Stationary Nonlinear Schr\"odinger Equation on Simplest Graphs: Boundary conditions and exact solutions

TL;DR

This paper derives exact solutions for the stationary nonlinear Schrödinger equation on simple graph structures, extending methods to various topologies and nonlinearities, with potential applications in quantum graph theory.

Contribution

It introduces a method for solving the stationary nonlinear Schrödinger equation on star, tree, and loop graphs, including boundary conditions and solutions for different nonlinearities.

Findings

Exact analytical solutions for star graphs with three bonds.

Method extension to arbitrary bonds and other simple topologies.

Separate treatment of repulsive and attractive nonlinearities.

Abstract

We treat the stationary (cubic) nonlinear Schr\"odinger equation (NSLE) on simplest graphs. Formulation of the problem and exact analytical solutions of NLSE are presented for star graphs consisting of three bonds. It is shown that the method can be extended for the case of arbitrary number of bonds of star graphs and for other simplest topologies such as tree and loop graphs. The case of repulsive and attractive nonlinearities are treated separately.