We present two non-equivalent families of hierarchies of non-Abelian compatible maps and we provide their Lax pair formulation. These maps are associated with families of hierarchies of non-Abelian Yang-Baxter maps, which we provide explicitly. In addition, these hierarchies correspond to integrable difference systems with variables defined on edges of an elementary cell of the $\mathbb{Z}^2$ graph, that in turn lead to hierarchies of difference systems with variables defined on vertices of the same cell. In that respect we obtain the non-Abelian lattice-modified Gel'fand-Dikii hierarchy, together with the explicit form of a non-Abelian hierarchy that we refer to as the lattice-NQC (or lattice-$(Q3)_0$) Gel'fand-Dikii hierarchy.
Reductions for systems of ODEs integrable via the standard factorization method (the Adler-Kostant-Symes scheme) or the generalized factorization method, developed by the authors earlier, are considered. Relationships between such reductions, operator Yang-Baxter equations, and some kinds of non-associative algebras are established.
Following the procedure, described in the paper nlin.SI/0003002, for the integrable DST chain we construct Baxter Q-operators as the traces of monodromy of some M-operators, that act in quantum and auxiliary spaces. Within this procedure we obtain two basic M-operators and derive some functional relations between them such as intertwining relations and wronskian-type relations between two basic Q-operators.
The Lax-Sato approach to the hierarchies of Manakov-Santini type is formalized in order to extend it to a more general class of integrable systems. For this purpose some linear operators are introduced, which must satisfy some integrability conditions, one of them is the Rota-Baxter identity. The theory is illustrated by means of the algebra of Laurent series, the related hierarchies are classified and examples, also new, of Manakov-Santini type systems are constructed, including those that are related to the dispersionless modified Kadomtsev-Petviashvili equation and so called dispersionless …
In the present paper we introduce a multi-dimensional version of the R-matrix approach to the construction of integrable hierarchies. Applying this method to the case of the Lie algebra of functions with respect to the contact bracket, we construct integrable hierarchies of (3+1)-dimensional dispersionless systems of the type recently introduced by one of us in arXiv:1401.2122.
A transforation between a hierarchy of integrable equations arising from the standard $R$-matrix construction on the algebra of differential operators and a hierarchy of integrable equations arising from a deformation of the standard $R$-matrix is given.
In this paper, we establish Liouville correspondences for the integrable two-component Camassa-Holm hierarchy, the two-component Novikov (Geng-Xue) hierarchy, and the two-component dual dispersive water wave hierarchy by means of the related Liouville transformations. This extends previous results on the scalar Camassa-Holm and KdV hierarchies, and the Novikov and Sawada-Kotera hierarchies to the multi-component case.
The quasi-classical $\bar{\partial}$-dressing method is used to derive compact generating equations for dispersionless hierarchies. Dispersionless Kadomtsev-Petviashvili (KP) and two-dimensional Toda lattice (2DTL) hierarchies are considered as illustrative examples.
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 propose a general scheme of constructing of soliton hierarchies from finite dimensional Stäckel systems and related separation relations. In particular, we concentrate on the simplest class of separation relations, called Benenti class, i.e. certain Stäckel systems with quadratic in momenta integrals of motion.
This paper develops the technique of constructing Lax representations for PDEs via non-central extensions generated by non-triivial exotic 2-cocycles of their contact symmetry algebras. We show that the method is applicable to the Lax representations with non-removable spectral parameters. Also we demonstrate that natural extensions of the symmetry algebras produce the integrable hierarchies associated to their PDEs.
