Semiclassical dynamics and coherent soliton condensates in self‐focusing nonlinear media with periodic initial conditions
Semiclassical dynamics and coherent soliton condensates in self‐focusing nonlinear media with periodic initial conditions
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DOI：
10.1111/sapm.12321
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发表时间：
2020-05
影响因子：
2.7
通讯作者：
G. Biondini;Jeffrey Oregero
中科院分区：
文献类型：
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作者：
G. Biondini;Jeffrey Oregero
用解析和数值方法研究了具有周期初始条件（IC）的聚焦非线性薛定谔方程的半经典（小色散）极限。首先，通过一组全面的数值模拟，它表明，解决方案所产生的某一类IC，被称为“周期性的单瓣”的潜力，共享相同的定性特征，这也符合那些解决方案所产生的本地化IC。在这些情况下，相关的散射问题的频谱，然后进行数值计算，它表明，这样的频谱被限制在半经典极限的频谱变量的真实的和虚轴。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采用形式的Wentzel-Kramers-Brillouin展开，得到了光谱中带隙的数量和位置的渐近表达式，以及相应的相对带宽和“有效孤子”数量的表达式。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。
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.