Quasi one dimensional anomalous (rogue) waves in multidimensional nonlinear Schrödinger equations 1: fission and fusion

In this paper we study the first nonlinear stage of modulation instability (NLSMI) of $x$-periodic AWs in multidimensional generalizations of the focusing nonlinear Schrödinger (NLS) equation, like the non-integrable elliptic and hyperbolic NLS equations in $2+1$ and $3+1$ dimensions. In the quasi one-dimensional (Q1D) regime, where the wavelength in the $x$ direction of propagation is significantly smaller than in the transversal directions, the behavior is universal, independent of the particular model at leading order, and described by adiabatic deformations of the Akhmediev breather solution of NLS. Varying the initial data, the first NLSMI shows various combinations of basic processes like AW growth from the unstable background, followed by fission in the slowly varying transversal directions, and the inverse process of fusion, followed by AW decay to the background. Fission and fusion are critical processes showing similarities with multidimensional wave breaking, and with phase transitions of second kind and critical exponent $1/2$. In $3+1$ dimensions with radial symmetry in the transversal slowly varying plane, fission consists in the formation of an opening smoke ring centered on the $x$ axis. In the long wave limit, the Q1D Akhmediev breather reduces to the Q1D analogue of the Peregrine instanton, rationally localized in space. Numerical experiments on the hyperbolic NLS equation show that the process of "AW growth + fission" is not restricted to the Q1D regime, extending to a finite area of the modulation instability domain. The universality of these processes suggests their observability in natural phenomena related to AWs in contexts such as water waves, nonlinear optics, and plasma physics.