### Citations

136 |
The solution to a generalized Toda lattice and representation theory
- Kostant
- 1979
(Show Context)
Citation Context ...triction to a sub-lattice for the fully discrete systems [13, 14]. The open Toda lattice has a long history going back to Moser [15] and has been extensively studied and generalised (see for instance =-=[16, 17, 18, 19, 20, 21]-=-), so is the better-known setting for the present work. One of our main motives is to understand better the ABS equations as prototypical equations whose natural integrability property is the multidim... |

116 | Finitely many mass points on the line under the influence of an exponential potential — an integrable system - Moser - 1975 |

89 | W.: Lax pair for the Adler (lattice Krichever-Novikov) system
- Nijhoff
- 2002
(Show Context)
Citation Context ...prototypical equations whose natural integrability property is the multidimensional consistency. In particular we are interested in the primary such model due to Adler [22] known as Q4 since [5], see =-=[23, 14]-=-. Remarkably Q4 can be characterised (up to some non-degeneracy) by the symmetry of its defining polynomial [24]. On a local level there is an intimate relationship between the singularities and the i... |

59 |
Nonlinear partial difference equations. I. A difference analogue of the Korteweg-de Vries equation
- Hirota, Satsuma
(Show Context)
Citation Context ...licable to discrete KdV-type equations and introduce a technique to obtain exact expressions for the solution of the reduced systems. The discrete KdV-type equations have a long history going back to =-=[1, 2, 3]-=-, cf. [4], and more recently classification results were obtained by Adler Bobenko and Suris (ABS) [5, 6]. Of these systems the type-Q subclass are roughly speaking the least degenerate, they occupy t... |

42 |
Bäcklund transformation for solutions of the Korteweg-de Vries equation Phys
- Wahlquist, Estabrook
- 1973
(Show Context)
Citation Context ...licable to discrete KdV-type equations and introduce a technique to obtain exact expressions for the solution of the reduced systems. The discrete KdV-type equations have a long history going back to =-=[1, 2, 3]-=-, cf. [4], and more recently classification results were obtained by Adler Bobenko and Suris (ABS) [5, 6]. Of these systems the type-Q subclass are roughly speaking the least degenerate, they occupy t... |

34 |
Integrable mappings and nonlinear integrable lattice equations,
- Papageorgiou, Nijhoff
- 1990
(Show Context)
Citation Context ...problems for discrete KdV-type equations have received a fair amount of attention, the earliest work establishing basic techniques based on the integrability is due to Papageorgiou, Nijhoff and Capel =-=[41]-=-, and since then the problem has generated a substantial literature, see for instance [42, 43, 44, 45, 46, 47, 48]. Note that assuming p and q are non-negative here does not lose generality due to ref... |

32 | Bäcklund transformation for the Krichever-Novikov equation Int
- Adler
(Show Context)
Citation Context ...and better the ABS equations as prototypical equations whose natural integrability property is the multidimensional consistency. In particular we are interested in the primary such model due to Adler =-=[22]-=- known as Q4 since [5], see [23, 14]. Remarkably Q4 can be characterised (up to some non-degeneracy) by the symmetry of its defining polynomial [24]. On a local level there is an intimate relationship... |

31 |
Theory of Elliptic Functions,
- Hancock
- 1958
(Show Context)
Citation Context ...ed to giving a set of intermediary steps which are easy to verify. Definitions of the Jacobi elliptic and theta functions involved in the calculation can be found in Chapter 5 of [34] or Chapter X of =-=[35]-=-. In this section Qα,β will denote the Q4 polynomial appearing in Table 1, and the associated function f (2.2) is taken to be f(ξ) := √ k sn(ξ) = H(ξ) Θ(ξ) . (8.1) Central are the following basic iden... |

19 |
Bobenko A I and Suris Yu B 2003 Classification of integrable equations on quad-graphs. The consistency approach Commun
- Adler
(Show Context)
Citation Context ...n of the reduced systems. The discrete KdV-type equations have a long history going back to [1, 2, 3], cf. [4], and more recently classification results were obtained by Adler Bobenko and Suris (ABS) =-=[5, 6]-=-. Of these systems the type-Q subclass are roughly speaking the least degenerate, they occupy the top level of the hierarchy of such equations. The basic reduction we consider is a finite degree-offre... |

19 | Bäcklund transformations for integrable lattice equations
- Atkinson
(Show Context)
Citation Context ...s α and β, k ∈ C \ {−1, 0, 1} is the modulus of the Jacobi sn function appearing in Q4 and for Q3 δ ∈ {0, 1}. really lose generality; all similar models can be obtained from it by limiting procedures =-=[14, 31]-=-. Nevertheless we find it useful to consider the other systems listed in the table in order to see explicit solutions for the full progression through the rational, trigonometric and elliptic models. ... |

