We study "circular net" (discrete orthogonal net) equations for angular data generalized by external spectral parameters. A criterion defining physical regimes of these Hamiltonian equations is the reality of Lagrangian density. There are four distinct regimes for fields and spectral parameters classified by four types of spherical or hyperbolic triangles. Non-zero external spectral parameters provide the existence of field-theoretical ground states and soliton excitations. Spectral parameters of a spherical triangle correspond to a statistical mechanics; spectral parameters of hyperbolic triangles correspond to three different field theories with massless anisotropic dispersion relations.
The problem of discretization of Darboux integrable equations is considered. Given a Darboux integrable continuous equation, one can obtain a Darboux integrable differential-discrete equation, using the integrals of the continuous equation. In the present paper, the discretization of the differential-discrete equations is done using the corresponding characteristic algebras. New examples of integrable discrete equations are obtained.
KdV6 equation can be described as the Kupershmidt deformation of the KdV equation (see 2008, Phys. Lett. A 372: 263). In this paper, starting from the bi-Hamiltonian structure of the discrete integrable system, we propose a generalized Kupershmidt deformation to construct new discrete integrable systems. Toda hierarchy, Kac-van Moerbeke hierarchy and Ablowitz-Ladik hierarchy are considered. The Lax representations for these new deformed systems are presented. The generalized Kupershmidt deformation for the discrete integrable systems provides a new way to construct new discrete integrable syst…
The Lie algebraic integrability test is applied to the problem of classification of integrable Klein-Gordon type equations on quad-graphs. The list of equations passing the test is presented containing several well-known integrable models. A new integrable example is found, its higher symmetry is presented.
A discrete analogue of the dressing method is presented and used to derive integrable nonlinear evolution equations, including two infinite families of novel continuous and discrete coupled integrable systems of equations of nonlinear Schrodinger type. First, a demonstration is given of how discrete nonlinear integrable equations can be derived starting from their linear counterparts. Then, starting from two uncoupled, discrete one-directional linear wave equations, an appropriate matrix Riemann-Hilbert problem is constructed, and a discrete matrix nonlinear Schrodinger system of equations is …
In the present paper we consider a discretization of hyperbolic systems of exponential type. We proved that, in the case of $2\times 2$ systems, the resulting semi-discrete system is Darboux integrable only if it corresponds to a Cartan matrix of a semi-simple Lie algebra.
We propose discrete mappings of second order that have a discrete analogue of Lyapunov function. The mappings are extensions of the integrable Quispel-Roberts-Thompson (QRT) mapping, and a discrete Lyapunov function of the mappings is identical to an explicit conserved quantity of the QRT mapping. Moreover we can obtain a differential and an ultradiscrete limit of the mappings preserving the existence of Lyapunov function. We also give applications of a mapping with an adjusted parameter, a probabilistic mapping and coupled mappings.
We present a method of constructing discrete integrable systems with crystallographic reflection group (Weyl) symmetries, thus clarifying the relationship between different discrete integrable systems in terms of their symmetry groups. Discrete integrable systems are associated with space-filling polytopes arise from the geometric representation of the Weyl groups in the $n$-dimensional real Euclidean space $\mathbb{R}^n$. The "multi-dimensional consistency" property of the discrete integrable system is shown to be inherited from the combinatorial properties of the polytope; while the dynamics…
In the paper we continue to consider symmetries related to the Ablowitz-Ladik hierarchy. We derive symmetries for the integrable discrete nonlinear Schrödinger hierarchy and discrete AKNS hierarchy. The integrable discrete nonlinear Schrödinger hierarchy are in scalar form and its two sets of symmetries are shown to form a Lie algebra. We also present discrete AKNS isospectral flows, non-isospectral flows and their recursion operator. In continuous limit these flows go to the continuous AKNS flows and the recursion operator goes to the square of the AKNS recursion operartor. These discrete AKN…
A relation between semi-direct sums of Lie algebras and integrable couplings of lattice equations is established, and a practicable way to construct integrable couplings is further proposed. An application of the resulting general theory to the generalized Toda spectral problem yields two classes of integrable couplings for the generalized Toda hierarchy of lattice equations. The construction of integrable couplings using semi-direct sums of Lie algebras provides a good source of information on complete classification of integrable lattice equations.
