Periodic anomalous (rogue) waves in the Ablowitz- Ladik lattice and in the 2 + 1 dimensional Davey-Stewartson 2 equation

Modulation instability and nonlinearity are the main causes of the appearance of anomalous (rogue) waves (AWs) in several physical contexts. The theory of periodic anomalous waves has been recently developed on the basic Nonlinear Schrödinger (NLS) model in $1+1$ dimensions, adapting the finite gap method to the Cauchy problem for periodic initial perturbations of the homogeneous background solution of NLS [1]. This theory allows one to express the solution of the Cauchy problem, to leading order, in terms of elementary functions of the unstable part of the initial data, and has already been tested in the nonlinear optics of a photorefractive crystal [2]. Also a perturbation theory of AWs allowing one to study the leading order effects of small perturbations of the NLS equation on the dynamics of AWs has been constructed [3]. In this lecture we show how this theory extends to lattices, using as basic model the Ablowitz-Ladik lattice [4,5], and to multidimensions, using as basic model the $2+1$ dimensional Davey - Stewartson 2 equation [6],[7]. Joint works with P. G. Grinevich and F. Coppini.