Download Origin of heavy tails in equations of the Nonliner Schro ̈dinger type

Origin of heavy tails in equations of the Nonliner Schro ̈dinger type:
an exact result
M. Onorato, D. Proment, G. El, S. Randoux, P. Suret,
The determination of the statistical properties of nonlinear
dispersive waves is an important issue in many physical systems.
Several research activities have been conducted in order to understand
the formation of rogue waves in systems described by envelope
equations, such as the Nonliner Schrodinger. Here, we derive an
identity that relates the evolution of the kurtosis (a measure of the
relevance of the tails in a probability density function) of the wave
amplitude to the rate of change of the width of Fourier spectrum of
the wave field. The result is exact for all dispersive systems
characterized by a nonlinear term of the form of the one contained in
the Nonliner Schro ̈dinger equa tion. Numerical simulations are also
performed to confirm our findings. Our work sheds some light on the
origin of rogue waves in dispersive nonlinear media ruled by local
cubic nonlinearity.