Wave instabilities in Nonlinear Schrödinger Systems with nonvanishing background

K Katterbauer,S Trillo,A Fratalocchi

doi:10.1364/cleo_at.2013.jw2a.16

Abstract

The generalized Nonlinear Schrodinger Equation (GNLSE): i∂ϕ/∂ t+1/2 Δ ϕ + F(| ϕ|2) ϕ = 0 is a fundamental equation for the universal propagation of dispersive and nonlinear waves [1-4]. In the presence of high order nonlinear responses, these equations exhibit instabilities that lead to wave collapse [1, 4]. The study of collapse has stirred significant interest in scientific community, especially in Optics, as it lead to the localization and trapping of energy in small spatial scales [4]. To date, most efforts have been directed to the study of localized pulses with vanishing boundary conditions, where collapse is demonstrated to occur when the field Hamiltonian is negative [4], while practically nothing is known in the presence of a nonzero background. The latter is a particularly important in Optics, due to the large interest stirred by the study of nonlinear waves with nonzero background, such as e.g., Dark/Gray solitons [1,3-4].

Highlights

We investigate wave collapse ruled by the generalized nonlinear Schrodinger (NLS) equation in 111 dimensions, for localized excitations with non-zero background, establishing through virial identities a new criterion for blow-up
Many physical classical and quantum systems need to be described in terms of generalized NLS equation which accounts for higher-order nonlinearities
Through virial identities and semiclassical analysis, we have demonstrated a sufficient criterion for collapse and unveiled a novel instability dynamics characterized by the emission of dispersive shocks

Summary

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Wave instabilities in the presence of non vanishing background in nonlinear Schrodinger systems. We investigate wave collapse ruled by the generalized nonlinear Schrodinger (NLS) equation in 111 dimensions, for localized excitations with non-zero background, establishing through virial identities a new criterion for blow-up. The general findings are illustrated with a model of interest to both classical and quantum physics (cubic-quintic NLS equation), demonstrating a radically novel scenario of instability, where solitons identify a marginal condition between blow-up and occurrence of shock waves, triggered by arbitrarily small mass perturbations of different sign. Our aim in this paper is to establish a novel criterion for collapse valid for solutions of this type, and study the dynamics across the threshold for blow-up This allows us to reveal a new instability scenario where opposite behaviors, such as blowup or decay into a dispersive shock wave, can be controlled by means of an arbitrarily weak perturbation which controls the variation of the power (or mass) integral of a launched perturbed solitary wavepacket. We start from the following gNLS equation in dimensionless units i

Ly Lt z

LyÃ Lx

LV Lt

Lu Ly z

Reduced energy functional

Conclusions

Additional information