7,248 research outputs found
NLS ground states on graphs
We investigate the existence of ground states for the subcritical NLS energy
on metric graphs. In particular, we find out a topological assumption that
guarantees the nonexistence of ground states, and give an example in which the
assumption is not fulfilled and ground states actually exist. In order to
obtain the result, we introduce a new rearrangement technique, adapted to the
graph where it applies. Owing to such a technique, the energy level of the
rearranged function is improved by conveniently mixing the symmetric and
monotone rearrangement procedures.Comment: 24 pages, 4 figure
Galilean Genesis: an alternative to inflation
We propose a novel cosmological scenario, in which standard inflation is
replaced by an expanding phase with a drastic violation of the Null Energy
Condition (NEC): \dot H >> H^2. The model is based on the recently introduced
Galileon theories, that allow NEC violating solutions without instabilities.
The unperturbed solution describes a Universe that is asymptotically Minkowski
in the past, expands with increasing energy density until it exits the regime
of validity of the effective field theory and reheats. This solution is a
dynamical attractor and the Universe is driven to it, even if it is initially
contracting. The study of perturbations of the Galileon field reveals some
subtleties, related to the gross violation of the NEC and it shows that
adiabatic perturbations are cosmologically irrelevant. The model, however,
suggests a new way to produce a scale invariant spectrum of isocurvature
perturbations, which can later be converted to adiabatic: the Galileon is
forced by symmetry to couple to the other fields as a dilaton; the effective
metric it yields on the NEC violating solution is that of de Sitter space, so
that all light scalars will automatically acquire a nearly scale-invariant
spectrum of perturbations.Comment: 25 pages, 1 figure. v2: minor changes, JCAP published versio
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.Comment: 47 pages, 44 figure. Lecture notes for a course given at the Summer
School "MMKT 2016, Methods and Models of Kinetic Theory, Porto Ercole, June
5-11, 2016. To be published in Riv. Mat. Univ. Parm
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