Talanov dynamics and chirp-controlled rogue-wave formation for the vector nonlinear Schrödinger equation

Abstract

We study the focusing vector nonlinear Schrödinger equation, or Manakov system, for a pair of coupled initial conditions with compactly supported, concave parabolic density profiles and phases depending quadratically on the spatial coordinate. We focus on the natural class of self-similar solutions generated by such profiles, whose dispersionless evolution we refer to as Talanov dynamics. We extend the criterion for blow-up in finite time, first established for a single field, to the Manakov case. For arbitrary initial chirps in the two fields, we provide closed-form expressions for their evolution, and an explicit expression for the time of the gradient catastrophe, if it occurs, and we classify the effect of the coupling on the catastrophe. Depending on the initial chirps, the two fields may catastrophise, or relax, as each would if decoupled. However, coupling can also force the catastrophe in fields that would otherwise relax if decoupled. For identical initial supports, and in the presence of dispersion, we numerically show that the same dispersionless criterion determines, in the semiclassical regime, whether the fully dispersive solution relaxes or develops a rogue wave in each field, whose onset time is accurately predicted by the corresponding dispersionless catastrophe time, in direct analogy with the scalar case. As chirp and amplitude are relatively easy to control experimentally, we expect these results to be relevant to experiments in nonlinear optics and Bose-Einstein condensates, extending to coupled systems observations made for the scalar equation.