Forced nonlinear Schrödinger equation with arbitrary nonlinearity

Abstract

We consider the nonlinear Schrödinger equation (NLSE) in 1 + 1 dimension with scalar-scalar self-interaction g 2 κ+1 (ψ ψ) κ+1 in the presence of the external forcing terms of the form re −i(kx+θ) − δψ. We find new exact solutions for this problem and show that the solitary wave momentum is conserved in a moving frame where v k = 2k. These new exact solutions reduce to the constant phase solutions of the unforced problem when r → 0. In particular we study the behavior of solitary wave solutions in the presence of these external forces in a variational approximation which allows the position, momentum, width, and phase of these waves to vary in time. We show that the stationary solutions of the variational equations include a solution close to the exact one and we study small oscillations around all the stationary solutions. We postulate that the dynamical condition for instability is that dp(t)/dq(t) < 0, where p(t) is the normalized canonical momentum p(t) = 1 M(t)∂L ∂q , anḋ q(t) is the solitary wave velocity. Here M(t) = dxψ (x,t)ψ(x,t). Stability is also studied using a "phase portrait" of the soliton, where its dynamics is represented by two-dimensional projections of its trajectory in the four-dimensional space of collective coordinates. The criterion for stability of a soliton is that its trajectory is a closed single curve with a positive sense of rotation around a fixed point. We investigate the accuracy of our variational approximation and these criteria using numerical simulations of the NLSE. We find that our criteria work quite well when the magnitude of the forcing term is small compared to the amplitude of the unforced solitary wave. In this regime the variational approximation captures quite well the behavior of the solitary wave.

Soliton stability criterion for generalized nonlinear Schrödinger equations

“…The solutions describe oscillating solitons, and the criterion Eq. (3) makes predictions for which IC unstable solitons are expected, which is confirmed by simulations [13].…”

“…(1), an NLSE with the nonlinearity g|u| 2κ u was investigated [13], with positive coupling constant g, arbitrary positive nonlinearity exponent κ, and the same driving as in case (i). For the unperturbed case (R ≡ 0), there exist exact one-soliton solutions [13] that were used as the ansatz for a CC theory. The resulting CC equations were solved analytically in the cases of stationary solutions and small oscillations about them.…”

Nonlinear Schrödinger solitons oscillate under a constant external force

“…This criterion has a wide validity. It also holds for the NLSE with the nonlinearity [21] g(ψ * ψ) κ ψ with κ = 1/2, and it also holds if K is not constant but a harmonic or biharmonic function of time [18]; here κ = 1.…”

“…Here, F = −j + ru + ru * , where j (y,t) is the current density of the unperturbed NLS Eq. (3) in the moving frame [21].…”

“…Two questions have not been clarified in the above work [17,18,21]: (1) Why do the solitons oscillate, although the driving force is time independent? (2) In the simulations, the spectrum of the oscillations in most cases exhibits two strong peaks, but only one of them is located near the frequency ω cc s which is predicted by the CC theory.…”