13 |
Theory of nonlinear lattices. Second edition.
- Toda
- 1989
(Show Context)
Citation Context ...triction to a sub-lattice for the fully discrete systems [13, 14]. The open Toda lattice has a long history going back to Moser [15] and has been extensively studied and generalised (see for instance =-=[16, 17, 18, 19, 20, 21]-=-), so is the better-known setting for the present work. One of our main motives is to understand better the ABS equations as prototypical equations whose natural integrability property is the multidim... |

10 |
E 2001 Discrete equations on planar graphs
- Adler
(Show Context)
Citation Context ...n1 . . . nd replacing n and m. This section is devoted to generalisation of the regular singularity-bounded strip, and its solution, to this higher dimensional setting. This is motivated by the works =-=[14, 13, 49]-=-: some related systems of interest are obtained by restriction to a sub-lattice in Zd with d > 2. We make the higher-dimensional extension explicit here because the solutions are not defined isotropic... |

9 |
Bobenko A I and Suris Yu B 2009 Discrete nonlinear hyperbolic equations. Classification of integrable cases Funct
- Adler
(Show Context)
Citation Context ...n of the reduced systems. The discrete KdV-type equations have a long history going back to [1, 2, 3], cf. [4], and more recently classification results were obtained by Adler Bobenko and Suris (ABS) =-=[5, 6]-=-. Of these systems the type-Q subclass are roughly speaking the least degenerate, they occupy the top level of the hierarchy of such equations. The basic reduction we consider is a finite degree-offre... |

8 |
Studies of a nonlinear lattice Phys
- Toda
- 1975
(Show Context)
Citation Context ...lar solution outside of a bounded domain, a situation which there was motivated physically. As remarked in [8], probably the nearest related examples come from the theory of Toda-type lattice systems =-=[10, 11]-=-, where the ‘open’ (or ‘finite non-periodic’) boundary conditions can be identified with what we call singular boundaries here1. In fact the relationship between KdV-type and Toda-type systems, which ... |

8 |
Hietarinta J 2009 Soliton solutions for ABS lattice equations: I Cauchy matrix approach
- Nijhoff, Atkinson
(Show Context)
Citation Context ...tifiable aspects of the Hirota-type polynomial, such as constants Y{i,j}, which are new and distinct from the analagous quantities present in the soliton-type solutions of these equations obtained in =-=[37, 38, 39, 40, 33]-=-. 11 Relation to periodic solutions The explicit solutions constructed here generally need not be defined outside of their natural domain, which is a diagonal strip of the lattice (cf. Section 3). How... |

6 |
A tau-function of the finite nonperiodic Toda lattice
- Nakamura
- 1988
(Show Context)
Citation Context ...triction to a sub-lattice for the fully discrete systems [13, 14]. The open Toda lattice has a long history going back to Moser [15] and has been extensively studied and generalised (see for instance =-=[16, 17, 18, 19, 20, 21]-=-), so is the better-known setting for the present work. One of our main motives is to understand better the ABS equations as prototypical equations whose natural integrability property is the multidim... |

5 |
Marikhin V G and Shabat A
- Adler
(Show Context)
Citation Context ...lar background. On the other hand, they can also be viewed as solutions of the open-boundary problems for the Toda-type systems connected to Q4, namely discrete analogues of the elliptic Toda lattice =-=[26, 27]-=- and the elliptic generalisation of the (Relativistic) Ruijsenaars-Toda lattice [14, 28, 29, 30]. We proceed as follows. Section 2 recalls the class of equations and their singularity structure, and i... |

5 |
A discrete-time relativistic Toda lattice
- B
- 1995
(Show Context)
Citation Context ...dary problems for the Toda-type systems connected to Q4, namely discrete analogues of the elliptic Toda lattice [26, 27] and the elliptic generalisation of the (Relativistic) Ruijsenaars-Toda lattice =-=[14, 28, 29, 30]-=-. We proceed as follows. Section 2 recalls the class of equations and their singularity structure, and in particular explains the idea of singular-boundary reduction. A particular reduction, the singu... |

5 |
Hietarinta J and Nijhoff F 2008 Soliton solutions for Q3
- Atkinson
(Show Context)
Citation Context ...tifiable aspects of the Hirota-type polynomial, such as constants Y{i,j}, which are new and distinct from the analagous quantities present in the soliton-type solutions of these equations obtained in =-=[37, 38, 39, 40, 33]-=-. 11 Relation to periodic solutions The explicit solutions constructed here generally need not be defined outside of their natural domain, which is a diagonal strip of the lattice (cf. Section 3). How... |