We establish connections between two cascades of integrable systems generated from the continuum limits of the Hirota-Miwa equation and its remarkable nonlinear counterpart under the Miwa transformation respectively. Among these equations, we are mainly concerned with the semi-discrete bilinear Kadomtsev-Petviashvili (KP) equation which is seldomly studied in literature. We present both of its Casorati and Grammian determinant solutions. Through the Pfaffianization procedure proposed by Hirota and Ohta, we are able to derive the coupled integrable system for the semi-discrete KP equation.
A new integrable discrete system is constructed and studied, based on the algebraization of the difference operator. The model is named the discrete generalized nonlinear Schrodinger (GNLS) equation for which can be reduced to classical discrete nonlinear Schrodinger (NLS) equation. To show the complete integrability of the discrete GNLS equation, the recursion operator, symmetries and conservation quantities are obtained. Furthermore, all of reductions for the discrete GNLS equation are given and the discrete NLS equation is obtained by one of the reductions. At the same time, the recursion o…
We introduce a family of order $N\in \mathbb{N}$ Lax matrices that is indexed by the natural number $k\in \{1,\ldots,N-1\}.$ For each value of $k$ they serve as strong Lax matrices of a hierarchy of integrable difference systems in edge variables that in turn lead to hierarchies of integrable difference systems in vertex variables or in a combination of edge and vertex variables. Furthermore, the entries of the Lax matrices are considered as elements of a division ring, so we obtain hierarchies of discrete integrable systems extended in the non-commutative domain.
We investigate indeterminate points in discrete integrable system. They appear in singularity confinement phenomenon naturally. We develop a method to analyse indeterminate points of dynamical maps and using this method we clarify behaviour of indeterminate points of some integrable maps. As a result, (1) we determine their indeterminacy, (2) find they are `confined' by the map, and (3) obtain a periodicity condition of the map from indeterminate point. Finally, we propose a new type of entropy using the indeterminacy in conclusion.
Fordy and Xenitidis [J. Phys. A: Math. Theor. 50 (2017) 165205] recently proposed a large class of discrete integrable systems which include a number of novel integrable difference equations, from the perspective of $\mathbb{Z}_\mathcal{N}$ graded Lax pairs, without providing solutions. In this paper, we establish the link between the Fordy-Xenitidis discrete systems in coprime case and linear integral equations in certain form, which reveals solution structure of these equations. The bilinear form of the Fordy-Xenitidis integrable difference equations is also presented together with the assoc…
We constructed involutions for a Halphen pencil of index 2, and proved that the birational mapping corresponding to the autonomous reduction of the elliptic Painlevé equation for the same pencil can be obtained as the composition of two such involutions.
We present the Darboux transformations for a novel class of two-dimensional discrete integrable systems named as $\mathbb{Z}_\mathcal{N}$ graded discrete integrable systems, which were firstly proposed by Fordy and Xenitidis within the framework of $\mathbb{Z}_\mathcal{N}$ graded discrete Lax pairs very recently. In this paper, the $\mathbb{Z}_\mathcal{N}$ graded discrete equations in coprime case and their corresponding Lax pairs are derived from the discrete Gel'fand-Dikii hierarchy by applying a transformation of the independent variables. The construction of the Darboux tranformations is r…
We consider a discrete equation, defined on the two-dimensional square lattice, which is linearizable, namely, of the Burgers type and depends on a parameter $α$. For any natural number $N$ we choose $α$ so that the equation becomes Darboux integrable and the minimal orders of its first integrals in both directions are greater or equal than $N$.
A consistent set of six integrable discrete and continuous dynamical systems are suggested corresponding to arbitrary affine Lie algebra. The set contains a system of partial differential equations which can be treated as a version of generalized Toda lattice while semi-discrete systems in the set define the Backlund transform for this Toda lattice and the fully discrete representative of the set can be obtained as a superposition of such kind Backlund transforms. Four linear Zakharov-Shabat type systems taken pairwise realize Lax pairs for these six dynamical systems.
All Darboux integrable difference equations on the quad-graph are described in the case of the equations that possess autonomous first-order integrals in one of the characteristics. A generalization of the discrete Liouville equation is obtained from a subclass of these equation via a non-point transformation. The general proposition on the symmetry structure of the quad-graph equations is proved as an auxiliary result.
