Riccardo Adami, Claudio Cacciapuoti, Domenico Finco, Diego Noja
Fast solitons on star graphs
(141K, AMS-TeX)
ABSTRACT. We define the Schroedinger equation with focusing, cubic
nonlinearity on one-vertex graphs. We prove global strong and weak well-posedness and conservation laws for some self-adjoint boundary conditions at the vertex, i.e. Kirchhoff and the so called delta and delta-prime. 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 [HMZ07] about scattering of fast solitons by a delta interaction on the line, by Holmer, Marzuola and Zworski; the present paper represents a generalization of their work to 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.