Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, , via R. Cozzi, 53, 20125 Milano, Italy.
Philos Trans A Math Phys Eng Sci. 2013 Dec 16;372(2007):20130002. doi: 10.1098/rsta.2013.0002. Print 2014 Jan 28.
In this paper, an introduction to the new subject of nonlinear dispersive Hamiltonian equations on graphs is given. The focus is on recently established properties of solutions in the case of the nonlinear Schrödinger (NLS) equation. Special consideration is given to the existence and behaviour of solitary solutions. Two subjects are discussed in some detail concerning the NLS equation on a star graph: the standing waves of the NLS equation on a graph with a δ interaction at the vertex, and the scattering of fast solitons through a Y-junction in the cubic case. The emphasis is on a description of concepts and results and on physical context, without reporting detailed proofs; some perspectives and more ambitious open problems are discussed.
本文介绍了图上非线性色散哈密顿方程这一新课题。重点是最近在非线性薛定谔（NLS）方程情况下解的性质。特别考虑了孤立解的存在性和行为。详细讨论了星型图上 NLS 方程的两个问题：顶点处具有 δ 相互作用的图上 NLS 方程的驻波，以及立方情形下 Y 结处快速孤子的散射。重点是描述概念和结果以及物理背景，而不报告详细的证明；讨论了一些观点和更具挑战性的开放性问题。
