Dynamic behaviour of a non-propagating soliton under a periodically modulated oscillation

Abstract

It has been found theoretically and experimentally that a non-propagating soliton in a small rectangular water tank manifests dynamic behaviour when subjected to a modulated oscillation. A modification of the cubic Schrödinger equation was generalized for this case and analysed by the inverse-scattering perturbation method. The problem was reduced to a lower-dimensional one, i.e. to a pair of first-order ordinary differential equations for the amplitude and phase of the soliton, which were solved numerically. It was found that the soliton executes multi-periodic and chaotic motions under the periodically modulated oscillation. Corresponding experiments were carried out and both qualitative and quantitative agreement was obtained for the phenomena predicted and the parameter ranges in which they occur.

Traveling solitons in the parametrically driven nonlinear Schrödinger equation

“…(1), and our condition V < c is essentially an exclusion principle ruling out a resonance between solitons and linear waves. In the undamped case the parametrically driven NLS equation (2) conserves the momentum,…”

“…In this section we will produce several more explicit solutions of the undamped, parametrically driven NLS equation (2). Writing ψ = u + iv, the stationary equation (3) transforms into the system…”

“…A number of nonstationary regimes were reported in the water tank experiments, including the formation of oscillating soliton pairs [2], but no steadily moving solitons were detected so far. On the other hand, numerical simulations of the undamped parametrically driven NLS equation [Eq.…”

“…was used to model the nonlinear Faraday resonance in a vertically oscillating water tank [1,2] and the effect of phase-sensitive parametric amplifiers on solitons in optical fibers [3]. The same equation describes an easy-plane ferromagnet with a combination of a static and hf field in the easy plane [4,5], and the planar weakly anisotropic XY model [6].…”

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

“…This equation describes nonlinear Faraday resonance in a vertically oscillating water trough [1,2,3,4,5,6,7,8,9,10,11,12]; an easy-plane ferromagnet with a combination of a stationary and a high-frequency magnetic field in the easy plane [13]; and the effect of phase-sensitive amplifiers on solitons propagating in optical fibres [14,15,16]. The equation (1)- (2) has two stationary soliton solutions, one of which is unstable for all h and γ and hence usually disregarded [13].…”