Solitary wave solutions of nonlocal sine-Gordon equations.

In this paper a nonlocal generalization of the sine-Gordon equation, u(tt)+sin u=( partial differential / partial differential x) integral (- infinity ) (+ infinity )G(x-x('))u(x(') )(x('),t)dx(') is considered. We present a brief review of the applications of such equations and show that involving such a nonlocality can change features of the model. In particular, some solutions of the sine-Gordon model (for example, traveling 2pi-kink solutions) may disappear in the nonlocal model; furthermore, some new classes of solutions such as traveling topological solitons with topological charge greater than 1 may arise. We show that the lack of Lorenz invariancy of the equation under consideration can lead to a phenomenon of discretization of kink velocities. We discussed this phenomenon in detail for the special class of kernels G(xi)= summation operator (j=1) (N)kappa(j)e(-eta(j)mid R:ximid R:), eta(j)>0, j=1,2, em leader,N. We show that, generally speaking, in this case the velocities of kinks (i) are determined unambiguously by a type of kink and value(s) of kernel parameter(s); (ii) are isolated i.e., if c(*) is the velocity of a kink then there are no other kink solutions of the same type with velocity c in (c(*)- varepsilon,c(*)+ varepsilon ) for a certain value of varepsilon. We also used this special class of kernels to construct approximations for analytical and numerical study of the problem in a more general case. Finally, we set forth results of the numerical investigation of the problem with the kernel that is the McDonald function G(xi) approximately K(0)(mid R:ximid R:/lambda) (lambda is a parameter) that have applications in the Josephson junction theory. (c) 1998 American Institute of Physics.

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.

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).

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.

Nonintegrable field equations and homoclinic loops of Hamiltonian systems.

Connections between soliton or self-localized states of nonlinear wave equations and special objects (homo- and heteroclinic trajectories) of dynamical systems are considered.