Exactly Solvable And Integrable Systems

The motion on the sphere S 2 with the potential V=( x 1 x 2 x 3 ) −2/3 is considered. The Lax representation and the linearisation procedure for this two-dimensional integrable system are discussed.

An integrable discretization of the inhomogeneous Ablowitz-Ladik model with a linear force is introduced. Conditions on parameters of the discretization which are necessary for reproducing Bloch oscillations are obtained. In particular, it is shown that the step of the discretization must be comensurable with the period of oscillations imposed by the inhomogeneous force. By proper choice of the step of the discretization the period of oscillations of a soliton in the discrete model can be made equal to an integer number of periods of oscillations in the underline continuous-time lattice.

We show that lumps (solitons) of the Davey--Stewartson II equation fail under small perturbations of initial data.

Pulse broadening for optical solitons due to birefringence is investigated. We present an analytical solution which describes the propagation of solitons in birefringent optical fibers. The special solutions consist of a combination of purely solitonic terms propagating along the principal birefringence axes and soliton-soliton interaction terms. The solitonic part of the solutions indicates that the decay of initially localized pulses could be due to different propagation velocities along the birefringence axes. We show that the disintegration of solitonic pulses in birefringent optical fibers can be caused by two effects. The first effect is similar as in linear birefringence and is related to the unequal propagation velocities of the modes along the birefringence axes. The second effect is related to the nonlinear soliton-soliton interaction between the modes, which makes the solitonic pulse-shape blurred.

Some soliton equation in 2+1 dimensions and their 1+1 and/or dimensional integrable reductions are considered.

In this work we consider superintegrable systems in the classical r -matrix method. By using other authomorphisms of the loop algebras we construct new superintegrable systems with rational potentials from geodesic motion on R 2n .

The Calogero model with negative harmonic term is shown to be equivalent to the set of negative harmonic oscillators. Two time-independent canonical transformations relating both models are constructed: one based on the recent results concerning quantum Calogero model and one obtained from dynamical $sL(2,\rr)$ algebra. The two-particle case is discussed in some detail.

In this paper we study the reductions of evolutionary PDEs on the manifold of the stationary points of time-dependent symmetries. In particular we describe how the finite dimensional Hamiltonian structure of the reduced system is obtained from the Hamiltonian structure of the initial PDE and we construct the time-dependent Hamiltonian function. We also present a very general Lagrangian formulation of the procedure of reduction. As an application we consider the case of the Painleve' equations PI, PII, PIII, PVI and also certain higher order systems appeared in the theory of Frobenius manifolds.

A new Lax operator is proposed from the viewpoint of constructing the integrable hierarchies related with N=2 super W n algebra. It is shown that the Poisson algebra associated to the second Hamiltonian structure for the resulted hierarchy contains the N=2 super Virasoro algebra as a proper subalgebra. The simplest cases are discussed in detail. In particular, it is proved that the supersymmetric two-boson hierarchy is one of N=2 supersymmetric KdV hierarchies. Also, a Lax operator is supplied for one of N=2 supersymmetric Boussinesq hierarchies.

The Lakshmanan equivalent counterparts of the some Myrzakulov equations are found.