The q-deformed analogue of the Onsager algebra: Beyond the Bethe ansatz approach

The spectral properties of operators formed from generators of the q-Onsager non-Abelian infinite-dimensional algebra are investigated. Using a suitable functional representation, all eigenfunctions are shown to obey a second-order q-difference equation (or its degenerate discrete version). In the algebraic sector associated with polynomial eigenfunctions (or their discrete analogues), Bethe equations naturally appear. Beyond this sector, where the Bethe ansatz approach is not applicable in related massive quantum integrable models, the eigenfunctions are also described. The spin-half XXZ open spin chain with general integrable boundary conditions is reconsidered in light of this approach: all the eigenstates are constructed. In the algebraic sector which corresponds to special relations among the parameters, known results are recovered.     

Integrable Evolution Equations on Associative Algebras

This paper surveys the classification of integrable evolution equations whose field variables take values in an associative algebra, which includes matrix, Clifford, and group algebra valued systems. A variety of new examples of integrable systems possessing higher order symmetries are presented. Symmetry reductions lead to an associative algebra-valued version of the Painlevé transcendent equations. The basic theory of Hamiltonian structures for associative algebra-valued systems is developed and the biHamiltonian structures for several examples are found. Received: 12 March 1997 / Accepted: 27 August 1997     

On an integrable deformed affinsphären equation. A reciprocal gasdynamic connection

The integrable affinsphären equation originally arose in a geometric context but has an interesting gasdynamic connection. Here, an integrable deformed version of the affinsphären equation is derived in a novel manner via the action of reciprocal transformations on a related anisentropic gasdynamics system. A linear representation for the deformed affinsphären equation is constructed by means of the reciprocal transformations. The latter are then employed to derive a class of exact solutions in parametric form.     

Multi-Soliton Solutions and Integrable Discretization for a Coupled Modified Volterra Lattice Equation

In this paper, we study a coupled modified Volterra lattice equation which is an integrable semidiscrete version of the coupled KdV and the coupled mKdV equation. By using the Darboux transformation, we obtain its new explicit solutions including multi-soliton and multi-positon. Furthermore, an integrable discretization of the coupled modified Volterra lattice equation is constructed.     

Multi-Soliton Solutions and Integrable Discretization for a Coupled Modified Volterra Lattice Equation

In this paper, we study a coupled modified Volterra lattice equation which is an integrable semidiscrete version of the coupled KdV and the coupled mKdV equation. By using the Darboux transformation, we obtain its new explicit solutions including multi-soliton and multi-positon. Furthermore, an integrable discretization of the coupled modified Volterra lattice equation is constructed.     

ON THE APPLICATION OF A GENERALIZED VERSION OF THE DRESSING METHOD TO THE INTEGRATION OF VARIABLE COEFFICIENT N-COUPLED NONLINEAR SCHRÖDINGER EQUATION

N-coupled nonlinear Schrödinger (NLS) equations have been proposed to describe N-pulse simultaneous propagation in optical fibers. When the fiber is nonuniform, N-coupled variable-coefficient NLS equations can arise. In this paper, a family of N-coupled integrable variable-coefficient NLS equations are studied by using a generalized version of the dressing method. We first extend the dressing method to the versions with (N + 1) × (N + 1) operators and (2N + 1) × (2N + 1) operators. Then, we obtain three types of N-coupled variable-coefficient equations (N-coupled NLS equations, N-coupled Hirota equations and N-coupled high-order NLS equations). Then, the compatibility conditions are given, which insure that these equations are integrable. Finally, the explicit solutions of the new integrable equations are obtained.     

Is the Classical Bukhvostov-Lipatov Model Integrable? A Painlevé Analysis

Abstract

In this work we apply the Weiss, Tabor and Carnevale integrability criterion (Painlevé analysis) to the classical version of the two dimensional Bukhvostov-Lipatov model. We are led to the conclusion that the model is not integrable classically, except at a trivial point where the theory can be described in terms of two uncoupled sine-Gordon models.     

A modified Schwarzian Korteweg-de Vries equation in 2 + 1 dimensions with lots of isochronous solutions

Physics of Atomic Nuclei - A modified version of the integrable Schwarzian Korteweg de Vries equation in 2 + 1 dimensions is introduced, and it is pointed out that it possesses lots of isochronous...     

Correlation functions for the Heisenberg XXZ-antiferromagnet

The general method for calculation of correlation functions in integrable quantum models has been given in papers [1, 2]. The correlation function of the third components of local spins for the Heisenberg one-dimensionalXXZ-antiferromagnet is calculated in this paper. The answer is a series which gives, in particular, an improved version of the usual perturbative expansion in the anisotropy parameter. The remarkable property of the series obtained is that the long-distance asymptotics of the correlator is given already by the first term. The arguments are given in favour of the convergence of the series.     

Multi-component hierarchies of double bracket equations

A new integrable hierarchy, with equations defined by double brackets of two matrix pseudo-differential operators (Lax pairs), is constructed. Some algebraic properties are demonstrated. It is also shown that each equation is equivalent to a certain gradient flow. A new version of the Zakharov-Shabat type equations is proved. Formal solutions of this hierarchy are constructed using a matrix “double bracket bilinear identity”.     

