For a fixed external potential field, the evolution of a highly localized finite-amplitude initial pulse
is investigated by numerically solving the corresponding two-dimensional generalized nonlinear Schrödinger
equation i∂tE +p ▽2E+[V (x,y )+q|E|2] E =0 . The evolution depends crucially on the complex group dispersion
coefficient p and the complex nonlinearity coefficient q of the equation. It is found that, in general, the wave field
first suffers modulational instability, followed by abrupt collapse into a turbulent state containing the shortest
wavelength modes allowed in the system. The latter is in turn followed by inverse cascade of the shortest
wavelength modes back to the longer wavelength ones, until a statistically stationary turbulent state is reached.
For p = 3.5 +0.5i and q = 8.0+ 0.9i , it is found that the energy is mainly concentrated in the wave vector region |k|≥100 of the energy spectrum, i.e., the inverse cascade is limited to the shorter wavelength modes with |k|≥100 . Furthermore, for the imaginary part pi (which represents viscous damping) of the group dispersion
coefficient in the range 0.1 < pi <1.0, it is found that the region of inverse cascade gradually shrinks with
increase of p i( <1.1) . That is, in the range considered, the viscous damping coefficient p i acts like a control
switch, and it can regulate the degree of inverse cascade and the final results.
