Soliton–radiation coupling in the parametrically driven, damped nonlinear Schrödinger equation

Abstract

We use the Riemann-Hilbert problem to study the interaction of the soliton with radiation in the parametrically driven, damped nonlinear Schrödinger equation. The analysis is reduced to the study of a finite-dimensional dynamical system for the amplitude and phase of the soliton and the complex amplitude of the long-wavelength radiation. In contrast to previously utilised Inverse Scattering-based perturbation techniques, our approach is valid for arbitrarily large driving strengths and damping coefficients. We show that, contrary to suggestions made in literature, the complexity observed in the soliton's dynamics cannot be accounted for just by its coupling to the long-wavelength radiation.

Soliton pair interaction law in parametrically driven Newtonian fluid

“…Increasing γ more, these localized states become unstable at the Arnold tongue (γ 2 = μ 2 + ν 2 and ν < 0). When one increases the modulus of the detuning-far from the tip of the Arnold tongue-the amplitude of the dissipative soliton exhibits an Andronov-Hopf bifurcation and increasing more the detuning the amplitude of the dissipative soliton exhibits double-period scenarios (Shchesnovich & Barashenkov 2002). Recently, these bifurcations have been verified experimentally (Zhang et al 2007).…”

Localized waves in a parametrically driven magnetic nanowire

“…We can recognize two different zones, outside (OS) and inside (IS); for instance in the left hand OS-zone there appear soliton solutions [21] and in the right hand OS-zone there appear extended waves [24]. Also, in the IS-zone spatio-temporal chaos appears [36] as well as oscillating states and localized waves, as we shall see later.…”

A quasi-periodic route to chaos in a parametrically driven nonlinear medium

“…where µ ≡ µ 0 /2, γ ≡ γ 0 /4, , and ψ is the complex conjugate of ψ. This equation is known to produce stationary, time-periodic, or chaotic solutions, including Faraday waves [50,62], soliton-like modes [47,52], two-soliton [56,57] and soliton-antisoliton [67] bound states, and spatiotemporal chaos [47,58]. 2)), at different values of the strength of the parametric forcing: γ 0 = (0.4, 0.5, 0.6, 0.7).…”

“…A generic model that describes periodically forced systems is based on the parametrically driven damped nonlinear Schrödinger (PDDNLS) equation [46][47][48]. This amplitude equation produces a variety of temporal behaviors, including stationary, periodic, and chaotic regimes, such as Faraday waves [3,[49][50][51], single-soliton [47,[52][53][54][55] and two-soliton [56,57] states, and spatiotemporal chaos [47,58]. Remarkable hydrodynamic modes are excited by the parametric instability in the form of standing (Faraday) waves on the surface of a vertically vibrated Newtonian fluid [3].…”