Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fast solitons on star graphs   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.

31
The categories of this paper are: ao-sci
18
The categories of this paper are: econ
32
The categories of this paper are: atom-ph
8
The categories of this paper are: hep-ex
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
36
The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
29
The categories of this paper are: funct-an
30
The categories of this paper are: q-alg
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas
1
The categories of this paper are: cs
11
The categories of this paper are: hep-lat
25
The categories of this paper are: patt-sol
3
The categories of this paper are: physics