School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom.
Fakultät für Physik, Universität Duisburg-Essen, Lotharstraße 1, 47048 Duisburg, Germany.
Phys Rev E. 2016 Mar;93(3):032204. doi: 10.1103/PhysRevE.93.032204. Epub 2016 Mar 4.
In this paper we present a general framework for solving the stationary nonlinear Schrödinger equation (NLSE) on a network of one-dimensional wires modeled by a metric graph with suitable matching conditions at the vertices. A formal solution is given that expresses the wave function and its derivative at one end of an edge (wire) nonlinearly in terms of the values at the other end. For the cubic NLSE this nonlinear transfer operation can be expressed explicitly in terms of Jacobi elliptic functions. Its application reduces the problem of solving the corresponding set of coupled ordinary nonlinear differential equations to a finite set of nonlinear algebraic equations. For sufficiently small amplitudes we use canonical perturbation theory, which makes it possible to extract the leading nonlinear corrections over large distances.
本文提出了一种在由度量图建模的一维线网络上求解定态非线性薛定谔方程（NLSE）的通用框架，在顶点处具有合适的匹配条件。给出了一个正式的解，该解以非线性方式表达了一条边（线）一端的波函数及其导数与另一端的值之间的关系。对于三次 NLSE，这种非线性传输操作可以用雅可比椭圆函数显式表示。它的应用将求解相应的耦合常微分非线性方程组的问题简化为有限个非线性代数方程组。对于足够小的振幅，我们使用正则摄动理论，这使得能够在大距离上提取主要的非线性修正。
