triangle (or cube-flip) transformation of the lattice, which is a direct consequence of Baxter's Z-invariance in the associated lattice models.

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“…The Lagrangian multiform structure has become one of the main research topics in the integrable systems after pioneer works were initiated by Lobb and Nijhoff [1][2][3]. In these works, the discrete Lagrangian 2-form and 3-form for the systems with infinite degrees of freedom had been shown to possess a remarkable property called the closure relation resulting from the variational principle on the space of independent variables.…”

Section: Introductionmentioning

confidence: 99%

On the lagrangian 1-form structure of the hyperbolic calogero–moser system

Jairuk

Танаситтикосол

Yoo-Kong

2017

Reports on Mathematical Physics

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In this work, we present another example of the Lagrangian 1-form structure for the hyperbolic Calogero-Moser system both in discrete-time level and continuous-time level. The discrete-time hyperbolic Calogero-Moser system is obtained by considering pole-reduction of the semi-discrete Kadomtsev-Petviashvili (KP) equation. The key relation called the discrete-time closure relation is directly obtained from the compatibility between the temporal Lax matrices. The continuous-time hierarchy of the hyperbolic Calogero-Moser system is obtained through two successive continuum limits. The continuous-time closure relation, which is a consequence of continuum limits on the discrete-time one, is also shown to hold.

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Lagrangian multiform structure for the lattice KP system

Cited by 47 publications

References 17 publications

Variational symmetries and pluri-Lagrangian systems in classical mechanics

Variational symmetries and pluri-Lagrangian systems in classical mechanics

Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs

On the lagrangian 1-form structure of the hyperbolic calogero–moser system

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