On chaotic dynamics in "pseudobilliard" Hamiltonian systems with two degrees of freedom.

A new class of Hamiltonian dynamical systems with two degrees of freedom is studied, for which the Hamiltonian function is a linear form with respect to moduli of both momenta. For different potentials such systems can be either completely integrable or behave just as normal nonintegrable Hamiltonian systems with two degrees of freedom: one observes many of the phenomena characteristic of the latter ones, such as a breakdown of invariant tori as soon as the integrability is violated; a formation of stochastic layers around destroyed separatrices; bifurcations of periodic orbits, etc. At the same time, the equations of motion are simply integrated on subsequent adjacent time intervals, as in billiard systems; i.e., all the trajectories can be calculated explicitly: Given an initial data, the state of the system is uniquely determined for any moment. This feature of systems in interest makes them very attractive models for a study of nonlinear phenomena in finite-dimensional Hamiltonian systems. A simple representative model of this class (a model with quadratic potential), whose dynamics is typical, is studied in detail. (c) 1997 American Institute of Physics.

One-dimensional mappings of the eigenelements of the Schrodinger problem.

An approach is proposed for analyzing the inverse spectral problems for the Schrodinger equation based on writing the equation for the analog of the number-of-quanta operator for a harmonic oscillator. This equation makes it possible to determine not only the one-dimensional mapping of the energy eigenvalues but also the linear equation for the point spectrum shift operator of the Schrodinger problem. The solvability conditions of the latter lead to a nonlinear equation that determines the class of allowable potentials. Two classes of potentials regular in R(1) and symmetrical are isolated on the basis of the proposed approach. The first of these leads to equidistant spectra with a gap of arbitrary size and location. The spectrum of the second potential class is a combination of three rigorously equidistant spectra with ground states that are shifted by an arbitrary amount. Generalizations to the case of essentially nonequidistant spectra are shown to be possible.

Dynamical systems in the theory of nonlinear waves with allowance for nonlocal interactions (the Whitham-Benjamin equation).

In this paper we briefly present a general approach to the description of the nonlinear and nonlocal Whitham-Benjamin model, based on the introduction of a system of auxiliary fields that interact locally with the initial nonlinear field. In the case of stationary waves a corresponding dynamical system is defined that admits of a Hamiltonian representation. Some results are presented of a qualitative and numerical analysis of the stationary solitary waves of the Whitham-Benjamin equation with a rapidly decreasing oscillatory kernel. An investigation is made into a phenomenon related to the loss of smoothness of the solution of the original equation and the noncontinuability of these solutions when the structural parameters of the system are changed (this phenomenon is analogous to the formation of limiting Stokes waves).

Equidistant spectra of anharmonic oscillators.

Some representative potentials of the anharmonic-oscillator type are constructed. Some corresponding spectra-shift operators are also constructed. These operators are a natural generalization of Fok creation and annihilation operators. The Schrodinger problem for these potentials leads to an equidistant energy spectrum for all excited states, which are separated from the ground state by an energy gap. The general properties of the dynamic system generated by spectral-shift operators of third degree are analyzed. Several examples of such anharmonic oscillators are discussed. The relationship between the eigenvectors of the Schrodinger problem and a certain type of nonclassical orthogonal polynomials is established.

Dynamics of topological solitons in models with nonlocal interactions.

A nondissipative generalization of the sine-Gordon equation to cases with nonlocal interactions is analyzed. A model of this sort is shown to describe signal propagation in a Josephson transmission line with a nonlocal inductive coupling. The incorporation of nonlocal interactions changes the properties of the model in a qualitative way, leading in particular to the appearance of some new soliton entities: 2kpi kinks, where k greater, similar 1. These entities do not arise in a local model. They are evolutionary, they interact with each other in a quasielastic fashion, and they can be generated in a corresponding transmission line.

Dynamical systems in the theory of solitons in the presence of nonlocal interactions.

The behavior of solitons in models which take into account complex dispersion or nonlocal interaction of nonlinear waves is examined. A method is proposed to reduce this problem to one involving special trajectories (homoclinic and heteroclinic) of the dynamic system. This method involves replacing the nonlinear integrodifferential equation with the differential equations which link the original nonlinear field with the auxiliary linear fields. The interaction of fields in such a model is a local interaction. The number of introduced linear fields is determined by the Laplace transform of the integral operator kernel of the basic integrodifferential equation. The problem involving topological solitons for the nonlocal generalization of the Klein-Gordon equation is considered. Nonlocal interactions are found to lead to a number of singularities (unrestricted increase in the slope of the topological soliton front, break in the solutions, and other singularities).

Bifurcations of the trajectories at the saddle level in a Hamiltonian system generated by two coupled Schrodinger equations.

Bifurcations of the complex homoclinic loops of an equilibrium saddle point in a Hamiltonian dynamical system with two degrees of freedom are studied. It arises to pick out the stationary solutions in a system of two coupled nonlinear Schrodinger equations. Their relation to bifurcations of hyperbolic and elliptic periodic orbits at the saddle level is studied for varying structural parameters of the system. Series of complex loops are described whose existence is related to periodic orbits.