Moving Gap Solitons in Deep Periodic Structures with Transversally Crossing Bands

该项目研究了脉冲在非线性周期性结构中的相干传播，例如光子晶体，具有任意大的周期性对比度。在具有渐进小对比度的周期性结构中，运动脉冲在数学上相对较好地理解。然而，真正的物理结构具有固定的有限对比度。特别是对于一维的立方非线性薛定谔方程和立方非线性波动方程，我们将研究窄谱隙中的运动孤波（移动隙孤子），窄谱隙是由谱带函数的横向交叉的周期结构的扰动打开的。扰动的幅度，或者等价地差距的宽度，将起到小的渐近参数的作用。在能带结构中具有谱函数的横向交叉的周期性介质的一个例子是由有限带势描述的。横截性预期将导致存在一个完整的家庭的间隙孤子的速度是独立的渐近参数的间隔。这与具有任意对比度的结构的标准情况相反，在标准情况下，只知道渐近慢的间隙孤子渐近接近于谱。首先，通过耦合模方程给出间隙孤子的渐近包络近似。然后将严格证明渐近性，并估计大但有限的时间间隔的近似误差。数值计算将证实分析，显示的例子，间隙孤子和检查的渐近收敛速度。第二，更具有挑战性的，部分将处理的存在性的确切移动广义呼吸的两个方程。这些呼吸是运动的脉冲，其形状在空间和时间上周期性地变化。基于现有的相关结果，只有广义呼吸者的预期，即这样的脉冲，其轮廓非常接近本地化的非常大，但有限的空间间隔。这是，再一次，目的是生产一个家庭的呼吸机与速度的间隔。

The project investigates coherent propagation of pulses in nonlinear periodic structures, e.g. photonic crystals, with an arbitrarily large contrast of the periodicity. Moving pulses are mathematically relatively well understood in periodic structures with an asymptotically small contrast. True physical structure have, however, a fixed finite contrast. The mathematical analysis for this case is markedly more difficult.In particular, for the one dimensional cubically nonlinear Schrödinger equation and for the cubically nonlinear wave equation we will study moving solitary waves (moving gap solitons) in narrow spectral gaps which open by a perturbation of a periodic structure with a transversal crossing of spectral band functions. The amplitude of the perturbation, or equivalently the width of the gap, will play the role of a small asymptotic parameter. An example of a periodic medium featuring transversal crossings of spectral functions in the band structure is one described by a finite band potential. The transversality is expected to lead to the existence of a whole family of gap solitons with an interval of velocities which are independent of the asymptotic parameter. This is in contrast with the standard case of structure with an arbitrary contrast, where only asymptotically slow gap solitons are known asymptotically close to the spectrum.Firstly, an asymptotic envelope approximation of gap solitons via coupled mode equations will be given. The asymptotics will then be rigorously justified and the approximation error estimated for large but finite time intervals. Numerical computations will then corroborate the analysis, show examples of gap solitons and check the asymptotic convergence rates. The second, and more challenging, part will deal with the existence of exact moving generalized breathers of the two equations. These breathers are moving pulses, whose shape changes periodically in space and time. Based on related existing results only generalized breathers are expected, i.e. such pulses whose profiles are very close to localized ones on very large but finite spatial intervals. It is, once again, the aim to produce a family of breathers with an interval of velocities.