Quasi integrable and single appearance anomalous waves in multidimensions; a preliminary phenomenological perspective (joint work with F. Coppini)

Modulation instability (MI) and nonlinearity are considered as the main causes for the appearance of anomalous (rogue) waves (AWs) in several physi- cal contexts. In $1+1$ dimensions, like in optical fibers, AWs are well described by the integrable nonlinear Schrödinger (NLS) equation and, in the periodic setting, the algebro-geometrical tools of the finite gap method play a key role in the theory. In $n + 1,\, n \ge 2$ dimensions, like in the ocean and in the nonlinear optics of crystals, the large majority of physically relevant NLS type models are non integrable, and it is not clear yet if the NLS AWs can be really observed. In this talk we consider integrable and non integrable multidimensional NLS type models, and we give a preliminary phenomenological look at quasi integrable and single appearance AW dynamics. The NLS segment of MI becomes an open bounded or unbounded domain in Fourier space, and we explore the rich phenomenology present in different regions of this domain: the quasi 1D region, the region in which AWs undergo fission and fusion, the no fission region, the long wave (Peregrine type) corner, the X wave region, and the asymptotic domain.