The small dispersion limit of the focusing nonlinear Schrodinger equation with periodic initial conditions is studied analytically and numerically. First, through a comprehensive set of numerical simulations, it is demonstrated that solutions arising from a certain class of initial conditions, referred to as "periodic single-lobe" potentials, share the same qualitative features, which also coincide with those of solutions arising from localized initial conditions. 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.

中文翻译：

---

具有周期性初始条件的自聚焦非线性介质中的半经典动力学和相干孤子凝聚

分析和数值研究了具有周期性初始条件的聚焦非线性薛定谔方程的小色散极限。首先，通过一组全面的数值模拟，证明了由某一类初始条件产生的解，称为“周期单瓣”势，具有相同的定性特征，这也与由局部初始条件。然后对每种情况下相关散射问题的频谱进行数值计算，结果表明这种频谱仅限于半经典极限中频谱变量的实轴和虚轴。这意味着来自输入的所有非线性激励都具有零速度，并形成相干非线性凝聚。最后，通过对散射特征函数采用正式的 Wentzel-Kramers-Brillouin 展开式，得到了光谱中能带和间隙的数量和位置的渐近表达式，以及相对带宽和“有效孤子数”的相应表达式”。这些结果与本征函数的直接数值计算结果非常吻合。特别是，获得了一个标度定律，表明有效孤子的数量与小色散参数成反比。