A bound state attractor in optical turbulence

Abstract

We study numerically the nonintegrable dynamics of coherent, solitonic, nonlinear waves, in a spatially nonlocal nonlinear Schrödinger equation relevant to realistic modelling of optical systems: the Schrödinger–Helmholtz equation. We observe a single oscillating, coherent solitary wave emerging from a variety of initial conditions. Using the direct scattering transform of the (integrable) cubic nonlinear Schrödinger equation, we find that this structure is a bound state, comprising of a primary and secondary soliton whose amplitudes oscillate in phase opposition. We interpret this as the solitons periodically exchanging mass. We also observe bound states comprising of three oscillating solitons, hinting at the existence of a family of multi-soliton bound states. Focusing on the two-soliton bound state, we observe it self-organising from an initial state of incoherent turbulence, and from solitonic structures launched into the system. When a single (primary) solitonic structure is launched, a resonance process between it and waves in the system generates the secondary soliton, resulting in the bound state. Further, when two solitons are initially launched, we show that they can merge if their phases are synchronised when they collide. When the system is launched from a turbulent state comprised of many initial solitons, we propose that the bound state formation is preceded by a sequence of binary collisions, in which the mass is transferred on average from the weak soliton to the strong one, with occasional soliton mergers. Both processes lead to increasingly stronger and fewer dominant solitons. The final state – a solitary bound state surrounded by weakly nonlinear waves – is robust and ubiquitous. We propose that for nonlocal media, a bound state comprising of at least two solitons is a more typical statistical attractor than the single-soliton attractor suggested in previous literature.