### Description

The aim of the project is mainly devoted to a deep understanding of propagation wave phenomena. We plan to work in many directions, however we shall be mainly focused in the analysis of qualitative properties of solutions to the nonlinear Schroedinger equations (NLS) and Dirac equations. The main points of our research can be summarized as follows:

1) Linear Dispersive Equations.

Many members of the project are well-known experts of dispersive estimates (Schrödinger, wave, Klein/Gordon, Dirac): time decay, Strichartz, Morawetz, local smoothing. Many results have been devoted to prove how the action of external electromagnetic fields can influence the dispersion. Some members of the project started a program concerned with the study of critical perturbations, that we plan to pursuit with this proposal. We wish to get a total comprehension of the dispersion for hamiltonians with scaling-critical potentials. The analysis will be based on the representation of solutions for the Schroedinger equation with critical magnetic potentials through the eigenfunctions of the angular part of the Hamiltonian. Also perturbations with electric potentials in the BV class will be analyzed.

2) NLS with Point Interactions.

The study of NLS in the presence of defects and singularities in the propagation media arises in several areas of physics and in applications. The present analysis is limited to the study of one-dimensional systems. In this context, defects are modeled by means of singular perturbations of the so-called point interactions. Many results are known in the case of the delta interaction, and some recent results have been obtained (by some members of the project) in the case of the delta-prime and dipole interaction. We plan to extend those results to a general point interaction in 1-D and to delta type interactions in 2-D and 3-D. We shall also consider NLS on graph, where we shall analyze the structure of the stationary states, as well as the dispersive and Strichartz estimates in the case of trees and graphs with non-trivial topology.

3) Asymptotic Stability of Standing Waves.

We plan to develop a suitable version of the conformal vector fields technique for NLS with variable coefficients. Recall that this equation arise naturally when studying the NLS (with constant coefficient) near ground states. Our target is the application of this new theory to study the asymptotic stability of ground states of NLS with pure power nonlinearities in the L^2 subcritical case. In fact this is a question still not well understood in the literature in contrast with the case of NLS with smooth nonlinearities.

4) Solitary Waves for a Family of NLS.

The so called "saturable" nonlinearity has been studied in the literature in view of its applications to the theory of optical fibers when the medium does not answer proportionally to the light impulse, but saturates. Different phenomena from the usual ones here appear. We will study saturable NLS, with the aim of classifying solitons, their qualitative properties and stability. Experimental data suggest the existence of many orbitally stable solitons, in contrast with the pure power NLS.

In addition, we plan to start a research project regarding the Levy-Schrödinger equations. Here, the hamiltonian contains a local term (the usual Laplace operator) and a lower-order non-local one, corresponding to the Levy-diffusion.

5) Nonlinear Dirac Equations.

Part of the project will be devoted to the implementation of the "concentration-compactness/rigidity" program by Kenig e Merle to the nonlinear Dirac equation. The first part (existence of the minimal element) should now follow by the linear profile decomposition proved by Fanelli and Visciglia. We will study the part concerning the rigidity, which is quite interesting and difficult by the following facts: the energy preserved along the evolution has not a sign and there are not "good" estimates of the virial type. The Dirac equation will be also analyzed from the variational point of view. At this stage two main issues arise: firstly, the associated Euler functional is strongly indefinite and all of its critical points have infinite Morse index; secondly, the equation is set in the whole 3D space, which is non-compact. We plan to attack the problem via some perturbative techniques.

6) Construction of Invariant Measures for Benjamin-Ono equation.

Although many results have been achieved in recent years concerning the local and global well posedness of the Benjamin-Ono equation with very low regular initial data, almost nothing is known about the long-time behavior of the solutions. A part of our project will be devoted to the analysis of those properties via the construction of invariant measures for the flow associated to Benjamin-Ono. Recall that the classical Recurrence Poincaré Theorem can be applied to any dynamical system endowed with an invariant probability measure. In particular the aforementioned Theorem gives some informations on the long-time behavior of the flow and suggests a kind of periodicity in time. To complete this part of the project it will be fundamental a combination of some key probabilistic cancellations discovered recently by Tzvetkov and Visciglia, in conjunction with a deep analysis of the local Cauchy theory up to the gauge transform (introduced by Tao).

1) Linear Dispersive Equations.

