Results 1  10
of
20
Soliton solutions for ABS lattice equations: I. Cauchy matrix approach
 J. Phys. A: Math Theor. Special
, 2009
"... Abstract. In recent years there have been new insights into the integrability of quadrilateral lattice equations, i.e. partial difference equations which are the natural discrete analogues of integrable partial differential equations in 1+1 dimensions. In the scalar (i.e. singlefield) case there no ..."
Abstract

Cited by 17 (2 self)
 Add to MetaCart
(Show Context)
Abstract. In recent years there have been new insights into the integrability of quadrilateral lattice equations, i.e. partial difference equations which are the natural discrete analogues of integrable partial differential equations in 1+1 dimensions. In the scalar (i.e. singlefield) case there now exist classification results by Adler, Bobenko and Suris (ABS) leading to some new examples in addition to the lattice equations “of KdV type ” that were known since the late 1970s and early 1980s. In this paper we review the construction of soliton solutions for the KdV type lattice equations and use those results to construct Nsoliton solutions for all lattice equations in the ABS list except for the elliptic case of Q4, which is left to a separate treatment. 1.
On Miura Transformations and VolterraType Equations Associated with the Adler–Bobenko–Suris Equations
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2008
"... We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler–Bobenko–Suris (ABS) list into the discrete Schrödinger spectral problem associated with Volterratype equations. We show that the ABS equations correspond to Bäcklund ..."
Abstract

Cited by 15 (8 self)
 Add to MetaCart
We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler–Bobenko–Suris (ABS) list into the discrete Schrödinger spectral problem associated with Volterratype equations. We show that the ABS equations correspond to Bäcklund transformations for some particular cases of the discrete Krichever–Novikov equation found by Yamilov (YdKN equation). This enables us to construct new generalized symmetries for the ABS equations. The same can be said about the generalizations of the ABS equations introduced by Tongas, Tsoubelis and Xenitidis. All of them generate Bäcklund transformations for the YdKN equation. The higher order generalized symmetries we construct in the present paper confirm their integrability.
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 ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
(Show Context)
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.
Linear quadrilateral lattice equations and multidimensional consistency
"... Abstract. It is shown that every scalar linear quadrilateral lattice equation lies within a family of similar equations, members of which are compatible between one another on a higher dimensional lattice. There turn out to be two such families, a natural parametrisation is given for each. PACS numb ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
(Show Context)
Abstract. It is shown that every scalar linear quadrilateral lattice equation lies within a family of similar equations, members of which are compatible between one another on a higher dimensional lattice. There turn out to be two such families, a natural parametrisation is given for each. PACS numbers: 02.30.Ik 1.
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 ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
(Show Context)
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).
Nonsymmetric discrete Toda systems from quadgraphs
, 908
"... For all nonsymmetric discrete relativistic Toda type equations we establish a relation to 3D consistent systems of quadequations. Unlike the more simple and better understood symmetric case, here the three coordinate planes of Z 3 carry different equations. Our construction allows for an algorithm ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
(Show Context)
For all nonsymmetric discrete relativistic Toda type equations we establish a relation to 3D consistent systems of quadequations. Unlike the more simple and better understood symmetric case, here the three coordinate planes of Z 3 carry different equations. Our construction allows for an algorithmic derivation of the zero curvature representations and yields analogous results also for the continuous time case. 1
Weak Lax pairs for lattice equations
, 2011
"... We consider various 2D lattice equations and their integrability, from the point of view of 3D consistency, Lax pairs and Bäcklund transformations. We show that these concepts, which are associated with integrability, are not strictly equivalent. In the course of our analysis, we introduce a number ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
We consider various 2D lattice equations and their integrability, from the point of view of 3D consistency, Lax pairs and Bäcklund transformations. We show that these concepts, which are associated with integrability, are not strictly equivalent. In the course of our analysis, we introduce a number of black and white lattice models, as well as variants of the functional YangBaxter equation. 1
Soliton Solutions for ABS Lattice Equations
 I Cauchy Matrix Approach, J. Phys. A: Math. Theor
"... Abstract. Elliptic Nsolitontype solutions, i.e. solutions emerging from the application of N consecutive Bäcklund transformations to an elliptic seed solution, are constructed for all equations in the ABS list of quadrilateral lattice equations, except for the case of the Q4 equation which is trea ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
Abstract. Elliptic Nsolitontype solutions, i.e. solutions emerging from the application of N consecutive Bäcklund transformations to an elliptic seed solution, are constructed for all equations in the ABS list of quadrilateral lattice equations, except for the case of the Q4 equation which is treated elsewhere. The main construction, which is based on an elliptic Cauchy matrix, is performed for the equation Q3, and by coalescence on certain auxiliary parameters, the corresponding solutions of the remaining equations in the list are obtained. Furthermore, the underlying linear structure of the equations is exhibited, leading, in particular, to a novel Lax representation of the Q3 equation. 1.
Idempotent biquadratics, YangBaxter maps and birational representations of Coxeter groups
, 2013
"... A transformation is obtained which completes the unification of quadrirational YangBaxter maps and known integrable multiquadratic quad equations. By combining theory from these two classes of quadgraph models we find an extension of the known integrability feature, and show how this leads subs ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
A transformation is obtained which completes the unification of quadrirational YangBaxter maps and known integrable multiquadratic quad equations. By combining theory from these two classes of quadgraph models we find an extension of the known integrability feature, and show how this leads subsequently to a natural extension of the associated lattice geometry. The extended lattice is encoded in a birational representation of a particular sequence of Coxeter groups. In this setting the usual quadgraph is associated with a subgroup of type BCn, and is part of a larger and more symmetric ambient space. The model also defines, for instance, integrable dynamics on a trianglegraph associated with a subgroup of type An, as well as finite degreeoffreedom dynamics, in the simplest cases associated with Ẽ6 and Ẽ8 affine subgroups. Underlying this structure is a class of biquadratic polynomials, that we call idempotent, which express the trisection of elliptic function periods algebraically via the addition law.