Title: Traveling waves in periodic metric graphs via spatial dynamics Show Chinese title

Title: 周期性度量图中的行波通过空间动力学

Abstract: The purpose of this work is to introduce a concept of traveling waves in the setting of periodic metric graphs. It is known that the nonlinear Schr{\"o}dinger (NLS) equation on periodic metric graphs can be reduced asymptotically on long but finite time intervals to the homogeneous NLS equation, which admits traveling solitary wave solutions. In order to address persistence of such traveling waves beyond finite time intervals, we formulate the existence problem for traveling waves via spatial dynamics. There exist no spatially decaying (solitary) waves because of an infinite-dimensional center manifold in the spatial dynamics formulation. Existence of traveling modulating pulse solutions which are solitary waves with small oscillatory tails at very long distances from the pulse core is proven by using a local center-saddle manifold. We show that the variational formulation fails to capture existence of such modulating pulse solutions even in the singular limit of zero wave speeds where true (standing) solitary waves exist. Propagation of a traveling solitary wave and formation of a small oscillatory tail outside the pulse core is shown in numerical simulations of the NLS equation on the periodic graph. Show Chinese abstract

Abstract: 本文工作的目的是在周期度量图的背景下引入行波的概念。众所周知，周期度量图上的非线性薛定谔（NLS）方程可以在长时间但有限的时间区间内渐近约化为齐次NLS方程，该方程承认行单孤波解。为了处理这种行波在有限时间区间之外的持久性问题，我们通过空间动力学方法提出了行波的存在性问题。由于空间动力学表述中存在无限维中心流形，不存在空间衰减的（孤立）波。通过局部中心-鞍形变流形证明了行调制脉冲解的存在，这些解是在脉冲核心非常远的距离上有微小振荡尾部的孤立波。我们展示了即使在零波速的奇异极限下，变分表述也无法捕捉到此类调制脉冲解的存在，而在该极限下确实存在真正的（静止）孤立波。数值模拟显示了周期图上NLS方程中的行孤立波传播以及在脉冲核心外形成的小振荡尾部。