In a recent paper [TMP, 200:1 (2019), 966--984] by the authors, a series of integrable discrete autonomous equations on a square lattice with a non-standard structure of generalized symmetries is constructed. We build modified series by using discrete non-point transformations. We use both non-invertible linearizable transformations and non-point transformations invertible on solutions of the discrete equation. As a result, we get several series of new examples of discrete equations along with their generalized symmetries and master symmetries. The generalized symmetries constructed give new i…
A method of classification of integrable equations on quad-graphs is discussed based on algebraic ideas. We assign a Lie ring to the equation and study the function describing the dimensions of linear spaces spanned by multiple commutators of the ring generators. For the generic case this function grows exponentially. Examples show that for integrable equations it grows slower. We propose a classification scheme based on this observation.
We introduce the sub-lattice approach, a procedure to generate, from a given integrable lattice, a sub-lattice which inherits its integrability features. We consider, as illustrative example of this approach, the discrete Moutard 4-point equation and its sub-lattice, the self-adjoint 5-point scheme on the star of the square lattice, which are relevant in the theory of the integrable Discrete Geometries and in the theory of discrete holomorphic and harmonic functions (in this last context, the discrete Moutard equation is called discrete Cauchy-Riemann equation). We use the sub-lattice point of…
Group based moving frames have a wide range of applications, from the classical equivalence problems in differential geometry to more modern applications such as computer vision. Here we describe what we call a discrete group based moving frame, which is essentially a sequence of moving frames with overlapping domains. We demonstrate a small set of generators of the algebra of invariants, which we call the discrete Maurer--Cartan invariants, for which there are recursion formulae. We show that this offers significant computational advantages over a single moving frame for our study of discrete…
We construct new integrable systems describing particles with internal spin from four-dimensional $\mathcal{N}=2$ quiver gauge theories. The models can be quantized and solved exactly using the quantum inverse scattering method and also using the Bethe/Gauge correspondence.
Based on the classical Plücker correspondence, we present algebraic and geometric properties of discrete integrable line complexes in $CP^3$. Algebraically, these are encoded in a discrete integrable system which appears in various guises in the theory of continuous and discrete integrable systems. Geometrically, the existence of these integrable line complexes is shown to be guaranteed by Desargues' classical theorem of projective geometry. A remarkable characterisation in terms of correlations of $CP^3$ is also recorded.
Integrable fractional equations such as the fractional Korteweg-deVries and nonlinear Schrödinger equations are key to the intersection of nonlinear dynamics and fractional calculus. In this manuscript, the first discrete/differential difference equation of this type is found, the fractional integrable discrete nonlinear Schrödinger equation. This equation is linearized; special soliton solutions are found whose peak velocities exhibit more complicated behavior than other previously obtained fractional integrable equations. This equation is compared with the closely related fractional averaged…
Based on our previous work to the Degasperis-Procesi equation (J. Phys. A 46 045205) and the integrable semi-discrete analogue of its short wave limit (J. Phys. A 48 135203), we derive an integrable semi-discrete Degasperis-Procesi equation by Hirota's bilinear method. Meanwhile, $N$-soliton solution to the semi-discrete Degasperis-Procesi equation is provided and proved. It is shown that the proposed semi-discrete Degasperis-Procesi equation, along with its $N$-soliton solution converge to ones of the original Degasperis-Procesi equation in the continuous limit.
A semi-discrete Lax pair formed from the differential system and recurrence relation for semi-classical orthogonal polynomials, leads to a discrete integrable equation for a specific semi-classical orthogonal polynomial weight. The main example we use is a semi-classical Laguerre weight to derive a third order difference equation with a corresponding Lax pair.
Discrete integrable equations over finite fields are investigated. The indeterminacy of the equation is resolved by treating it over a field of rational functions instead of the finite field itself. The main discussion concerns a generalized discrete KdV equation related to a Yang-Baxter map. Explicit forms of soliton solutions and their periods over finite fields are obtained. Relation to the singularity confinement method is also discussed.
