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Lagrangian multiforms and multidimensional consistency
"... We show that wellchosen Lagrangians for a class of twodimensional 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 ..."
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Cited by 22 (5 self)
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We show that wellchosen Lagrangians for a class of twodimensional 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.
Symmetries of integrable difference equations on the quadgraph
"... This paper describes symmetries of all integrable difference equations that belong to the famous AdlerBobenkoSuris classification. For each equation, the characteristics of symmetries satisfy a functional equation, which we solve by reducing it to a system of partial differential equations. In thi ..."
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Cited by 18 (1 self)
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This paper describes symmetries of all integrable difference equations that belong to the famous AdlerBobenkoSuris classification. For each equation, the characteristics of symmetries satisfy a functional equation, which we solve by reducing it to a system of partial differential equations. In this way, all fivepoint symmetries of integrable equations on the quadgraph are found. These include mastersymmetries, which allow one to construct infinite hierarchies of local symmetries. We also demonstrate a connection between the symmetries of quadgraph equations and those of the corresponding Toda type difference equations. 1
Searching for integrable lattice maps using factorization
 P. Phys. A
"... We analyze the factorization process for lattice maps, searching for integrable cases. The maps were assumed to be at most quadratic in the dependent variables, and we required minimal factorization (one linear factor) after 2 steps of iteration. The results were then classified using algebraic entr ..."
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Cited by 8 (4 self)
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We analyze the factorization process for lattice maps, searching for integrable cases. The maps were assumed to be at most quadratic in the dependent variables, and we required minimal factorization (one linear factor) after 2 steps of iteration. The results were then classified using algebraic entropy. Some new models with polynomial growth (strongly associated with integrability) were found. One of them is a nonsymmetric generalization of the homogeneous quadratic maps associated with KdV (modified and Schwarzian), for this new model we have also verified the “consistency around a cube”. 1
Multiquadratic quad equations: integrable cases from a factorised discriminant hypothesis
 Intl. Math. Res. Not
"... Abstract. We give integrable quad equations which are multiquadratic (degreetwo) counterparts of the wellknown multiaffine (degreeone) equations classified by Adler, Bobenko and Suris (ABS). These multiquadratic equations define multivalued evolution from initial data, but our construction is ..."
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Cited by 7 (2 self)
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Abstract. We give integrable quad equations which are multiquadratic (degreetwo) counterparts of the wellknown multiaffine (degreeone) equations classified by Adler, Bobenko and Suris (ABS). These multiquadratic equations define multivalued evolution from initial data, but our construction is based on the hypothesis that discriminants of the defining polynomial factorise in a particular way that allows to reformulate the equation as a singlevalued system. Such reformulation comes at the cost of introducing auxiliary (edge) variables and augmenting the initial data. Like the multiaffine equations listed by ABS, these new models are consistent in multidimensions. We clarify their relationship with the ABS list by obtaining Bäcklund transformations connecting all but the primary multiquadratic model back to equations from the multiaffine class. 1.
Hierarchies of nonlinear integrable qdifference equations from series of Lax pairs
 J. Phys. A
"... Abstract. We present, for the first time, two hierarchies of nonlinear, integrable qdifference equations, one of which includes a qdifference form of each of the second and fifth Painlevé equations, qP II and qP V , the other includes qP III . All the equations have multiple free parameters. A me ..."
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Cited by 5 (3 self)
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Abstract. We present, for the first time, two hierarchies of nonlinear, integrable qdifference equations, one of which includes a qdifference form of each of the second and fifth Painlevé equations, qP II and qP V , the other includes qP III . All the equations have multiple free parameters. A method to calculate a 2 × 2 Lax pair for each equation in the hierarchy is also given.
On a TwoParameter Extension of the Lattice KdV System Associated with an Elliptic Curve
, 2003
"... A general structure is developed from which a system of integrable partial di#erence equations is derived generalising the lattice KdV equation. The construction is based on an infinite matrix scheme with as key ingredient a (formal) elliptic Cauchy kernel. ..."
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Cited by 4 (2 self)
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A general structure is developed from which a system of integrable partial di#erence equations is derived generalising the lattice KdV equation. The construction is based on an infinite matrix scheme with as key ingredient a (formal) elliptic Cauchy kernel.
Symmetries and special solutions of reductions of the lattice potential KdV equation
"... Abstract. We identify a periodic reduction of the nonautonomous lattice potential Korteweg–de Vries equation with the additive discrete Painlevé equation with E (1) 6 symmetry. We present a description of a set of symmetries of the reduced equations and their relations to the symmetries of the disc ..."
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Cited by 3 (0 self)
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Abstract. We identify a periodic reduction of the nonautonomous lattice potential Korteweg–de Vries equation with the additive discrete Painlevé equation with E (1) 6 symmetry. We present a description of a set of symmetries of the reduced equations and their relations to the symmetries of the discrete Painlevé equation. Finally, we exploit the simple symmetric form of the reduced equations to find rational and hypergeometric solutions of this discrete Painlevé equation. Key words: difference equations; integrability; reduction; isomonodromy
On nonmultiaffine consistent around the cube lattice equations
"... We show that integrable involutive maps, due to the fact they admit three integrals in separated form, cangiverisetoequationswhichareconsistentaroundthecubeandwhicharenotinthemultiaffine form assumed in papers [1, 2]. In the examples of maps presented here the equations are related to lattice potent ..."
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Cited by 3 (2 self)
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We show that integrable involutive maps, due to the fact they admit three integrals in separated form, cangiverisetoequationswhichareconsistentaroundthecubeandwhicharenotinthemultiaffine form assumed in papers [1, 2]. In the examples of maps presented here the equations are related to lattice potential KdV equation by nonlocal transformations (discrete quadratures).