Abstract:
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.
