Exactly Solvable And Integrable Systems

We derive and classify all regular solutions of the boundary Yang-Baxter equation for 19-vertex models known as Zamolodchikov-Fateev or A (1) 1 model, Izergin-Korepin or A (2) 2 model, sl(2|1) model and osp(2|1) model. We find that there is a general solution for A (1) 1 and sl(2|1) models. In both models it is a complete K-matrix with three free parameters. For the A (2) 2 and osp(2|1) models we find three general solutions, being two complete reflection K-matrices solutions and one incomplete reflection K-matrix solution with some null entries. In both models these solutions have two free parameters. Integrable spin-1 Hamiltonians with general boundary interactions are also presented. Several reduced solutions from these general solutions are presented in the appendices.

We establish a simple algebraic relationship between the energy eigenstates of the rational Calogero-Sutherland model with harmonic oscillator and Coulomb-like potentials. We show that there is an underlying SU(1,1) algebra in both of these models which plays a crucial role in such an identification. Further, we show that our analysis is in fact valid for any many-particle system in arbitrary dimensions whose potential term (apart from the oscillator or the Coulomb-like potential) is a homogeneous function of coordinates of degree -2. The explicit coordinate transformation which maps the Coulomb-like problem to the oscillator one has also been determined in some specific cases.

We consider dynamical systems depending on one or more real parameters, and assuming that, for some ``critical'' value of the parameters, the eigenvalues of the linear part are resonant, we discuss the existence -- under suitable hypotheses -- of a general class of bifurcating solutions in correspondence to this resonance. These bifurcating solutions include, as particular cases, the usual stationary and Hopf bifurcations. The main idea is to transform the given dynamical system into normal form (in the sense of Poincaré-Dulac), and to impose that the normalizing transformation is convergent, using the convergence conditions in the form given by A. Bruno. Some specially interesting situations, including the cases of multiple-periodic solutions, and of degenerate eigenvalues in the presence of symmetry, are also discussed with some detail.

The discrete Painlevé property is precisely defined, and basic discretization rules to preserve it are stated. The discrete Painlevé test is enriched with a new method which perturbs the continuum limit and generates infinitely many no-log conditions. A general, direct method is provided to search for discrete Lax pairs.

We introduce the Schlesinger transformations of the Gambier equation. The latter can be written, in both the continuous and discrete cases, as a system of two coupled Riccati equations in cascade involving an integer parameter n. In the continuous case the parameter appears explicitly in the equation while in the discrete case it corresponds to the number of steps for singularity confinement. Two Schlesinger transformations are obtained relating the solutions for some value n to that corresponding to either n+1 or n+2.

Schlesinger transformations are discrete monodromy preserving symmetry transformations of the classical Schlesinger system. Generalizing well-known results from the Riemann sphere we construct these transformations for isomonodromic deformations on genus one Riemann surfaces. Their action on the system's tau-function is computed and we obtain an explicit expression for the ratio of the old and the transformed tau-function.

We study self-similar solutions of NLS-type dynamical systems. Lagrangian approach is used to show that they can be reduced to three canonical forms, which are related by Miura transformations. The fourth Painleve equation (PIV) is central in our consideration - it connects Heisenberg model, Volterra model and Toda model to each other. The connection between the rational solutions of PIV and Coulomb gas in a parabolic potential is established. We discuss also the possibility to obtain an exact solution for optical soliton i.e. of the NLS equation with time-dependent dispersion.

Similarity symmetries of the factorization chains for one-dimensional differential and finite-difference Schrödinger equations are discussed. Properties of the potentials defined by self-similar reductions of these chains are reviewed. In particular, their algebraic structure, relations to q -special functions, infinite soliton systems, supersymmetry, coherent states, orthogonal polynomials, one-dimensional Ising chains and random matrices are outlined.

For the elliptic Gaudin model (a degenerate case of XYZ integrable spin chain) a separation of variables is constructed in the classical case. The corresponding separated coordinates are obtained as the poles of a suitably normalized Baker-Akhiezer function. The classical results are generalized to the quantum case where the kernel of separating integral operator is constructed. The simplest one-degree-of-freedom case is studied in detail.

Binary constrained flows of soliton equations admitting 2×2 Lax matrices have 2N degrees of freedom, which is twice as many as degrees of freedom in the case of mono-constrained flows. For their separation of variables only N pairs of canonical separated variables can be introduced via their Lax matrices by using the normal method. A new method to introduce the other N pairs of canonical separated variables and additional separated equations is proposed. The Jacobi inversion problems for binary constrained flows are established. Finally, the factorization of soliton equations by two commuting binary constrained flows and the separability of binary constrained flows enable us to construct the Jacobi inversion problems for some soliton hierarchies.