Fast solitons on star graphs

TL;DR

This paper studies the behavior of fast solitons on star graphs governed by the nonlinear Schrödinger equation, analyzing well-posedness, boundary conditions, and soliton scattering, extending previous line interaction results to graph structures.

Contribution

It extends the analysis of soliton scattering from line interactions to star graphs, including boundary conditions and collision dynamics.

Findings

Global well-posedness established for certain boundary conditions.

Fast solitons split into reflected and transmitted parts upon collision.

Reflection and transmission coefficients match linear case over logarithmic timescale.

Abstract

We define the Schr\"odinger equation with focusing, cubic nonlinearity on one-vertex graphs. We prove global well-posedness in the energy domain and conservation laws for some self-adjoint boundary conditions at the vertex, i.e. Kirchhoff boundary condition and the so called $\delta$ and $\delta'$ boundary conditions. Moreover, in the same setting we study the collision of a fast solitary wave with the vertex and we show that it splits in reflected and transmitted components. The outgoing waves preserve a soliton character over a time which depends on the logarithm of the velocity of the ingoing solitary wave. Over the same timescale the reflection and transmission coefficients of the outgoing waves coincide with the corresponding coefficients of the linear problem. In the analysis of the problem we follow ideas borrowed from the seminal paper \cite{[HMZ07]} about scattering of fast solitons by a delta interaction on the line, by Holmer, Marzuola and Zworski; the present paper represents an extension of their work to the case of graphs and, as a byproduct, it shows how to extend the analysis of soliton scattering by other point interactions on the line, interpreted as a degenerate graph.