Exactly Solvable And Integrable Systems

We study the additional symmetries associated with the q -deformed Kadomtsev-Petviashvili ( q -KP) hierarchy. After identifying the resolvent operator as the generator of the additional symmetries, the q -KP hierarchy can be consistently reduced to the so-called q -deformed constrained KP ( q -cKP) hierarchy. We then show that the additional symmetries acting on the wave function can be viewed as infinitesimal Bäcklund transformations by acting the vertex operator on the tau-function of the q -KP hierarchy. This establishes the Adler-Shiota-van Moerbeke formula for the q -KP hierarchy.

It is shown that quadratic constraints are compatible with the geometric integrability scheme of the multidimensional quadrilateral lattice equation. The corresponding Ribaucour reduction of the fundamental transformation of quadrilateral lattices is found as well, and superposition of the Ribaucour transformations is presented in the vectorial framework. Finally, the quadratic reduction approach is illustrated on the example of multidimensional circular lattices.

The space of realizations of a finite-dimensional Lie algebra by first order differential operators is naturally isomorphic to H^1 with coefficients in the module of functions. The condition that a realization admits a finite-dimensional invariant subspace of functions seems to act as a kind of quantization condition on this H^1. It was known that this quantization of cohomology holds for all realizations on 2-dimensional homogeneous spaces, but the extent to which quantization of cohomology is true in general was an open question. The present article presents the first known counter-examples to quantization of cohomology; it is shown that quantization can fail even if the Lie algebra is semi-simple, and even if the homogeneous space in question is compact. A explanation for the quantization phenomenon is given in the case of semi-simple Lie algebras. It is shown that the set of classes in H^1 that admit finite-dimensional invariant subspaces is a semigroup that lies inside a finitely-generated abelian group. In order for this abelian group be a discrete subset of H^1, i.e. in order for quantization to take place, some extra conditions on the isotropy subalgebra are required. Two different instances of such necessary conditions are presented.

A quantum evolution model in 2+1 discrete space - time, connected with 3D fundamental map R, is investigated. Map R is derived as a map providing a zero curvature of a two dimensional lattice system called "the current system". In a special case of the local Weyl algebra for dynamical variables the map appears to be canonical one and it corresponds to known operator-valued R-matrix. The current system is a kind of the linear problem for 2+1 evolution model. A generating function for the integrals of motion for the evolution is derived with a help of the current system. The subject of the paper is rather new, and so the perspectives of further investigations are widely discussed.

For the integrable case of the discrete self-trapping (DST) model we construct a Backlund transformation. The dual Lax matrix and the corresponding dual Backlund transformation are also found and studied. The quantum analog of the Backlund transformation (Q-operator) is constructed as the trace of a monodromy matrix with an infinite-dimensional auxiliary space. We present the Q-operator as an explicit integral operator as well as describe its action on the monomial basis. As a result we obtain a family of integral equations for multivariable polynomial eigenfunctions of the quantum integrable DST model. These eigenfunctions are special functions of the Heun class which is beyond the hypergeometric class. The found integral equations are new and they shall provide a basis for efficient analytical and numerical studies of such complicated functions.

In this letter, I have considered one-dimensional quantum system with different masses m and M , which does not appear integrable in general. However I have found an exact two-body wave function and due to the extension of the integrable system to more general system, it was concluded that the rapidity or quasi-momentum in the integrable system should be regarded as a modification of velocity rather than that of momentum. I have also considered the three-body wave function and argued its integrable condition.

We show explicitly how to construct the quantum Lax pair for systems with open boundary conditions. We demonstrate the method by applying it to the Heisenberg XXZ model with general integrable boundary terms.

We develop a quantum Lax scheme for IRF models and difference versions of Calogero-Moser-Sutherland models introduced by Ruijsenaars. The construction is in the spirit of the Adler-Kostant-Symes method generalized to the case of face Hopf algebras and elliptic quantum groups with dynamical R-matrices.

Two quasi--biHamiltonian systems with three and four degrees of freedom are presented. These systems are shown to be separable in terms of Nijenhuis coordinates. Moreover the most general Pfaffian quasi-biHamiltonian system with an arbitrary number of degrees of freedom is constructed (in terms of Nijenhuis coordinates) and its separability is proved.

Systems of Newton equations of the form q ¨ =−1/2 A −1 (q)∇k with an integral of motion quadratic in velocities are studied. These equations generalize the potential case (when A=I, the identity matrix) and they admit a curious quasi-Lagrangian formulation which differs from the standard Lagrange equations by the plus sign between terms. A theory of such quasi-Lagrangian Newton (qLN) systems having two functionally independent integrals of motion is developed with focus on two-dimensional systems. Such systems admit a bi-Hamiltonian formulation and are proved to be completely integrable by embedding into five-dimensional integrable systems. They are characterized by a linear, second-order PDE which we call the fundamental equation. Fundamental equations are classified through linear pencils of matrices associated with qLN systems. The theory is illustrated by two classes of systems: separable potential systems and driven systems. New separation variables for driven systems are found. These variables are based on sets of non-confocal conics. An effective criterion for existence of a qLN formulation of a given system is formulated and applied to dynamical systems of the Henon-Heiles type.