x-periodic Quasi One Dimensional Anomalous (Rogue) Waves in Multidimensional Nonlinear Schr\"odinger Equations: Fission, Fusion, and Recurrence

Abstract

In a recent work we studied the first nonlinear stage of modulation instability (NLSMI) of x-periodic anomalous (rogue, freak, extreme) waves (AWs) of physically relevant multidimensional (generalizations of the focusing) nonlinear Schr\"odinger (MNLS) equation, like the non integrable elliptic and hyperbolic nonlinear Schr\"odinger (NLS) equations in d+1 dimensions, d = 2, 3, in the quasi one dimensional (Q1D) regime in which the wavelength in the direction of propagation x is small with respect to the wavelengths in the transversal directions. We showed that, at leading order, the first NLSMI is universal, independent of the particular MNLS model, and described by suitable adiabatic deformations of the quasi-homoclinic Akhmediev breather solution of NLS, in excellent agreement with numerical simulations. In the present work we focus on the recurrence of x-periodic AWs in the Q1D regime. We show that, although the first nonlinear stage of MI is essentially universal for all MNLS equations, the recurrence dynamics exhibit significant O(1) differences among different models. Moreover, successive nonlinear stages generally display increasingly complex combinations of fission and fusion processes, leading to progressively richer dynamical choreographies. Since MNLS equations in the Q1D regime can be viewed as multidimensional perturbations of the integrable NLS equation, we use the recently developed finite gap perturbation theory of NLS AWs to give an analytic and quantitative description of the recurrence of Q1D AWs, in excellent agreement with numerical simulations. Due to the physical relevance of the MNLS equations considered in this work, and due to the universality of the processes discussed in this paper, it is plausible that they be observable in many fields of physics, like water waves, nonlinear optics, plasma physics, Bose-Einstein condensates, etc . . .