孤立波Maslov指数的计算方面
Computational aspects of the Maslov index of solitary waves
AI总结：
针对有限维哈密顿系统中的孤立波，提出基于外代数表示的Maslov指数鲁棒数值算法，并应用于反应扩散方程、五阶KdV方程和长短波共振方程。
摘要：
当孤立波被表征为有限维哈密顿系统的同宿轨道时，它们具有一个整数值拓扑不变量——Maslov指数。我们致力于开发一个鲁棒的数值算法来计算Maslov指数，理解其性质，并研究其对孤立波稳定性的影响。本文报告的算法是在外代数表示中开发的，这导致了一个具有一些新颖性质的鲁棒且快速的算法。我们使用两种不同的Maslov指数表示：一种基于交指数，另一种基于用周期轨道序列逼近同宿轨道。给出了反应扩散方程、五阶Korteweg-De Vries方程和长短波共振方程的孤立波解的Maslov指数新结果。第一部分考虑四维相空间的情况，第二部分考虑$2n$维相空间且$n>2$的情况。
英文摘要：
When solitary waves are characterized as homoclinic orbits of a finite-dimensional Hamiltonian system, they have an integer-valued topological invariant, the Maslov index. We are interested in developing a robust numerical algorithm to compute the Maslov index, to understand its properties, and to study the implications for the stability of solitary waves. The algorithms reported here are developed in the exterior algebra representation, which leads to a robust and fast algorithm with some novel properties. We use two different representations for the Maslov index, one based on an intersection index and one based on approximating the homoclinic orbit by a sequence of periodic orbits. New results on the Maslov index for solitary wave solutions of reaction-diffusion equations, the fifth-order Korteweg-De Vries equation, and the longwave-shortwave resonance equations are presented. Part 1 considers the case of four-dimensional phase space, and Part 2 considers the case of $2n-$dimensional phase space with $n>2$.
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