In this paper, we study explicit correspondences between the integrable Novikov and Sawada-Kotera hierarchies, and between the Degasperis-Procesi and Kaup-Kupershmidt hierarchies. We show how a pair of Liouville transformations between the isospectral problems of the Novikov and Sawada-Kotera equations, and the isospectral problems of the Degasperis-Procesi and Kaup-Kupershmidt equations relate the corresponding hierarchies, in both positive and negative directions, as well as their associated conservation laws. Combining these results with the Miura transformation relating the Sawada-Kotera a…
These lecture notes are devoted to the integrability of discrete systems and their relation to the theory of Yang-Baxter (YB) maps. Lax pairs play a significant role in the integrability of discrete systems. We introduce the notion of Lax pair by considering the well-celebrated doubly-infinite Toda lattice. In particular, we present solution of the Cauchy initial value problem via the method of the inverse scattering transform, provide a review of scattering theory of Jacobi matrices, and give the Riemann-Hilbert formulation of the inverse scattering transform. On the other hand, the Lax-Darbo…
A class of multidimensional integrable hierarchies connected with commutation of general (unreduced) (N+1)-dimensional vector fields containing derivative over spectral variable is considered. They are represented in the form of generating equation, as well as in the Lax-Sato form. A dressing scheme based on nonlinear vector Riemann problem is presented for this class. The hierarchies connected with Manakov-Santini equation and Dunajski system are considered as illustrative examples.
The two matrix spectral problems of Ablowitz-Kaup-Newell-Segur (AKNS) and Kaup-Newell (KN) types associated with so(3,R) are generalized. The corresponding hierarchies of generalized soliton equations are derived by the standard procedure using the zero curvature formulation. Recursion operators and bi-Hamiltonian structures are explicitly constructed for the resulting two generalized soliton hierarchies of AKNS and KN types, which shows their Liouville integrability.
In this article, by means of using discrete zero curvature representation and constructing opportune time evolution problems, two new discrete integrable lattice hierarchies with n-dependent coefficients are proposed, which related to a new discrete Schrödinger nonisospectral operator equation. The relation of the two new lattice hierarchies with the Volterra hierarchy is discussed. It has been shown that one lattice hierarchy is equivalent to the positive Volterra hierarchy with n-dependent coefficients and another lattice hierarchy with isospectral problem is equivalent to the negative Volte…
The quasi-classical limit of the scalar nonlocal dbar-problem is derived and a quasi-classical version of the dbar-dressing method is presented. Dispersionless KP, mKP and 2DTL hierarchies are discussed as illustrative examples. It is shown that the universal Whitham hierarchy it is nothing but the ring of symmetries for the quasi-classical dbar-problem. The reduction problem is discussed and, in particular, the d2DTL equation of B type is derived.
A connection between the Yang-Baxter relation for maps and the multi-dimensional consistency property of integrable equations on quad-graphs is investigated. The approach is based on the symmetry analysis of the corresponding equations. It is shown that the Yang-Baxter variables can be chosen as invariants of the multi-parameter symmetry groups of the equations. We use the classification results by Adler, Bobenko and Suris to demonstrate this method. Some new examples of Yang-Baxter maps are derived in this way from multi-field integrable equations.
We transfer the scheme of constructing differential reductions, developed recently for the case of the Manakov-Santini hierarchy, to the general multidimensional case. We consider in more detail the four-dimensional case, connected with the second heavenly equation and its generalization proposed by Dunajski. We give a characterization of differential reductions in terms of the Lax-Sato equations as well as in the framework of the dressing method based on nonlinear Riemann-Hilbert problem.
In this note we study `a la Baxter [1] the possible integrable manifolds of the asymmetric eight-vertex model. As expected they occur when the Boltzmann weights are either symmetric or satisfy the free-fermion condition but our analysis clarify the reason both manifolds need to share a universal invariant. We also show that the free-fermion condition implies three distinct classes of integrable models.
Starting from known solutions of the functional Yang-Baxter equations, we exhibit Miura type of transformations leading to various known integrable quad equations. We then construct, from the same list of Yang-Baxter maps, a series of non-autonomous solvable recurrences of order two.
