Strong NLS soliton–defect interactions

Abstract

We consider the interaction of a nonlinear Schrödinger soliton with a spatially localized (point) defect in the medium through which it travels. Using numerical simulations, we find parameter regimes under which the soliton may be reflected, transmitted, or captured by the defect. We propose a mechanism of resonant energy transfer to a nonlinear standing wave mode supported by the defect. Extending Forinash et. al. [1], we then derive a finite-dimensional model for the interaction of the soliton with the defect via a collective coordinates method. The resulting system is a three degree-of-freedom Hamiltonian with an additional conserved quantity. We study this system both numerically and using the tools of dynamical systems theory, and find that it exhibits a variety of interesting behaviors, largely determined by the structures of stable and unstable manifolds of special classes of periodic orbits. We use this geometrical understanding to interpret the simulations of the finitedimensional model, compare them with the nonlinear Schrödinger simulations, and comment on differences due to the finite-dimensional ansatz.

Statics and dynamics of atomic dark-bright solitons in the presence of impurities

“…[23][24][25][26]30,31]). Furthermore, it would be interesting to study similar problems but for impurities that have spatial scales larger than the ones of the soliton, and investigate possible changes in the stability and dynamics.…”

“…In this context, the interaction of either bright or dark solitons with δ-like impurities has been investigated in the framework of the nonlinear Schrödinger (NLS) equation (see, e.g., Refs. [23][24][25][26]), while relevant studies have also appeared in the physics of atomic BECs (see, e.g., [27][28][29][30][31]). In the latter context, localized impurities may easily be created as sharply focused far-detuned laser beams and can be used to manipulate matter-wave dynamics (see, e.g., Chap.…”

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

“…This solution is even and is a ground state (see, for example, [4,7,29] for such results). For γ = 0, an explicit solution of (3) was presented in [13,22] and the following was proved in [12,13]. …”

“…(2) was studied previously by several authors. In [6,22,[25][26][27]40,41], the phenomenon of soliton scattering by the effect of the defect was observed, namely, interactions between the defect and the homogeneous medium soliton. For example, varying amplitude and velocity of the soliton, they studied how the defect is separating the soliton into two parts: one part is transmitted past the defect, the other one is captured at the defect.…”

Scattering of a flat top solitons of cubic - quintic nonlinear Shrödinger equation by a linear delta potential

“…Also the controlled management of soliton dynamics by application of external fields become a fascinating field of research [4]. One of the approaches to manage a solitons is a scattering on a localized potential barriers and wells [6,7,8,9]. It was revealed that solitons of one dimensional NLSE behave like classical particles with internal structure in the process of scattering on the low energy barriers, namely they could be reflected or transmitted depending on the initial velocity of soliton, and also the shape oscillations can be excited.…”