A.A. Balyakin, N.M. Ryskin have shown a numerical solution of NLS explaining MI regime in their paper of "Nonlinear dynamics of modulation instability in distributed resonance under external harmonic driving"; their numerical results state that a convective MI regime can be considered as a route to chaos; however, they have not clarified in which physical condition, such behavior is attainable; moreover, they have also assumed absolute MI criterion to construct their numerical results; why?
http://link.springer.com/article/10.1007%2Fs11141-007-0064-2#
Convective instability cannot be considered as a route to chaos, since chaos is a kind of self-oscillation which occurs in self-excited oscillatory system. In a medium with convective MI, if we set a small modulation of the input signal, it will grow in space along the system. However, if we switch this modulation off, i.e. if the input signal is a purely monochromatic wave with constant amplitude, we see no modulation at the output. On the contrary, in a media with absolute MI, even if the input signal is purely single-frequency, small internal fluctuations grow in time as well as in space, and finally evolve into a self-modulated oscillation which may be periodic, quasi-periodic, or chaotic (of course, under certain conditions, which usually means that the input signal should be large enough).
A criterion of absolute/convective MI for NLS has been derived in our papers co-authored with Dr. A A Balyakin (Tech. Phys. Lett. 2004) and (Nonlin. Phenom. Complex Syst. 2004). It reads that if the input signal amplitude exceeds a threshold value $A_c$ the MI is absolute, otherwise, if $A
Nonlin. Phenom. Complex Syst. 7, 34 (2004).
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