Analytic-bilinear approach for construction and study of integrable hierarchies is discussed. Generalized multicomponent KP and 2D Toda lattice hierarchies are considered. This approach allows to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations. Generalized hierarchy incorporates basic hierarchy, modified hierarchy, singularity manifold equation hierarchy and corresponding linear problems. Different levels of generalized hierarchy are connected via invariants of Combescure symmetry transformation. Resolution of functional equations al…
Using classical double G of a Lie algebra g equipped with a classical R-operator we define two sets of mutually commuting functions with respect to the initial Lie-Poisson bracket on g* and its extensions. We consider in details examples of the Lie algebras g with the "Adler--Kostant--Symes" R-operators and the corresponding two sets of mutually commuting functions. Using the constructed commutative hamiltonian flows on different extensions of g we obtain zero-curvature equations with g-valued U-V pairs. Among such the equations are so-called "negative flows" of soliton hierarchies. We illlust…
Unifying hierarchies of integrable equations are discussed. They are constructed via generalized Hirota identity. It is shown that the Combescure transformations, known for a long time for the Darboux system and having a simple geometrical meaning, are in fact the symmetry transformations of generalized integrable hierarchies. Generalized equation written in terms of invariants of Combescure transformations are the usual integrable equations and their modified partners. The KP-mKP, DS-mDS hierarchies and Darboux system are considered.
We demonstrate that statistics of certain classes of set partitions is described by generating functions related to the Burgers, Ibragimov--Shabat and Korteweg--de Vries integrable hierarchies.
This paper connects the quadrirational Yang-Baxter maps, which are two-dimensional integrable discrete systems of KdV type, and the elliptic Cremona system, which is a higher analogue of discrete Painlevé equations associated with $\tilde{E}_8$ symmetry. This is a natural connection between integrable systems in different dimensions that is outside of the usual paradigm of reductions. Our approach is based on formulation of both systems in terms of birational Coxeter groups.
We give a representation--theoretic interpretation of recent discovered coupled soliton equations using vertex operators construction of affinization of not simple but quadratic Lie algebras. In this setup we are able to obtain new integrable hierarchies coupled to each Drinfeld--Sokolov of $A$, $B$, $C$, $D$ hierarchies and to construct their soliton solutions.
A class of nonlinear problems on the plane, described by nonlinear inhomogeneous $\bar{\partial}$-equations, is considered. It is shown that the corresponding dynamics, generated by deformations of inhomogeneous terms (sources) is described by Hamilton-Jacobi type equations associated with hierarchies of dispersionless integrable systems. These hierarchies are constructed by applying the quasiclassical $\bar{\partial}$-dressing method.
We demonstrate that statistics for several types of set partitions are described by generating functions which appear in the theory of integrable equations.
Consider the statement "Every Yang-Baxter integrable system is defined to be exactly-solvable". To formalise this statement, definitions and axioms are introduced. Then, using a specific Yang-Baxter integrable bosonic system, it is shown that a paradox emerges. A generalisation for completely integrable bosonic systems is also developed.
The tau-function formalism for a class of generalized ``zero-curvature'' integrable hierarchies of partial differential equations, is constructed. The class includes the Drinfel'd-Sokolov hierarchies. A direct relation between the variables of the zero-curvature formalism and the tau-functions is established. The formalism also clarifies the connection between the zero-curvature hierarchies and the Hirota-type hierarchies of Kac and Wakimoto.
Starting from a so-called flat exact semisimple bihamiltonian structures of hydrodynamic type, we arrive at a Frobenius manifold structure and a tau structure for the associated principal hierarchy. We then classify the deformations of the principal hierarchy which possess tau structures.
Analytic-bilinear approach for construction and study of integrable hierarchies, in particular, the KP hierarchy is discussed. It is based on the generalized Hirota identity. This approach allows to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations. Resolution of these functional equations leads to the $τ$-function and addition formulae to it. General discrete transformations of the $τ$-function are presented in the determinant form. Closed one-form and other formulae also arise naturally within the approach proposed. Generalized KP hi…
The $6 = 3\times 2$ huge Lie algebra $Ξ$ of all local and non local differential operators on a circle is applied to the standard Adler-Kostant-Symes (AKS) R-bracket sckeme. It is shown in particular that there exist three additional Lie structures, associated to three graded modified classical Yang-Baxter(GMCYB) equations. As we know from the standard case, these structures can be used to classify in a more consitent way a wide class of integrable systems. Other algebraic properties are also presented.