4 |
Veselov A P 2004 Cauchy Problem for Integrable Discrete
- Adler
(Show Context)
Citation Context ...lst the constant γ is given by γ = λ1+ . . .+λN . The generic (finite and with no self-intersecting characteristics) quad-graph can be obtained by restriction to a subset of quadrilaterals in Zd (cf. =-=[7]-=-), so this multidimensional solution yields a natural singular-boundary reduction of the type-Q ABS equations in that setting. Of particular relevance here is the further restriction to the odd (or ev... |

4 |
Nijhoff F W 2007 Seed and soliton solutions for Adler’s lattice equation
- Atkinson, Hietarinta
(Show Context)
Citation Context ...tifiable aspects of the Hirota-type polynomial, such as constants Y{i,j}, which are new and distinct from the analagous quantities present in the soliton-type solutions of these equations obtained in =-=[37, 38, 39, 40, 33]-=-. 11 Relation to periodic solutions The explicit solutions constructed here generally need not be defined outside of their natural domain, which is a diagonal strip of the lattice (cf. Section 3). How... |

3 |
Nijhoff F W and Capel H W 2005 Exact solutions of quantum mappings from the lattice KdV as multi-dimensional operator difference equations
- Field
(Show Context)
Citation Context ... the boundaries that resolves the expected inconsistency and legitimises such reductions. This kind of reduction applied to a discrete equation of KdV-type was proposed by Field, Nijhoff and Capel in =-=[8]-=- where the lattice potential KdV equation (or H1 in ABS classification) was considered. The authors linearise the dynamics of the reduced system and give a finite iterative procedure to construct its ... |

3 |
partial difference equations II
- Hirota
- 1977
(Show Context)
Citation Context ...lar solution outside of a bounded domain, a situation which there was motivated physically. As remarked in [8], probably the nearest related examples come from the theory of Toda-type lattice systems =-=[10, 11]-=-, where the ‘open’ (or ‘finite non-periodic’) boundary conditions can be identified with what we call singular boundaries here1. In fact the relationship between KdV-type and Toda-type systems, which ... |

3 |
Suris Yu B 2004 Q4: Integrable Master Equation Related to an Elliptic Curve Int
- Adler
(Show Context)
Citation Context ... here1. In fact the relationship between KdV-type and Toda-type systems, which is well known since Flaschka [12], is remarkably nothing but restriction to a sub-lattice for the fully discrete systems =-=[13, 14]-=-. The open Toda lattice has a long history going back to Moser [15] and has been extensively studied and generalised (see for instance [16, 17, 18, 19, 20, 21]), so is the better-known setting for the... |

3 |
M 2000 Elliptic analog of the Toda lattice Int
- Krichever
(Show Context)
Citation Context ...lar background. On the other hand, they can also be viewed as solutions of the open-boundary problems for the Toda-type systems connected to Q4, namely discrete analogues of the elliptic Toda lattice =-=[26, 27]-=- and the elliptic generalisation of the (Relativistic) Ruijsenaars-Toda lattice [14, 28, 29, 30]. We proceed as follows. Section 2 recalls the class of equations and their singularity structure, and i... |

3 |
A J 2001 The Discrete and
- Nijhoff, Walker
(Show Context)
Citation Context ...the equation (2.1) itself. This system determines a new solution u∗ of (2.1) from an old solution u, the free parameter λ∗ is the Bäcklund parameter. Such Bäcklund transformations were described in =-=[32, 13]-=-, the lattice and Bäcklund directions are distinguished only by the difference of a parameter. In fact the equation, its Bäcklund transformations, and the superposition principle for pairs of commut... |

3 |
Zhang D-J 2009 Soliton solutions for ABS lattice equations II
- Hietarinta
(Show Context)
Citation Context |

3 |
Volkov A Yu 1994 Hirota Equation as an Example of an Integrable Symplectic Map
- Faddeev
(Show Context)
Citation Context ...liest work establishing basic techniques based on the integrability is due to Papageorgiou, Nijhoff and Capel [41], and since then the problem has generated a substantial literature, see for instance =-=[42, 43, 44, 45, 46, 47, 48]-=-. Note that assuming p and q are non-negative here does not lose generality due to reflection symmetry of the considered systems. The most basic system constructed by imposing both periodicity and sin... |

3 |
der Kamp P H and Quispel G R W 2009 Closed-form expressions for integrals of traveling wave reductions of integrable lattice equations
- Tran, van
(Show Context)
Citation Context ...liest work establishing basic techniques based on the integrability is due to Papageorgiou, Nijhoff and Capel [41], and since then the problem has generated a substantial literature, see for instance =-=[42, 43, 44, 45, 46, 47, 48]-=-. Note that assuming p and q are non-negative here does not lose generality due to reflection symmetry of the considered systems. The most basic system constructed by imposing both periodicity and sin... |

