Semiclassical dynamics and coherent soliton condensates in self-focusing nonlinear media with periodic initial conditions

The semiclassical (small dispersion) limit of the focusing nonlinear Schrödinger equation with periodic initial conditions (ICs) is studied analytically and numerically. First, through a comprehensive set of numerical simulations, it is demonstrated that solutions arising from a certain class of ICs, referred to as “periodic single-lobe” potentials, share the same qualitative features, which also coincide with those of solutions arising from localized ICs. The spectrum of the associated scattering problem in each of these cases is then numerically computed, and it is shown that such spectrum is confined to the real and imaginary axes of the spectral variable in the semiclassical limit. This implies that all nonlinear excitations emerging from the input have zero velocity, and form a coherent nonlinear condensate. Finally, by employing a formal Wentzel-Kramers-Brillouin expansion for the scattering eigenfunctions, asymptotic expressions for the number and location of the bands and gaps in the spectrum are obtained, as well as corresponding expressions for the relative band widths and the number of “effective solitons.” These results are shown to be in excellent agreement with those from direct numerical computation of the eigenfunctions. In particular, a scaling law is obtained showing that the number of effective solitons is inversely proportional to the small dispersion parameter.