We study additional non-isospectral symmetries of constrained (reduced) N=2 supersymmetric KP hierarchies of integrable ``soliton''-like evolution equations. These symmetries are shown to form an infinite-dimensional non-Abelian superloop superalgebra. Furthermore we study the general Darboux-Backlund (DB) transformations (including adjoint-DB and binary DB) of N=2 super- KP hierarchies preserving (most of) the additional symmetries. Also we derive the explicit form of the general DB (N=2 ``super- soliton''-like) solutions in the form of generalized Wronskian-like super-determinants.
Integrable lattice equations arising in the context of singular manifold equations for scalar, multicomponent KP hierarchies and 2D Toda lattice hierarchy are considered. These equation generate the corresponding continuous hierarchy of singular manifold equations, its Bäcklund transformations and different forms of superposition principles. They possess rather special form of compatibility representation. The distinctive feature of these equations is invariance under the action of Möbius transformation. Geometric interpretation of these discrete equations is given.
We generalize the Drinfeld-Sokolov formalism of bosonic integrable hierarchies to superspace, in a way which systematically leads to the zero curvature formulation for the supersymmetric integrable systems starting from the Lax equation in superspace. We use the method of symmetric space as well as the non-Abelian gauge technique to obtain the supersymmetric integrable hierarchies of the AKNS type from the zero curvature condition in superspace with the graded algebras, sl(n+1,n), providing the Hermitian symmetric space structure.
A variety of Yang-Baxter maps are obtained from integrable multi-field equations on quad-graphs. A systematic framework for investigating this connection relies on the symmetry groups of the equations. The method is applied to lattice equations introduced by Adler and Yamilov and which are related to the nonlinear superposition formulae for the Bäcklund transformations of the nonlinear Schrödinger system and specific ferromagnetic models.
We consider a general framework for integrable hierarchies in Lax form and derive certain universal equations from which `functional representations' of particular hierarchies (like KP, discrete KP, mKP, AKNS), i.e. formulations in terms of functional equations, are systematically and quite easily obtained. The formalism genuinely applies to hierarchies where the dependent variables live in a noncommutative (typically matrix) algebra. The obtained functional representations can be understood as `noncommutative' analogs of `Fay identities' for the KP hierarchy.
A simple construction of Whitham type hierarchies in all genera is suggested. Potentials of these hierarchies are written as integrals of hypergeometric type. Possible generalization for universal moduli space is also briefly discussed.
For an arbitrary semisimple Frobenius manifold we construct Hodge integrable hierarchy of Hamiltonian partial differential equations. In the particular case of quantum cohomology the tau-function of a solution to the hierarchy generates the intersection numbers of the Gromov--Witten classes and their descendents along with the characteristic classes of Hodge bundles on the moduli spaces of stable maps. For the one-dimensional Frobenius manifold the Hodge hierarchy is a deformation of the Korteweg--de Vries hierarchy depending on an infinite number of parameters. Conjecturally this hierarchy is…
An operator deformed quantum algebra is discovered exploiting the quantum Yang-Baxter equation with trigonometric R-matrix. This novel Hopf algebra along with its $q \to 1$ limit appear to be the most general Yang-Baxter algebra underlying quantum integrable systems. Three different directions of application of this algebra in integrable systems depending on different sets of values of deforming operators are identified. Fixed values on the whole lattice yield subalgebras linked to standard quantum integrable models, while the associated Lax operators generate and classify them in an unified w…
Yang-Baxter (YB) map systems (or set-theoretic analoga of entwining YB structures) are presented. They admit zero curvature representations with spectral parameter depended Lax triples L1, L2, L3 derived from symplectic leaves of 2 x 2 binomial matrices equipped with the Sklyanin bracket. A unique factorization condition of the Lax triple implies a 3-dimensional compatibility property of these maps. In case L1 = L2 = L3 this property yields the se--theoretic quantum Yang-Baxter equation, i.e. the YB map property. By considering periodic 'staircase' initial value problems on quadrilateral latti…
A family of mappings from the solution spaces of certain generalized Drinfeld-Sokolov hierarchies to the self-dual Yang-Mills system on R^{2,2} is described. This provides an extension of the well-known relationship between self-dual connections and integrable hierarchies of AKNS and Drinfeld-Sokolov type.