Contact flows and integrable systems

We consider Hamiltonian systems restricted to the hypersurfaces of contact type and obtain a partial version of the Arnold–Liouville theorem: the system need not be integrable on the whole phase space, while the invariant hypersurface is foliated on an invariant Lagrangian tori. In the second part of the paper we consider contact systems with constraints. As an example, the Reeb flows on Brieskorn manifolds are considered.     

Integrable systems,symmetries, and quantization

These notes are an expanded version of a mini-course given at the Poisson 2016 conference in Geneva. Starting from classical integrable systems in the sense of Liouville, we explore the notion of non-degenerate singularities and expose recent research in connection with semi-toric systems. The quantum and semiclassical counterparts are also presented, in the viewpoint of the inverse question: from the quantum mechanical spectrum, can one recover the classical system?     

N = 2 Supercomplexification of the Korteweg-de Vries,Sawada-Kotera and Kaup-Kupershmidt Equations

The supercomplexification is a special method of N = 2 supersymmetrization of the integrable equations in which the bosonic sector can be reduced to the complex version of these equations. The N = 2 supercomplex Korteweg-de Vries, Sawada-Kotera and Kaup-Kupershmidt equations are defined and investigated. The common attribute of the supercomplex equations is appearance of the odd Hamiltonian structures and superfermionic conservation laws. The odd bi-Hamiltonian structure, Lax representation and superfermionic conserved currents for new N = 2 supersymmetric Korteweg-de Vries equation and for Sawada-Kotera one, are given.     

Matter Creation by Geometry in an Integrable Weyl-Dirac Theory

An integrable version of the Weyl-Dirac geometry is presented. This framework is a natural generalization of the Riemannian geometry, the latter being the basis of the classical general relativity theory. The integrable Weyl-Dirac theory is both coordinate covariant and gauge covariant (in the Weyl sense), and the field equations and conservation laws are derived from an action integral. In this framework matter creation by geometry is considered. It is found that a spatially confined, spherically symmetric formation made of pure geometric quantities is a massive entity. This may be treated either as a fundamental particle or as a cosmic body. In an F-R-W universe at the very beginning of the expansion phase the cosmic matter is created from an initial Planckian egg made of geometry, and during the following expansion geometric fields continue to stimulate the matter production.     

The supersymmetric soliton correlation matrix

Supersymmetric systems in (2/2) dimensions integrable by the supersymmetric generalization of the Zakharov-Shabat ?dressing? method are studied. The supersymmetric version of the ?soliton correlation matrix? is used to obtain multi- soliton solutions to generic supersymmetric systems of Zakharov-Mikhailov- Shabat type, together with their reductions under finite automorphism groups. The sypersymmetric S² sigma model is worked out as an explicit application of the method.     

Exact g-function flows from the staircase model

Equations are found for exact g  -functions corresponding to integrable bulk and boundary flows between successive unitary c<1$c<1$ minimal conformal field theories in two dimensions, confirming and extending previous perturbative results. These equations are obtained via an embedding of the flows into a boundary version of Al. Zamolodchikov's staircase model.     

Mechanical systems with Poincaré invariance

Some years ago Ruijsenaars and Schneider initiated the study of mechanical systems exhibiting an action of the Poincaré algebra. The systems they discovered were far richer: their models were actually integrable and possessed a natural quantum version. We follow this early work finding and classifying mechanical systems with such an action. New solutions are found together with a new class of models exhibiting an action of the Galilean algebra. These are related to the functional identities underlying the various Hirzebruch genera. The quantum mechanics is also discussed.     

Continuum Limit of the Volterra Model, Separation of Variables and Non-Standard Realizations of the Virasoro Poisson Bracket

The classical Volterra model, equipped with the Faddeev-Takhtajan Poisson bracket provides a lattice version of the Virasoro algebra. The Volterra model being integrable, we can express the dynamical variables in terms of the so-called separated variables. Taking the continuum limit of these formulae, we obtain the Virasoro generators written as determinants of infinite matrices, the elements of which are constructed with a set of points lying on an infinite genus Riemann surface. The coordinates of these points are separated variables for an infinite set of Poisson commuting quantities including L ₀. The scaling limit of the eigenvector can also be calculated explicitly, so that the associated Schroedinger equation is in fact exactly solvable.     

Scaling of dynamical properties of the Fermi-Ulam accelerator

We investigate numerically the chaotic sea of the complete Fermi-Ulam model (FUM) and of its simplified version (SFUM). We perform a scaling analysis near the integrable to non-integrable transition to describe the average energy as function of time t and as function of iteration (or collision) number n. When t is employed as independent variable, the exponents of FUM and SFUM are different. However, when n is used, the exponents are the same for both FUM and SFUM. In the collision number analysis, we present analytical arguments supporting the values of the exponents related to the control paramenter and to the initial velocity. We describe also how the scaling exponents obtained by using t as independent variable are related to the ones obtained with n. In contrast to SFUM, the average energy in FUM saturates for long times. We discuss the origin of the observed differences and similarities between FUM and its simplified version.     

On the structure of the commutative Z2 graded algebra valued integrable equations

Partial differential equations integrable by the linear matrix spectral problem of arbitrary order are considered for the case that the “potentials” take their values in the commutative infinite-dimensional Z₂ graded algebra (superalgebra). The general form of the integrable equations and their Bäcklund transformations are found. The infinite sets of the integrals of the motion are constructed. The hamiltonian character of the integrable equations is proved.