A connection between the Yang-Baxter relation for maps and the multidimensional consistency property of integrable equations on quad-graphs is investigated. The approach is based on the symmetry analysis of the corresponding equations. It is shown that the Yang-Baxter variables can be chosen as invariants of the multiparameter symmetry groups of the equations. We use the classification results by Adler, Bobenko, and Suris to demonstrate this method. Some new examples of Yang-Baxter maps are derived in this way from multifield integrable equations.

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Yang-Baxter maps and symmetries of integrable equations on quad-graphs

Using generalized hydrodynamics (GHD), we develop the Euler hydrodynamics of classical integrable field theory. Classical field GHD is based on a known formalism for Gibbs ensembles of classical fields, that resembles the thermodynamic Bethe ansatz of quantum models, which we extend to generalized Gibbs ensembles (GGEs). In general, GHD must take into account both solitonic and radiative modes of classical fields. We observe that the quasi-particle formulation of GHD remains valid for radiative modes, even though these do not display particle-like properties in their precise dynamics. We point out that because of a UV catastrophe similar to that of black body radiation, radiative modes suffer from divergences that restrict the set of finite-average observables; this set is larger for GGEs with higher conserved charges. We concentrate on the sinh-Gordon model, which only has radiative modes, and study transport in the domain-wall initial problem as well as Euler-scale correlations in GGEs. We confirm a variety of exact GHD predictions, including those coming from hydrodynamic projection theory, by comparing with Metropolis numerical evaluations.

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Generalized hydrodynamics of classical integrable field theory: the sinh-Gordon model

We show that well-chosen Lagrangians for a class of two-dimensional integrable lattice equations obey a closure relation when embedded in a higher dimensional lattice. On the basis of this property we formulate a Lagrangian description for such systems in terms of Lagrangian multiforms. We discuss the connection of this formalism with the notion of multidimensional consistency, and the role of the lattice from the point of view of the relevant variational principle.

/pdf/lagrangian-multiforms-and-multidimensional-consistency-1za9cbt2go.pdf

Lagrangian multiforms and multidimensional consistency

We consider 3D consistent systems of six possibly different quad-equations assigned to the faces of a cube. The well-known classification of 3D consistent quad-equations, the so-called ABS-list, is included in this situation. The extension of these equations to the whole lattice ℤ3 is possible by reflecting the cubes. For every quad-equation we will give at least one system included leading to a Backlund transformation and a zero-curvature representation which means that they are integrable.

https://www.tandfonline.com/doi/pdf/10.1142/S1402925111001647

Classification of 3d consistent quad-equations

The generalized symmetry method is applied to a class of completely discrete equations including the Adler–Bobenko–Suris list. Assuming the existence of a generalized symmetry, we derive a few integrability conditions suitable for testing and classifying the equations of this class. Those conditions are used at the end to test for integrability discretizations of some well-known hyperbolic equations.

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The generalized symmetry method for discrete equations

We consider the partial difference equations of the Adler-Bobenko-Suris classification, which are characterized as multidimensionally consistent. The latter property leads naturally

Integrability and Symmetries of Difference Equations: the Adler-Bobenko-Suris Case

We study the deformations of the H equations, presented recently by Adler, Bobenko and Suris, which are naturally defined on a black–white lattice For each one of these equations, two different three-leg forms are constructed, leading to two different discrete Toda-type equations Their multidimensional consistency leads to Backlund transformations relating different members of this class as well as to Lax pairs Their symmetry analysis is presented yielding infinite hierarchies of generalized symmetries

Symmetries and integrability of discrete equations defined on a black?white lattice

We attempt to propose an algebraic approach to the theory of integrable difference equations. We define the concept of a recursion operator for difference equations and show that it generates an infinite sequence of symmetries and canonical conservation laws for a difference equation. As in the case of partial differential equations, these canonical densities can serve as integrability conditions for difference equations. We obtain the recursion operators for the Viallet equation and all the Adler-Bobenko-Suris equations.

Recursion operators, conservation laws, and integrability conditions for difference equations

In this paper we make an attempt to give a consistent background and definitions suitable for the theory of integrable difference equations. We adapt a concept of recursion operator to difference equations and show that it generates an infinite sequence of symmetries and canonical conservation laws for a difference equation. Similar to the case of partial differential equations these canonical densities can serve as integrability conditions for difference equations. We have found the recursion operators for the Viallet and all ABS equations.

Recursion operators, conservation laws and integrability conditions for difference equations

Integrability conditions for difference equations admitting a second order formal recursion operator are presented and the derivation of symmetries and canonical conservation laws are discussed. In a generic case, some of these conditions yield nonlocal conservation laws. A new integrable equation satisfying the second order integrability conditions is presented and its integrability is established by the construction of symmetries, conservation laws and a 3 × 3 Lax representation. Finally, via the relation of the symmetries of this equation to the Bogoyavlensky lattice, an integrable asymmetric quad equation and a consistent pair of difference equations are derived.

/pdf/second-order-integrability-conditions-for-difference-2a40ty31w6.pdf

Pavlos Xenitidis

Papers

Integrability and Symmetries of Difference Equations: the Adler-Bobenko-Suris Case

Symmetries and integrability of discrete equations defined on a black?white lattice

Recursion operators, conservation laws, and integrability conditions for difference equations

Recursion operators, conservation laws and integrability conditions for difference equations

Second order integrability conditions for difference equations. An integrable equation.