Physical dynamics of quasi-particles in nonlinear wave equations

Journal of Differential Equations, 2010

In this paper we investigate the dynamics of solitons occurring in the nonlinear Schroedinger equation when a parameter h → 0. We prove that under suitable assumptions, the the soliton approximately follows the dynamics of a point particle, namely, the motion of its barycenter q h (t) satisfies the equation

2006

Abstract: Profound advances have recently interested nonlinear field theories and their exact or approximate solutions. We review the last results and point out some important unresolved questions. It is well known that quantum field theories are based upon Fourier series and the identification of plane waves with free particles. On the contrary, nonlinear field theories admit the existence of coherent solutions (dromions, solitons and so on). Moreover, one can construct lower dimensional chaotic patterns, periodic-chaotic patterns, chaotic soliton and dromion patterns. In a similar way, fractal dromion and lump patterns as well as stochastic fractal excitations can appear in the solution. We discuss in some detail a nonlinear Dirac field and a spontaneous symmetry breaking model that are reduced by means of the asymptotic perturbation method to a system of nonlinear evolution equations integrable via an appropriate change of variables. Their coherent, chaotic and fractal solutions ...

International Journal of Theoretical Physics, 2005

The dynamics of interacting solitons of a system of two coupled nonlinear partial differential equations is studied numerically using a finite difference method in bidimensional spacetime. Stable, static topological solitons are obtained from an iterativevariational method, and used as initial solutions for the dynamical calculations. Some of the static solutions decay to stable solitons. Some subtle aspects of topological charges for the system under consideration are also discussed.

Progress of Theoretical Physics, 1987

2009

An analytical model for the soliton-potential interaction is presented, by constructing a collective coordinate for the system. Most of the characters of the interaction are derived analytically while they are calculated by other models numerically. We will find that the behaviour of the soliton is like a point particle ’living’ under the influence of a complicated potential which is a function of soliton velocity and the potential parameters. The analytic model does not have a clear prediction for the islands of initial velocities in which the soliton may reflect back or escape over the potential well.

Contrary to the common understanding, the sine-Gordon equation in (1 + 2) dimensions does have N-soliton solutions for any N. The Hirota algorithm allows for the construction of static N-soliton solutions (i.e., solutions that do not depend on time) of that equation for any N. Lorentz transforming the static solutions yields N-soliton solutions in any moving frame. They are scalar functions under Lorentz transformations. In an N-soliton solution in a moving frame, (N-2) of the (1 + 2)-dimensional momentum vectors of the solitons are linear combinations of the two remaining vectors. C 2013 American Institute of Physics. [http://dx.

Physics Letters A, 1995

We present a method to obtain soliton solutions to relativistic system of coupled scalar fields. This is done by examining the energy associated to static field configurations. In this case we derive a set of first-order differential equations that solve the equations of motion when the energy saturates its lower bound. To illustrate the general results, we investigate some systems described by polynomial interactions in the coupled fields.

Physical Review E, 2014

A collective coordinate theory is developed for soliton ratchets in the damped discrete sine-Gordon model driven by a biharmonic force. An ansatz with two collective coordinates, namely the center and the width of the soliton, is assumed as an approximated solution of the discrete nonlinear equation. The dynamical equations of these two collective coordinates, obtained by means of the generalized travelling wave method, explain the mechanism underlying the soliton ratchet and capture qualitatively all the main features of this phenomenon. The numerical simulation of these equations accounts for the existence of a nonzero depinning threshold, the nonsinusoidal behavior of the average velocity as a function of the relative phase between the harmonics of the driver, the nonmonotonic dependence of the average velocity on the damping, and the existence of nontransporting regimes beyond the depinning threshold. In particular, it provides a good description of the intriguing and complex pattern of subspaces corresponding to different dynamical regimes in parameter space.

Physics Letters A, 1991

The N Z-soliton solution of the Davey-Stewartson equation is considered. It is shown that the boundaries fix the kinematics of solitons, while the dynamics of their mutual interaction is determined by the chosen initial condition. The solution can simulate quantum effects as inelastic scattering fusion and fission, creation and annihilation.

Physical Review E, 2011

The Hirota algorithm for solving several integrable nonlinear evolution equations is suggestive of a simple quantized representation of these equations and their soliton solutions over a Fock space of bosons or of fermions. The classical nonlinear wave equation becomes a nonlinear equation for an operator. The solution of this equation is constructed through the operator analog of the Hirota transformation. The classical N-solitons solution is the expectation value of the solution operator in an N-particle state in the Fock space.