非线性晶格中的孤立波及其线性稳定性
Solitary waves and their linear stability in nonlinear lattices
摘要：
以具有空间周期非线性系数的非线性薛定谔方程为模型，讨论了一般非线性晶格中的孤立波。长孤立波的渐近理论，跨越大量的晶格周期。在这个极限中，孤立波相对于晶格的允许位置，以及它们的线性稳定性，取决于一定的递归关系，其中包含的信息超出了通常的双尺度扰动展开的所有阶次。因此，只有两个这样的位置是允许的，而这两个孤立波，一个是线性稳定的，另一个不稳定。对于余弦晶格，特别是，两个可能的孤立波集中在晶格的最大值或最小值，与前者是稳定的，和相关的线性稳定性特征值的分析预测与数值结果非常吻合。此外，一个可数的多孤立波束缚态的解析构造。尽管有相当不同的物理设置，指数渐近方法在这里是惊人的相似，在早期的研究孤立波包，涉及一个周期性的载波和一个缓慢变化的信封，这强调了一般价值的处理多尺度孤立波问题的程序。
英文摘要：
Solitary waves in a general nonlinear lattice are discussed, employing as a model the nonlinear Schrödinger equation with a spatially periodic nonlinear coefficient. An asymptotic theory is developed for long solitary waves, that span a large number of lattice periods. In this limit, the allowed positions of solitary waves relative to the lattice, as well as their linear stability properties, hinge upon a certain recurrence relation which contains information beyond all orders of the usual two-scale perturbation expansion. It follows that only two such positions are permissible, and of those two solitary waves, one is linearly stable and the other unstable. For a cosine lattice, in particular, the two possible solitary waves are centered at a maximum or minimum of the lattice, with the former being stable, and the analytical predictions for the associated linear stability eigenvalues are in excellent agreement with numerical results. Furthermore, a countable set of multi-solitary-wave bound states are constructed analytically. In spite of rather different physical settings, the exponential asymptotics approach followed here is strikingly similar to that taken in earlier studies of solitary wavepackets involving a periodic carrier and a slowly-varying envelope, which underscores the general value of this procedure for treating multi-scale solitary-wave problems.
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