Instability-induced dynamics of dark solitons

Abstract

Nonlinear theory describing the instability-induced dynamics of dark solitons in the generalized nonlinear Schrödinger equation is presented. Equations for the evolution of an unstable dark soliton, including its transformation into a stable soliton, are derived using a multiscale asymptotic technique valid near the soliton instability threshold. Results of the asymptotic theory are applied to analyze dark solitons in physically important models of optical nonlinearities, including competing, saturable, and transiting nonlinearities. It is shown that in all these models dark solitons may become unstable, and two general ͑bounded and unbounded͒ scenarios of the instability development are investigated analytically. Results of direct numerical simulations of the generalized nonlinear Schrödinger equation are also presented, which confirm predictions of the analytical approach and display main features of the instability-induced dynamics of dark solitons beyond the applicability limits of the multiscale asymptotic theory. ͓S1063-651X͑96͒09707-3͔

Traveling solitons in the parametrically driven nonlinear Schrödinger equation

“…This criterion is known in the context of dark solitons of the undriven nonlinear Schrödinger equations; see [10], [11]. Here we simply adapt the proof given in [10] to the case of the equation with the parametric forcing.…”

“…As far as solitons in Hamiltonian systems are concerned, the example of dark solitons in the nonlinear Schrödinger equations suggests that they have even greater mobility than dissipative fronts and pulses. Although in this case the galilean invariance is broken by the presence of the nonvanishing background, the dark solitons can propagate with arbitrary speeds bounded only by the velocity of sound waves [9][10][11].…”

Dark soliton dynamics in spatially inhomogeneous media: Application to Bose–Einstein condensates

“…Careful consideration demonstrates that the soliton escapes from the trap with some acceleration [this may be seen in the inset of Fig. 6 (left)], thus continuously losing its energy [37] (see also discussion in Ref. [14]).…”

Refined empirical stability criterion for nonlinear Schrödinger solitons under spatiotemporal forcing

“…In the case of dark solitons of the form u(x,t) = u(x − V t), with |u| 2 → ρ 0 as |x| → ∞, a similar criterion [25][26][27][28] involves the (renormalized) field momentum…”