On the Variational Interpretation of the Discrete KP Equation

Boll, Raphael; Petrera, Matteo; Suris, Yuri B.

doi:10.1007/978-3-662-50447-5_12

Cited by 5 publications

(8 citation statements)

References 18 publications

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“…while the closure relation (3.3) was proven in [15]. A geometrical interpretation of the Lagrangian structure of the discrete KP equation was given in the recent paper [7].…”

Section: Example: Bilinear Discrete Kpmentioning

confidence: 94%

“…In that paper a description of the system was given in terms of a discrete Lagrangian 3-form structure, and its closure relation was proven. In a recent paper [7] a more geometric interpretation of that result was provided. Here we define the general set-up for the variational approach, in terms of Lagrangian multiforms, of 3D discrete systems.…”

Section: Discrete 3-dimensional Systemsmentioning

confidence: 94%

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A Variational Principle for Discrete Integrable Systems

Lobb¹,

Nijhoff²

2018

SIGMA

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For integrable systems in the sense of multidimensional consistency (MDC) we can consider the Lagrangian as a form, which is closed on solutions of the equations of motion. For 2-dimensional systems, described by partial difference equations with two independent variables, MDC allows us to define an action on arbitrary 2-dimensional surfaces embedded in a higher dimensional space of independent variables, where the action is not only a functional of the field variables but also the choice of surface. It is then natural to propose that the system should be derived from a variational principle which includes not only variations with respect to the dependent variables, but also with respect to variations of the surface in the space of independent variables. Here we derive the resulting system of generalized Euler-Lagrange equations arising from that principle. We treat the case where the equations are 2 dimensional (but which due to MDC can be consistently embedded in higher-dimensional space), and show that they can be integrated to yield relations of quadrilateral type. We also derive the extended set of Euler-Lagrange equations for 3dimensional systems, i.e., those for equations with 3 independent variables. The emerging point of view from this study is that the variational principle can be considered as the set of equations not only encoding the equations of motion but as the defining equations for the Lagrangians themselves.

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“…while the closure relation (3.3) was proven in [15]. A geometrical interpretation of the Lagrangian structure of the discrete KP equation was given in the recent paper [7].…”

Section: Example: Bilinear Discrete Kpmentioning

confidence: 94%

Section: Discrete 3-dimensional Systemsmentioning

confidence: 94%

A Variational Principle for Discrete Integrable Systems

Lobb¹,

Nijhoff²

2018

SIGMA

View full text Add to dashboard Cite

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“…is the variational derivative in the (t i , t j )-plane. Note that the multi-time Euler-Lagrange equations contain the classical Euler-Lagrange equations in each (t i , t j )plane (8), where derivatives with respect to other times are considered as additional components of the field, plus additional equations (9)-(10) coming from choices of that are not coordinate planes.…”

Section: Pluri-lagrangian Problemsmentioning

confidence: 99%

“…The pluri-Lagrangian property requires the action to be critical also when this plane is replaced by any other 2-dimensional discrete surface in a higher-dimensional lattice. This remarkable property has been considered as a defining feature of integrability of 2-dimensional discrete equations [2,4,6,9,[13][14][15]32] as well as in the 1-dimensional [5,7,33] and 3-dimensional [8,16] cases.…”

Section: Introductionmentioning

confidence: 99%

Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs

Petrera

Vermeeren

2020

European Journal of Mathematics

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We investigate the relation between pluri-Lagrangian hierarchies of 2-dimensional partial differential equations and their variational symmetries. The aim is to generalize to the case of partial differential equations the recent findings in Petrera and Suris (Nonlinear Math. Phys. 24(suppl. 1):121–145, 2017) for ordinary differential equations. We consider hierarchies of 2-dimensional Lagrangian PDEs (many of which have a natural $$(1\,{+}\,1)$$ ( 1 + 1 ) -dimensional space-time interpretation) and show that if the flow of each PDE is a variational symmetry of all others, then there exists a pluri-Lagrangian 2-form for the hierarchy. The corresponding multi-time Euler–Lagrange equations coincide with the original system supplied with commuting evolutionary flows induced by the variational symmetries.

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“…The pluri-Lagrangian property requires the action to be critical also when this plane is replaced by any other 2-dimensional discrete surface in a higher-dimensional lattice. This remarkable property has been considered as a defining feature of integrability of 2-dimensional discrete equations [2,4,6,9,13,14,15,31] as well as in the 1-dimensional [5,7,32] and 3-dimensional [8,16] cases.…”

Section: Introductionmentioning

confidence: 99%

Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs

Petrera,

Vermeeren

2019

Preprint

Self Cite

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We investigate the relation between pluri-Lagrangian hierarchies of 2-dimensional partial differential equations and their variational symmetries. The aim is to generalize to the case of partial differential equations the recent findings in [Petrera, Suris. J. Nonlinear Math. Phys. 24:sup1, 121-145 (2017)] for ordinary differential equations. We consider hierarchies of 2-dimensional Lagrangian PDEs (many of which have a natural (1 + 1)-dimensional space-time interpretation) and show that if the flow of each PDE is a variational symmetry of all others, then there exists a pluri-Lagrangian 2-form for the hierarchy. The corresponding multi-time Euler-Lagrange equations coincide with the original system supplied with commuting evolutionary flows induced by the variational symmetries.

show abstract

On the Variational Interpretation of the Discrete KP Equation

Cited by 5 publications

References 18 publications

A Variational Principle for Discrete Integrable Systems

A Variational Principle for Discrete Integrable Systems

Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs

Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs

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