Exactly Solvable And Integrable Systems

Using the KMS condition and exchange algebras we discuss the monodromy and modular properties of two-point KMS states of the critical Ising model.

The stability of the recently discovered compacton solutions is studied by means of both linear stability analysis as well as Lyapunov stability criteria. From the results obtained it follows that, unlike solitons, all the allowed compacton solutions are stable, as the stability condition is satisfied for arbitrary values of the nonlinearity parameter. The results are shown to be true even for the higher order nonlinear dispersion equations for compactons. Some new conservation laws for the higher order nonlinear dispersion equations are also presented.

We present a novel integrable non-autonomous partial differential equation of the Schwarzian type, i.e. invariant under Möbius transformations, that is related to the Korteweg-de Vries hierarchy. In fact, this PDE can be considered as the generating equation for the entire hierarchy of Schwarzian KdV equations. We present its Lax pair, establish its connection with the SKdV hierarchy, its Miura relations to similar generating PDEs for the modified and regular KdV hierarchies and its Lagrangian structure. Finally we demonstrate that its similarity reductions lead to the {\it full} Painlevé VI equation, i.e. with four arbitary parameters.

It is shown that a class of dynamical systems (encompassing the one recently considered by F. Calogero [J. Math. Phys. 37 (1996) 1735]) is both quasi-bi-Hamiltonian and bi-Hamiltonian. The first formulation entails the separability of these systems; the second one is obtained trough a non canonical map whose form is directly suggested by the associated Nijenhuis tensor.

Equations of geodesic deviation for the 3-dimensional and 4-dimensional Riemann spaces are discussed. Availability of wide classes of exact solutions of such equations, due to recent results for the matrix Schrödinger equation, is demonstrated. Particular classes of exact solutions for the geodesic deviation equation as well as for the Raychaudhuri and generalized Raychaudhuri equation are presented. Solutions of geodesic deviation equation for the Schwarzshild and Kasner metrics are found.

We investigate some classical evolution model in the discrete 2+1 space-time. A map, giving an one-step time evolution, may be derived as the compatibility condition for some systems of linear equations for a set of auxiliary linear variables. Dynamical variables for the evolution model are the coefficients of these systems of linear equations. Determinant of any system of linear equations is a polynomial of two numerical quasimomenta of the auxiliary linear variables. For one, this determinant is the generating functions of all integrals of motion for the evolution, and on the other hand it defines a high genus algebraic curve. The dependence of the dynamical variables on the space-time point (exact solution) may be expressed in terms of theta functions on the jacobian of this curve. This is the main result of our paper.

We discuss some algebraic setting of chiral SU(N ) k models in terms of the statistical dimensions of their fields. In particular, the conformal dimensions and the central charge of the chiral SU(N ) k models are calculated from their braid matrices. Futhermore, at level K=2, we present the characteristic polynomials of their fusion matrices in a factored form.

We point out that the N=4 supersymmetric KdV hierarchy, when written through the prepotentials of the bosonic chiral and antichiral N=2 supercurrents, exhibits a freedom related to the possibility to choose different gauges for the prepotentials. In particular, this implies that the Lax operator for the N=4 SKdV system and the associated realization of N=4 supersymmetry obtained in solv-int/9802003 are reduced to the previously known ones. We give the prepotential form of the `small' N=4 superconformal algebra, the second hamiltonian structure algebra of the N=4 SKdV hierarchy, for two choices of gauge.

A (2+1)-dimensional perturbed KdV equation, recently introduced by W.X. Ma and B. Fuchssteiner, is proven to pass the Painlevé test for integrability well, and its 4 × 4 Lax pair with two spectral parameters is found. The results show that the Painlevé classification of coupled KdV equations by A. Karasu should be revised.

We consider an integrability test for ultradiscrete equations based on the singularity confinement analysis for discrete equations. We show how singularity pattern of the test is transformed into that of ultradiscrete equation. The ultradiscrete solution pattern can be interpreted as a perturbed solution. We can also check an integrability of a given equation by a perturbation growth of a solution, namely Lyapunov exponent. Therefore, singularity confinement test and Lyapunov exponent are related each other in ultradiscrete equations and we propose an integrability test from this point of view.