Soliton synchronization with randomness: rogue waves and universality

We consider an N-soliton solution of the focusing nonlinear Schr & ouml;dinger equations. We give conditions for the synchronous collision of these N solitons. When the solitons velocities are well separated and the solitons have equal amplitude, we show that the local wave profile at the collision point scales as the sinc(x) function. We show that this behaviour persists when the amplitudes of the solitons are i.i.d. sub-exponential random variables. Namely the central collision peak exhibits universality: its spatial profile converges to the sinc(x) function, independently of the distribution. We derive Central Limit Theorems for the fluctuations of the profile in the near-field regime (near the collision point) and in the far-field regime.