论文标题
半经典动力学和连贯的孤子在自我关注的非线性培养基中凝结具有周期性初始条件
Semiclassical dynamics and coherent soliton condensates in self-focusing nonlinear media with periodic initial conditions
论文作者
论文摘要
在分析和数值上研究了焦点非线性schrodinger方程的较小分散极限。首先，通过一组全面的数值模拟，证明了由特定类别的初始条件（称为“周期性的单杆”电势所引起的解决方案，共享相同的定性特征，这也与局部初始条件产生的解决方案相吻合。然后，在每种情况下，相关散射问题的光谱都进行数值计算，并显示出这种光谱仅限于半经典限制的频谱变量的真实和虚轴。这意味着从输入中出现的所有非线性激发都具有零速度，并形成相干的非线性冷凝物。最后，通过采用正式的多维策 - kramer-brillouin膨胀来进行散射本征函数，可以获得光谱中频带和间隙的数量和位置的渐近表达，以及相对带宽度的相应表达式，以及“有效溶剂的数量”。这些结果证明与本征函数直接数值计算的结果非常吻合。特别是，获得了缩放定律，表明有效孤子的数量与小分散参数成反比。
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.
