Page 464 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 464
ANALYSIS OF PDES ON NETWORKS (MS-26)
Towards nonlinear hybrids: the planar NLS with point interactions
Riccardo Adami, adami.riccardo@gmail.com
Politecnico di Torino, Italy
A natural, but not straightforward, generalization of networks is provided by hybrids, namely
systems made by gluing together pieces of different dimension. The simplest hybrid is made
of a plane connected to a halfline. Suitable matching conditions at the contact point are to
be imposed in order to define a well-posed dynamics, and this procedure leads to studying
propagation of waves in the presence of point interactions in the plane, subject to a possibly
nonlinear dynamics. We discuss the case of the Nonlinear Schroedinger Equation, where the
nonlinearity is either self-consistent, or external and concentrated at a point in the plane, morally
the connection point of the plane and the halfline. This is a joint project with Filippo Boni,
Raffaele Carlone, and Lorenzo Tentarelli.
Ground states of the NLSE with standard and delta nonlinearities on star
graphs
Filippo Boni, filippo.boni@polito.it
Politecnico di Torino, Italy
We study the existence of ground states at fixed mass of a Schrödinger equation on star graphs
with two subcritical power-type nonlinear terms: a pointwise one, located at the vertex of the
graph, and a standard one. We show that existence and non-existence results strongly depend
on the interplay between the two nonlinearities. In particular, we see that when one nonlinearity
prevails the other, existence of ground states depends both on the mass and on the number of
halflines in the graph, whereas if the two nonlinearities are in a specific balance, then existence
of ground states is only determined by the number of halflines of the graph. This is a joint work
with R. Adami and S. Dovetta.
Bound states for nonlinear Dirac equations on metric graphs with
localized nonlinearities
William Borrelli, william.borrelli@unicatt.it
Università Cattolica del Sacro Cuore, Italy
Coauthors: Raffaele Carlone, Lorenzo Tentarelli
The investigation of evolution equations on metric graphs has become very popular nowadays,
as they represent effective models for the dynamics of physical systems confined in branched
spatial domains. In particular, the Dirac operator
D := −ıcσ1 d + mc2σ3, , σ1 = 01 , σ3 = 10 (1)
dx 10 0 −1
on metric graphs has attracted a growing interest for the description of systems where confined
particles exhibit a ‘relativistic behavior’. Here m > 0 is the mass of the particle whose (ef-
fective) hamiltonian is given by (1) and c > 0 is a phenomenological parameter, playing the
role of the speed of light. In this talk, I will discuss nonlinear Dirac equations with localized
462
Towards nonlinear hybrids: the planar NLS with point interactions
Riccardo Adami, adami.riccardo@gmail.com
Politecnico di Torino, Italy
A natural, but not straightforward, generalization of networks is provided by hybrids, namely
systems made by gluing together pieces of different dimension. The simplest hybrid is made
of a plane connected to a halfline. Suitable matching conditions at the contact point are to
be imposed in order to define a well-posed dynamics, and this procedure leads to studying
propagation of waves in the presence of point interactions in the plane, subject to a possibly
nonlinear dynamics. We discuss the case of the Nonlinear Schroedinger Equation, where the
nonlinearity is either self-consistent, or external and concentrated at a point in the plane, morally
the connection point of the plane and the halfline. This is a joint project with Filippo Boni,
Raffaele Carlone, and Lorenzo Tentarelli.
Ground states of the NLSE with standard and delta nonlinearities on star
graphs
Filippo Boni, filippo.boni@polito.it
Politecnico di Torino, Italy
We study the existence of ground states at fixed mass of a Schrödinger equation on star graphs
with two subcritical power-type nonlinear terms: a pointwise one, located at the vertex of the
graph, and a standard one. We show that existence and non-existence results strongly depend
on the interplay between the two nonlinearities. In particular, we see that when one nonlinearity
prevails the other, existence of ground states depends both on the mass and on the number of
halflines in the graph, whereas if the two nonlinearities are in a specific balance, then existence
of ground states is only determined by the number of halflines of the graph. This is a joint work
with R. Adami and S. Dovetta.
Bound states for nonlinear Dirac equations on metric graphs with
localized nonlinearities
William Borrelli, william.borrelli@unicatt.it
Università Cattolica del Sacro Cuore, Italy
Coauthors: Raffaele Carlone, Lorenzo Tentarelli
The investigation of evolution equations on metric graphs has become very popular nowadays,
as they represent effective models for the dynamics of physical systems confined in branched
spatial domains. In particular, the Dirac operator
D := −ıcσ1 d + mc2σ3, , σ1 = 01 , σ3 = 10 (1)
dx 10 0 −1
on metric graphs has attracted a growing interest for the description of systems where confined
particles exhibit a ‘relativistic behavior’. Here m > 0 is the mass of the particle whose (ef-
fective) hamiltonian is given by (1) and c > 0 is a phenomenological parameter, playing the
role of the speed of light. In this talk, I will discuss nonlinear Dirac equations with localized
462