We develop the concept of pluri-Lagrangian structures for integrable hierarchies. This is a continuous counterpart of the pluri-Lagrangian (or Lagrangian multiform) theory of integrable lattice systems. We derive the multi-time Euler Lagrange equations in their full generality for hierarchies of two-dimensional systems, and construct a pluri-Lagrangian formulation of the potential Korteweg-de Vries hierarchy.
We introduce primitive forms with or without higher residue structure and explore their connection with the flat structures with or without a metric and integrable hierarchies of KdV type. Just as the classical case of primitive forms with metric arXiv:1311.1659, the primitive forms without metrics are constructed as the positive part of the Birkhoff decomposition of formal oscillatory integrals with respect to the descendent variable. The oscilating integrals of a primitive form without metric give rise to a hierarchy of commuting PDE of the KdV type as in the case of primitive forms with met…
We present a one dimensional reversible block cellular automaton, where the time evolution is dictated by a period 3 cycle of update rules. At each time step a subset of the cells is updated using a four site rule with two control bits and two action bits. The model displays rich dynamics. There are three types of stable particles, left movers, right movers and "frozen" bound states that only move as an effect of scattering with the left and right movers. Multi-particle scattering in the system is factorized. We embed the model into the canonical framework of Yang-Baxter integrability by rigor…
Hirota bilinear identity for Cauchy-Baker-Akhieser (CBA) kernel is introduced as a basic tool to construct integrable hierarchies containing lattice and q-difference times. Determinant formula for the action of meromorphic function on CBA kernel is derived. This formula gives opportunity to construct generic solutions for integrable lattice equations.
Applying braided Yang-Baxter equation quantum integrable and Bethe ansatz solvable 1D anyonic lattice and field models are constructed. Along with known models we discover novel lattice anyonic and $q$-anyonic models as well as nonlinear Schrödinger equation (NLS) and the derivative NLS quantum field models involving anyonic operators, $N$-particle sectors of which yield the well known anyon gases, interacting through $δ$ and derivative $δ$-function potentials.
We present some general results on properties of the bihamiltonian cohomologies associated to bihamiltonian structures of hydrodynamic type, and compute the third cohomology for the bihamiltonian structure of the dispersionless KdV hierarchy. The result of the computation enables us to prove the existence of bihamiltonian deformations of the dispersionless KdV hierarchy starting from any of its infinitesimal deformations.
We propose a systematic treatment of symmetries of KP integrable systems, including constrained (reduced) KP models ${\sl cKP}_{R,M}$, and their multi-component (matrix) generalizations. Any such integrable hierarchy is shown to possess an additional $({\hat U}(1)\oplus{\hat {SL}}(M))_{+} \oplus ({\hat {SL}}(M+R))_{-}$ loop-algebra symmetry. Also we provide a systematic construction of the full algebra of Virasoro additional symmetries in the case of constrained KP models which requires a nontrivial modification of the known Orlov-Schulman construction for the general unconstrained KP hierarch…
We introduce integrable KdV type hierarchy associated naturally with arbitrary semi-simple Frobenius manifold. We present hierarchy in a Lax form and show that it admits bihamiltonian description.
We here present two different hierarchies of PDEs in 1+1 dimensions whose first and second member are the shallow water wave Camassa-Holm and Qiao equations, correspondingly. These two hierarchies can be transformed by reciprocal methods into the Calogero-Bogoyanlevski-Schiff equation (CBS) and its modified version (mCBS), respectively. Considering that there exists a Miura transformation between the CBS and mCBS, we shall obtain a relation between the initial hierarchies by means of a composition of a Miura and reciprocal transforms.
We discuss the origin of the associativity (WDVV) equations in the context of quasiclassical or Whitham hierarchies. The associativity equations are shown to be encoded in the dispersionless limit of the Hirota equations for KP and Toda hierarchies. We show, therefore, that any tau-function of dispersionless KP or Toda hierarchy provides a solution to associativity equations. In general, they depend on infinitely many variables. We also discuss the particular solution to the dispersionless Toda hierarchy that describes conformal mappings and construct a family of new solutions to the WDVV equa…