There exist many four dimensional integrable theories. They include self-dual gauge and gravity theories, all their extended supersymmetric generalisations, as well the full (non-self-dual) N=3 super Yang-Mills equations. We review the harmonic space formulation of the twistor transform for these theories which yields a method of producing explicit connections and metrics. This formulation uses the concept of harmonic space analyticity which is closely related to that of quaternionic analyticity. (Talk by V. Ogievetsky at the Gürsey Memorial Conference I, Istanbul, June 1994)
We show that the following geometric properties of the motion of discrete and continuous curves select integrable dynamics: i) the motion of the curve takes place in the N dimensional sphere of radius R, ii) the curve does not stretch during the motion, iii) the equations of the dynamics do not depend explicitly on the radius of the sphere. Well known examples of integrable evolution equations, like the nonlinear Schroedinger and the sine-Gordon equations, as well as their discrete analogues, are derived in this general framework.
A fairly complete list of Toda-like integrable lattice systems, both in the continuous and discrete time, is given. For each system the Newtonian, Lagrangian and Hamiltonian formulations are presented, as well as the 2x2 Lax representation and r-matrix structure. The material is given in the "no comment" style, in particular, all proofs are omitted.
Inspired by the connection between the Dodgson's condensation algorithm and Hirota's difference equation, we consider condensation algorithms for Pfaffians from the perspectives of discrete integrable systems. The discretisation of Pfaffian elements demonstrate its effectiveness to the Pfaffian $τ$-functions and discrete integrable systems. The free parameter in the discretisation allows us in particular to obtain explicit, one-parameter condensation algorithms for the Pfaffians.
Interrelations between discrete deformations of the structure constants for associative algebras and discrete integrable systems are reviewed. A theory of deformations for associative algebras is presented. Closed left ideal generated by the elements representing the multiplication table plays a central role in this theory. Deformations of the structure constants are generated by the Deformation Driving Algebra and governed by the central system of equations. It is demonstrated that many discrete equations like discrete Boussinesq equation, discrete WDVV equation, discrete Schwarzian KP and BK…
We show that the following elementary geometric properties of the motion of a discrete (i.e. piecewise linear) curve select the integrable dynamics of the Ablowitz-Ladik hierarchy of evolution equations: i) the set of points describing the discrete curve lie on the sphere S^3, ii) the distance between any two subsequant points does not vary in time, iii) the dynamics does not depend explicitly on the radius of the sphere. These results generalize to a discrete context our previous work on continuous curves.
We present the first steps of a procedure which discretises surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved. We propose a canonical frame in terms of which the associated projective Gauss-Weingarten and Gauss-Mainardi-Codazzi equations adopt compact forms. Based on a scaling symmetry which injects a parameter into the linear Gauss-Weingarten equations, we set down an algebraic classification scheme of discrete projective minimal surfaces which turns out to admit a geometric counterpart formulated in terms of discre…
Following a recent proposal for integrable theories in higher dimensions based on zero curvature, new Lorentz invariant submodels of the principal chiral model in 2+1 dimensions are found. They have infinite local conserved currents, which are explicitly given for the su(2) case. The construction works for any Lie algebra and in any dimension, and it is given explicitly also for su(3). We comment on the application to supersymmetric chiral models.
This note gives a general construction of an integrable lattice model (and a solution of the Yang-Baxter equation with spectral parameter) from a four-dimensional field theory which is a mixture of topological and holomorphic. Spin-chain models arise in this way from a twisted, deformed version of N=1 gauge theory.
The general framework for integrable discrete systems on R in particular containing lattice soliton systems and their q-deformed analogues is presented. The concept of regular grain structures on R, generated by discrete one-parameter groups of diffeomorphisms, through which one can define algebras of shift operators is introduced. Two integrable hierarchies of discrete chains together with bi-Hamiltonian structures are constructed. Their continuous limit and the inverse problem based on the deformation quantization scheme are considered.
Discrete and q-difference deformations of the structure constants for a class of associative noncommutative algebras are studied. It is shown that these deformations are governed by a central system of discrete or q-difference equations which in particular cases represent discrete and q-difference versions of the oriented associativity equation. It is demonstrated also that the celebrated Hirota-Miwa bilinear equation for the AKP and BKP hierarchies describes discrete deformations of certain finite-dimensional algebras.
We construct local zero curvature representations for non-linear sigma models on homogeneous spaces, defined on a space-time of any dimension, following a recently proposed approach to integrable theories in dimensions higher than two. We present some sufficient conditions for the existence of integrable submodels possessing an infinite number of local conservation laws. Examples involving symmetric spaces and group manifolds are given. The $CP^N$ models are discussed in detail.
