Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Nonlinear dynamics on branched structures and networks   Nonlinear dynamics on graphs has rapidly become a topical issue with many
physical applications, ranging from nonlinear optics to Bose-Einstein
condensation. Whenever in a physical experiment a ramified structure is
involved, it can prove useful to approximate such a structure by a metric
graph, or network. For the Schroedinger equation it turns out that the sixth
power in the nonlinear term of the energy is critical in the sense that below
that power the constrained energy is lower bounded irrespectively of the value
of the mass (subcritical case). On the other hand, if the nonlinearity power
equals six, then the lower boundedness depends on the value of the mass: below
a critical mass, the constrained energy is lower bounded, beyond it, it is not.
For powers larger than six the constrained energy functional is never lower
bounded, so that it is meaningless to speak about ground states (supercritical
case). These results are the same as in the case of the nonlinear Schrodinger
equation on the real line. In fact, as regards the existence of ground states,
the results for systems on graphs differ, in general, from the ones for systems
on the line even in the subcritical case: in the latter case, whenever the
constrained energy is lower bounded there always exist ground states (the
solitons, whose shape is explicitly known), whereas for graphs the existence of
a ground state is not guaranteed. For the critical case, our results show a
phenomenology much richer than the analogous on the line.

6
The categories of this paper are: astro-ph
17
The categories of this paper are: q-fin
33
The categories of this paper are: plasm-ph
29
The categories of this paper are: funct-an
21
The categories of this paper are: adap-org
12
The categories of this paper are: nucl-th
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
26
The categories of this paper are: bayes-an
18
The categories of this paper are: econ
34
The categories of this paper are: chem-ph
13
The categories of this paper are: quant-ph
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
15
The categories of this paper are: nucl-ex
36
The categories of this paper are: cmp-lg