3 |
Quispel G R W 2010 The staircase method: integrals for periodic reductions of integrable lattice equations
- Kamp
(Show Context)
Citation Context ...liest work establishing basic techniques based on the integrability is due to Papageorgiou, Nijhoff and Capel [41], and since then the problem has generated a substantial literature, see for instance =-=[42, 43, 44, 45, 46, 47, 48]-=-. Note that assuming p and q are non-negative here does not lose generality due to reflection symmetry of the considered systems. The most basic system constructed by imposing both periodicity and sin... |

2 |
The Discrete Korteweg-de Vries Equation Act
- Nijhoff, Capel
- 1995
(Show Context)
Citation Context ...rete KdV-type equations and introduce a technique to obtain exact expressions for the solution of the reduced systems. The discrete KdV-type equations have a long history going back to [1, 2, 3], cf. =-=[4]-=-, and more recently classification results were obtained by Adler Bobenko and Suris (ABS) [5, 6]. Of these systems the type-Q subclass are roughly speaking the least degenerate, they occupy the top le... |

2 |
Singularities of type-Q
- Atkinson
(Show Context)
Citation Context ...rovides an explanation of the spectral curve and the natural parameterisation of the equation in terms of points on that curve [6]. The results reported here stem from the global singularity analysis =-=[25]-=-, in particular the solutions obtained can be viewed as soliton-type solutions built on a singular background. On the other hand, they can also be viewed as solutions of the open-boundary problems for... |

2 |
Nijhoff F W 2010 A constructive approach to the soliton solutions of integrable quadrilateral lattice equations Commun
- Atkinson
(Show Context)
Citation Context ... first Bäcklund transformation trivialises in the even-M case). 5 Integrating the Bäcklund scheme The problem of integrating a Bäcklund chain of length N for this class of equations was reduced in =-=[33]-=- to solving a set of N independent first-order, homogeneous, linear, but multidimensional equations. The method relies on the existence of 2N particular solutions of the Bäcklund chain which arise na... |

2 |
translated from the Russian by McFaden H H and edited by Silver B
- Akhiezer
- 1970
(Show Context)
Citation Context ...this section is devoted to giving a set of intermediary steps which are easy to verify. Definitions of the Jacobi elliptic and theta functions involved in the calculation can be found in Chapter 5 of =-=[34]-=- or Chapter X of [35]. In this section Qα,β will denote the Q4 polynomial appearing in Table 1, and the associated function f (2.2) is taken to be f(ξ) := √ k sn(ξ) = H(ξ) Θ(ξ) . (8.1) Central are the... |

2 |
Grammaticos B, Tamizhmani T and Ramani A 2006 From Integrable Lattices to NonQRT
- Joshi
(Show Context)
Citation Context |

1 |
P and Zabrodin A 1996 Quantum Integrable Models and Discrete Classical Hirota Equations Commun
- Krichever, Lipan, et al.
(Show Context)
Citation Context ...ems have more commonly been studied for integrable equations outside the KdV class. In particular the discrete Hirota equation was considered 1 ar X iv :1 10 8. 45 02 v1s[ nli n.S I]s23sA ugs20 11 in =-=[9]-=- with prescribed singular solution outside of a bounded domain, a situation which there was motivated physically. As remarked in [8], probably the nearest related examples come from the theory of Toda... |

1 |
The Toda lattice II. Existence of integrals Phys
- Flashka
- 1973
(Show Context)
Citation Context ...e non-periodic’) boundary conditions can be identified with what we call singular boundaries here1. In fact the relationship between KdV-type and Toda-type systems, which is well known since Flaschka =-=[12]-=-, is remarkably nothing but restriction to a sub-lattice for the fully discrete systems [13, 14]. The open Toda lattice has a long history going back to Moser [15] and has been extensively studied and... |

1 |
The spectrum of Jacobi matrices Invent
- van
- 1976
(Show Context)
Citation Context |

1 |
Habibullin I T 1995 Integrable boundary conditions for the Toda lattice
- Adler
(Show Context)
Citation Context |

1 |
Vaninsky K L 2000 The periodic and open Toda lattice arXiv:hep-th/0010184v1
- Krichever
(Show Context)
Citation Context |

1 |
Discrete-time generalized Toda lattices: complete integrability and relation with relativistic Toda lattices Phys
- B
- 1990
(Show Context)
Citation Context ...dary problems for the Toda-type systems connected to Q4, namely discrete analogues of the elliptic Toda lattice [26, 27] and the elliptic generalisation of the (Relativistic) Ruijsenaars-Toda lattice =-=[14, 28, 29, 30]-=-. We proceed as follows. Section 2 recalls the class of equations and their singularity structure, and in particular explains the idea of singular-boundary reduction. A particular reduction, the singu... |