Neck instability of bright solitary waves in hyperbolic Kerr media

Symmetry-breaking instability of solitons has been studied in different areas of physics and appears as a common feature shared by most solitons despite the diversity of the physical systems that support them [1–3]. In optics, the break-up of self guided light beams through transverse and/or modulational instability has been demonstrated for solitons in self-focussing and defocusing Kerr media [1,4] as well as in second harmonic generation media [5]. In particular, for the case of focusing Kerr nonlinearity, the dynamics is governed by the nonlinear Schrödinger (NLS) equation that possesses solutions in the form of bright (1+1)D spatial solitons. When propagating in a (2+1)D media, these solitons are known to be unstable against periodic perturbations in the extra transverse dimension in which they are uniform (regardless of the relative sign between the dispersion (diffraction) terms, i.e. in both elliptic and hyperbolic systems) [6,7]. In hyperbolic systems, such as normally dispersive planar waveguides, the transverse instability in (2+1)D leads to a spontaneous breaking of spatial bright soliton stripe into a spatiotemporal snake like pattern. However, beside the well known snake instability branch in the cubic soliton spectrum, another instability branch associated with a neck type instability has recently been identified theoretically [8] and experimentally [9]. Neck instability of spatially localized waves has also been reported in other media with normal dispersion, such as bulk media [10], coupled nonlinear waveguides [11] or quadratic media [12]. However, this neck instability is counterintuitive since it is well known that continuous waves are stable when propagating in a (1+1)D normally dispersive focusing Kerr medium. This is the reason why we propose in the present Letter a detailed theoretical and experimental analysis of the neck instability of solitons of the (2+1)D hyperbolic NLS equation. As we shall see, this instability shares some properties with light filament dynamics that occur in spatially localized beams propagating in normally dispersive bulk media [13]. Let us consider the hyperbolic nonlinear Schrödinger equation with normal time dispersion: i ∂ψ ∂Z + 1 2 ∂2ψ ∂X2 − 1 2 ∂2ψ ∂T 2 + |ψ|ψ = 0. (1)