Soliton stabilization on top of a hill: a nonlinear-wave inverted pendulum

Abstract

We construct a nonlinear-wave analogue of the celebrated inverted, Kapitza, pendulum. We consider bright soliton solutions of the one-dimensional nonlinear Schrödinger (NLS) equation under the influence of a repelling, when stationary, hump (or ‘hill’) potential and a confining harmonic trap. Using the variational approximation methodology, we reduce the soliton dynamics to an effective Newton-type model on the soliton position and study its behaviour when the hump potential is made to vibrate vertically. We corroborate that the effective model is able to qualitatively and quantitatively describe the statics, stability and dynamical properties of the original NLS bright soliton. Using the effective model, we find the stability regions as a function of the hump parameters (height and vertical shaking frequency), which correspond to the stability tongues of the Mathieu equation. We show that these theoretically predicted regions accurately describe the corresponding, numerically computed, Floquet stability regions of the original soliton on top of the vertically vibrating hill. Inside these stability regions, we are able to fully stabilize the soliton on top of the hill as a result of the vertical shaking in a manner akin to the inverted, Kapitza, pendulum.