Many members of the project are well-known experts of dispersive estimates (Schrödinger, wave, Klein/Gordon, Dirac): time decay, Strichartz, Morawetz, local smoothing. Many results have been devoted to prove how the action of external electromagnetic fields can influence the dispersion. Some members of the project started a program concerned with the study of critical perturbations, that we plan to pursuit with this proposal. We wish to get a total comprehension of the dispersion for hamiltonians with scaling-critical potentials. The analysis will be based on the representation of solutions for the Schroedinger equation with critical magnetic potentials through the eigenfunctions of the angular part of the Hamiltonian. Also perturbations with electric potentials in the BV class will be analyzed.

2) NLS with Point Interactions.

The study of NLS in the presence of defects and singularities in the propagation media arises in several areas of physics and in applications. The present analysis is limited to the study of one-dimensional systems. In this context, defects are modeled by means of singular perturbations of the so-called point interactions. Many results are known in the case of the delta interaction, and some recent results have been obtained (by some members of the project) in the case of the delta-prime and dipole interaction. We plan to extend those results to a general point interaction in 1-D and to delta type interactions in 2-D and 3-D. We shall also consider NLS on graph, where we shall analyze the structure of the stationary states, as well as the dispersive and Strichartz estimates in the case of trees and graphs with non-trivial topology.

3) Asymptotic Stability of Standing Waves.

We plan to develop a suitable version of the conformal vector fields technique for NLS with variable coefficients. Recall that this equation arise naturally when studying the NLS (with constant coefficient) near ground states. Our target is the application of this new theory to study the asymptotic stability of ground states of NLS with pure power nonlinearities in the L^2 subcritical case. In fact this is a question still not well understood in the literature in contrast with the case of NLS with smooth nonlinearities.

4) Solitary Waves for a Family of NLS.

The so called "saturable" nonlinearity has been studied in the literature in view of its applications to the theory of optical fibers when the medium does not answer proportionally to the light impulse, but saturates. Different phenomena from the usual ones here appear. We will study saturable NLS, with the aim of classifying solitons, their qualitative properties and stability. Experimental data suggest the existence of many orbitally stable solitons, in contrast with the pure power NLS.

In addition, we plan to start a research project regarding the Levy-Schrödinger equations. Here, the hamiltonian contains a local term (the usual Laplace operator) and a lower-order non-local one, corresponding to the Levy-diffusion.

5) Nonlinear Dirac Equations.

Part of the project will be devoted to the implementation of the "concentration-compactness/rigidity" program by Kenig e Merle to the nonlinear Dirac equation. The first part (existence of the minimal element) should now follow by the linear profile decomposition proved by Fanelli and Visciglia. We will study the part concerning the rigidity, which is quite interesting and difficult by the following facts: the energy preserved along the evolution has not a sign and there are not "good" estimates of the virial type. The Dirac equation will be also analyzed from the variational point of view. At this stage two main issues arise: firstly, the associated Euler functional is strongly indefinite and all of its critical points have infinite Morse index; secondly, the equation is set in the whole 3D space, which is non-compact. We plan to attack the problem via some perturbative techniques.

6) Construction of Invariant Measures for Benjamin-Ono equation.

Although many results have been achieved in recent years concerning the local and global well posedness of the Benjamin-Ono equation with very low regular initial data, almost nothing is known about the long-time behavior of the solutions. A part of our project will be devoted to the analysis of those properties via the construction of invariant measures for the flow associated to Benjamin-Ono. Recall that the classical Recurrence Poincaré Theorem can be applied to any dynamical system endowed with an invariant probability measure. In particular the aforementioned Theorem gives some informations on the long-time behavior of the flow and suggests a kind of periodicity in time. To complete this part of the project it will be fundamental a combination of some key probabilistic cancellations discovered recently by Tzvetkov and Visciglia, in conjunction with a deep analysis of the local Cauchy theory up to the gauge transform (introduced by Tao).

Status | Active |
---|---|

Effective start/end date | 1/1/12 → … |

### Funding

- MIUR - FIRB

### Fingerprint

Nonlinear equations

Dirac equation

Point interactions

Invariant measure

Nonlinearity

Interaction

Dispersive estimates

Benjamin-Ono equation

Qualitative properties

Long-time behavior

Asymptotic stability

Rigidity

Ground state

Solitons

Defects

Perturbation

Graph in graph theory

Theorem

Poincaré recurrences

Conformal vector field