Modeling the Dynamics of Soliton Collisions

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, 2008

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.

Progress of Theoretical Physics, 1987

The aim of this paper was the numerical computing of the first conservation law present in the interactions between solitons generated from numerical solutions of the Korteweg de Vries (KdV) equation. As follows from the theory of partial differential equations in which appears solitons, like some solutions of the KdV, many interesting behaviors come associated with them, in particular the conservation laws derived from it. Only the first three ones have direct interpretation or are associated with physical quantities, such as conservation of energy, momentum or mass density. The remaining are conservation laws in the strict sense of physics but do not have a clearly associated with magnitudes that usually appear in physical processes. The KdV equation constitutes one of the called "integrable systems" and in this way the conservations laws derived from it, have special interest for applications on the study of solitons's propagation. These non-linear traveling waves that can be found from tsunamis in the oceans, transmission of information by transoceanic fiber optic communications, transmitted information through neural microtubules in living beings have a relevant technological and scientific interest. So the knowledge of conservation laws in the interaction between solitons becomes very interesting from both point of view technological and for the study of certain process in living beings. To obtain the explicit form of the conservation laws derived from the KdV equation, mainly due to the large number of terms that arise in the mathematical expression of them, there were developed a computational code implemented in a symbolic calculus platform that facilitated their algebraic handling. The conservation laws thus obtained were numerically evaluated during the computing integration of the KdV equation, based on an spectral type scheme. Their solutions represent the interaction of two solitons. Among the obtained results it can be pointed out that if the generation of solitons by solving the KdV equation has a high degree of approximation, the integrals of motion does not require highly sophisticated numerical algorithms to be satisfied and also make a numerical validation of the integration schemes of the nonlinear equation under study.

Physics Letters A, 2009

We study the dynamics of bright and dark matter-wave solitons in the presence of a spatially varying nonlinearity. When the spatial variation does not involve zero crossings, a transformation is used to bring the problem to a standard nonlinear Schrödinger form, but with two additional terms: an effective potential one and a non-potential term. We illustrate how to apply perturbation theory of dark and bright solitons to the transformed equations. We develop the general case, but primarily focus on the non-standard special case whereby the potential term vanishes, for an inverse square spatial dependence of the nonlinearity. In both cases of repulsive and attractive interactions, appropriate versions of the soliton perturbation theory are shown to accurately describe the soliton dynamics.

Low Temperature Physics

This article offers an introduction to the vast area of experimental and theoretical studies of solitons. It is composed of two large parts. The first one provides a review of effectively one-dimensional (1D) settings. The body of theoretical and experimental results accumulated for 1D solitons is really large, the most essential among them being overviewed here. The second part of the article provides a transition to the realm of multidimensional solitons. These main parts are split into a number of sections, which clearly define particular settings and problems addressed by them. This article may be used by those who are interested in a reasonably short, but, nevertheless, sufficiently detailed introduction to the modern “soliton science”. It addresses, first, well-known “traditional” topics. In particular, these are the integrable Korteweg–de Vries, sine-Gordon, and nonlinear Schrödinger (NLS) equations in 1D, as well as the Kadomtsev–Petviashvili equations in 2D, and basic physi...

Nature Physics, 2014

Solitons are localised wave disturbances that propagate without changing shape, a result of a nonlinear interaction which compensates for wave packet dispersion. Individual solitons may collide, but a defining feature is that they pass through one another and emerge from the collision unaltered in shape, amplitude, or velocity. This remarkable property is mathematically a consequence of the underlying integrability of the one-dimensional (1D) equations, such as the nonlinear Schrödinger equation, that describe solitons in a variety of wave contexts, including matter-waves 1, 2 . Here we explore the nature of soliton collisions using Bose-Einstein condensates of atoms with attractive interactions confined to a quasione-dimensional waveguide. We show by real-time imaging that a collision between solitons is a complex event that differs markedly depending on the relative phase between the solitons. Yet, they emerge from the collision unaltered in shape or amplitude, but with a new trajectory reflecting a discontinuous jump. By controlling the strength of the nonlinearity we shed new light on these fundamental features of soliton collisional dynamics, and explore the 1 arXiv:1407.5087v1 [cond-mat.quant-gas] 18 Jul 2014 implications of collisions that bring the wave packets out of the realm of integrability, where they may undergo catastrophic collapse.

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.

2014

The cross section for scattering of x-rays by solitons is calculated. The authors consider solitons corresponding to the formation of a kink in a system of adatoms on the surface of a substrate, or of a crowdion in a chain of atoms in a crystal that are described by the sine-Gordon equation. It is shown that investigation of the x-ray scattering makes it possible to obtain information about the static and dynamic properties of the solitons.

Physical Review E, 2014

We study n-pulse interaction in fast collisions of N solitons of the cubic nonlinear Schrödinger (NLS) equation in the presence of generic weak nonlinear loss. We develop a reduced model that yields the contribution of n-pulse interaction to the amplitude shift for collisions in the presence of weak (2m + 1)-order loss, for any n and m. We first employ the reduced model and numerical solution of the perturbed NLS equation to analyze soliton collisions in the presence of septic loss (m = 3). Our calculations show that three-pulse interaction gives the dominant contribution to the collision-induced amplitude shift already in a full-overlap four-soliton collision, and that the amplitude shift strongly depends on the initial soliton positions. We then extend these results for a generic weak nonlinear loss of the form G(|ψ| 2)ψ, where ψ is the physical field and G is a Taylor polynomial of degree m c. Considering m c = 3, as an example, we show that three-pulse interaction gives the dominant contribution to the amplitude shift in a six-soliton collision, despite the presence of low-order loss. Our study quantitatively demonstrates that n-pulse interaction with high n values plays a key role in fast collisions of NLS solitons in the presence of generic nonlinear loss. Moreover, the scalings of n-pulse interaction effects with n and m and the strong dependence on initial soliton positions lead to complex collision dynamics, which is very different from the one observed in fast NLS soliton collisions in the presence of cubic loss.