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17
Continuous symmetric reductions of the Adler–Bobenko–Suris equations
, 2009
"... Continuously symmetric solutions of the AdlerBobenkoSuris class of discrete integrable equations are presented. Initially defined by their invariance under the action of both of the extended three point generalized symmetries admitted by the corresponding equations, these solutions are shown to be ..."
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Continuously symmetric solutions of the AdlerBobenkoSuris class of discrete integrable equations are presented. Initially defined by their invariance under the action of both of the extended three point generalized symmetries admitted by the corresponding equations, these solutions are shown
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 ..."
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Cited by 15 (8 self)
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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
Extension of the AdlerBobenkoSuris classification of integrable lattice equations
, 2008
"... The classification of lattice equations that are integrable in the sense of higherdimensional consistency is extended by allowing directed edges. We find two cases that are not transformable via the ‘admissible transformations ’ to the lattice equations in the existing classification. PACS: 02.30.Ik ..."
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Cited by 2 (0 self)
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The classification of lattice equations that are integrable in the sense of higherdimensional consistency is extended by allowing directed edges. We find two cases that are not transformable via the ‘admissible transformations ’ to the lattice equations in the existing classification. PACS: 02.30.Ik, 04.60.Nc, 05.45.Yv 1
Integrability and Symmetries of Difference Equations: the
, 2009
"... We consider the partial difference equations of the AdlerBobenkoSuris classification, which are characterized as multidimensionally consistent. The latter property leads naturally to the construction of autoBäcklund transformations and Lax pairs for all the equations in this class. Their symmetry ..."
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We consider the partial difference equations of the AdlerBobenkoSuris classification, which are characterized as multidimensionally consistent. The latter property leads naturally to the construction of autoBäcklund transformations and Lax pairs for all the equations in this class
Conservation laws for integrable difference equations
 J.Phys.A,Math. Theor
, 2007
"... Abstract. This paper deals with conservation laws for all integrable difference equations that belong to the famous AdlerBobenkoSuris classification. All inequivalent threepoint conservation laws are found, as are three fivepoint conservation laws for each equation. We also describe a method of ..."
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Cited by 16 (1 self)
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Abstract. This paper deals with conservation laws for all integrable difference equations that belong to the famous AdlerBobenkoSuris classification. All inequivalent threepoint conservation laws are found, as are three fivepoint conservation laws for each equation. We also describe a method
The generalized symmetry method for discrete equations
, 2009
"... The generalized symmetry method is applied to a class of completely discrete equations including the AdlerBobenkoSuris list. Assuming the existence of a generalized symmetry, we derive a few integrability conditions suitable for testing and classifying equations of this class. Those conditions are ..."
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Cited by 13 (4 self)
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The generalized symmetry method is applied to a class of completely discrete equations including the AdlerBobenkoSuris list. Assuming the existence of a generalized symmetry, we derive a few integrability conditions suitable for testing and classifying equations of this class. Those conditions
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
Symmetries and integrability of discrete equations defined on a black–white lattice
, 2009
"... We study the deformations of the H equations, presented recently by Adler, Bobenko and Suris, which are naturally defined on a blackwhite lattice. For each one of these equations, two different threeleg forms are constructed, leading to two different discrete Toda type equations. Their multidimens ..."
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Cited by 6 (0 self)
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We study the deformations of the H equations, presented recently by Adler, Bobenko and Suris, which are naturally defined on a blackwhite lattice. For each one of these equations, two different threeleg forms are constructed, leading to two different discrete Toda type equations
Symmetry, Integrability and Geometry: Methods and Applications On Quadrirational Yang–Baxter Maps ⋆
, 2009
"... doi:10.3842/SIGMA.2010.033 Abstract. We use the classification of the quadrirational maps given by Adler, Bobenko and Suris to describe when such maps satisfy the Yang–Baxter relation. We show that the corresponding maps can be characterized by certain singularity invariance condition. This leads to ..."
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doi:10.3842/SIGMA.2010.033 Abstract. We use the classification of the quadrirational maps given by Adler, Bobenko and Suris to describe when such maps satisfy the Yang–Baxter relation. We show that the corresponding maps can be characterized by certain singularity invariance condition. This leads
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 ..."
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Cited by 17 (2 self)
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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
Results 1  10
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17