We consider integrable discretizations of some soliton equations associated with the motions of plane curves: the Wadati-Konno-Ichikawa elastic beam equation, the complex Dym equation, and the short pulse equation. They are related to the modified KdV or the sine-Gordon equations by the hodograph transformations. Based on the observation that the hodograph transformations are regarded as the Euler-Lagrange transformations of the curve motions, we construct the discrete analogues of the hodograph transformations, which yield integrable discretizations of those soliton equations.
The quantum integrability of a class of massive perturbations of the parafermionic conformal field theories associated to compact Lie groups is established by showing that they have quantum conserved densities of scale dimension 2 and 3. These theories are integrable for any value of a continuous vector coupling constant, and they generalize the perturbation of the minimal parafermionic models by their first thermal operator. The classical equations-of-motion of these perturbed theories are the non-abelian affine Toda equations which admit (charged) soliton solutions whose semi-classical quant…
In this paper we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg-de Vries-type (KdV-type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symple…
The paper presents two results. First it is shown how the discrete potential modified KdV equation and its Lax pairs in matrix form arise from the Hirota-Miwa equation by a 2-periodic reduction. Then Darboux transformations and binary Darboux transformations are derived for the discrete potential modified KdV equation and it is shown how these may be used to construct exact solutions.
We present an integrable Hamiltonian which describes the sinh-Gordon model on the half line coupled to a non-linear oscillator at the boundary. We explain how we apply Sklyanin's formalism to a dynamical reflection matrix to obtain this model. This method can be applied to couple other integrable field theories to dynamical systems at the boundary. We also show how to find the dynamical solution of the quantum reflection equation corresponding to our particular example.
These lecture notes concern the semi-holomorphic 4d Chern-Simons theory and its applications to classical integrable field theories in 2d and in particular integrable sigma-models. After introducing the main properties of the Chern-Simons theory in 3d, we will define its 4d analogue and explain how it is naturally related to the Lax formalism of integrable 2d theories. Moreover, we will explain how varying the boundary conditions imposed on this 4d theory allows to recover various occurences of integrable sigma-models through this construction, in particular illustrating this on two simple exa…
We study 2D discrete integrable equations of order 1 with respect to one independent variable and $m$ with respect to another one. A generalization of the multidimensional consistency property is proposed for this type of equations. The examples are related to the Bäcklund--Darboux transformations for the lattice equations of Bogoyavlensky type.
Functional relations play a key role in the study of integrable models. We argue in this paper that for massless field theories at zero temperature, these relations can in fact be interpreted as monodromy relations. Combined with a recently discovered duality, this gives a way to bypass the Bethe ansatz, and compute directly physical quantities as solutions of a linear differential equation, or as integrals over a hyperelliptic curve. We illustrate these ideas in details in the case of the $c=1$ theory, and the associated boundary sine-Gordon model.
A relationship between the tetrahedron equation for maps and the consistency property of integrable discrete equations on $\mathbb{Z}^3$ is investigated. Our approach is a generalization of a method developed in the context of Yang-Baxter maps, based on the invariants of symmetry groups of the lattice equations. The method is demonstrated by a case-by-case analysis of the octahedron type lattice equations classified recently, leading to some new examples of tetrahedron maps and integrable coupled lattice equations.
We construct integrable discrete nonautonomous quad-equations as Bäcklund auto-transformations for known Volterra and Toda type semidiscrete equations, some of which are also nonautonomous. Additional examples of this kind are found by using transformations of discrete equations which are invertible on their solutions. In this way we obtain integrable examples of different types: discrete analogs of the sine-Gordon equation, the Liouville equation and the dressing chain of Shabat. For Liouville type equations we construct general solutions, using a specific linearization. For sine-Gordon type …
By employing CFT techniques, we show how to compute in the context of λ-deformations of current algebras and coset CFTs the exact in the deformation parameters C-function for a wide class of integrable theories that interpolate between a UV and an IR point. We explicitly consider RG flows for integrable deformations of left-right asymmetric current algebras and coset CFTs. In all cases, the derived exact C-functions obey all the properties asserted by Zamolodchikov's c-theorem in two-dimensions.
A unified framework is presented for the solution structure of three-dimensional discrete integrable systems, including the lattice AKP, BKP and CKP equations. This is done through the so-called direct linearising transform which establishes a general class of integral transforms between solutions. As a particular application, novel soliton-type solutions for the lattice CKP equation are obtained.
