周期初始条件下自聚焦非线性介质中的半经典动力学和相干孤子凝聚
Semiclassical dynamics and coherent soliton condensates in self-focusing nonlinear media with periodic initial conditions
摘要：
对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进行了解析和数值研究。首先，通过一套全面的数值模拟，它表明，解决方案所产生的某一类初始条件，被称为“周期性的单瓣”的潜力，共享相同的定性特征，这也与本地化的初始条件所产生的解决方案相吻合。在这些情况下，相关的散射问题的频谱，然后数值计算，它表明，这样的频谱被限制在半经典极限的频谱变量的实轴和虚轴。这意味着从输入中出现的所有非线性激励具有零速度，并且形成相干非线性凝聚。最后，通过对散射本征函数进行Wentzel-Kramers-Brillouin展开，得到了谱带和带隙的数目和位置的渐近表达式，以及相应的相对带宽和“有效孤子”数目的表达式.这些结果与直接数值计算本征函数的结果非常吻合。特别地，得到了一个标度律，表明有效孤子的数目与小色散参数成反比。
英文摘要：
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.
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