Simona Rota Nodari

Maître de conférences in Mathematics

Maître de conférences in Mathematics

## Research Interests

### Ongoing Works

#### Nonrelativistic limit of a mean-field model for nucleons

##### with Maria J. Esteban, Loïc Le Treust and Mathieu Lewin

One of my research topics is devoted to the study of a particular nonrelativistic limit of model from the relativistic mean-field theory of the atomic nucleus. This limit is interesting because some qualitative properties of the relativistic equations can be deduced from its solutions. In collaboration with Maria J. Esteban and Loïc Le Treust, we studied this problem in the particular case of one particle. We showed the existence of solutions for a wide class of parameters and we found numerically that some qualitative properties, including the shape of the meson potential inside and outside the atomic nucleus, are well described by the solutions of our model. This justifies the study of this new model. More recently, in collaboration with Mathieu Lewin, we proved that the positive radial solution is unique and non-degenerate which allows us to build, starting from this solution and by a bifurcation argument, a solution of the relativistic mean-field model underlying.

This model still has many open questions. The goal is to provide a full rigorous mathematical analysis of it (study of the Cauchy problem, problem with several particles, limit when the number of particles tends to infinity, etc.).

#### Renormalized energy equidistribution and local charge balance in 2D Coulomb systems

##### with Sylvia Serfaty

The Coulomb renormalized energy $W$ is a logarithmic interaction between points in the plane, calculated via a renormalization. This energy introduced by Sandier-Serfaty, is an important tool in the study of minimizers of the energy of the Ginzburg-Landau model of superconductivity and one-dimensional and two-dimensional Coulomb gases. Physical experiments on superconductors and the results obtained by Sandier-Serfaty led to conjecture that the triangular Abrikosov lattice is a minimizer of $W$ among all possible configurations of points. Solving this conjecture seems for the moment very difficult. However, intermediate results in this direction can be obtained. A first step is to study the distribution per unit volume of the renormalized energy of minimizers. In collaboration with Sylvia Serfaty, we showed that the renormalized energy of a minimizer and the number of points are "equidistributed". For the Ginzburg-Landau model, this means that the vortices are uniformly distributed and energy too. In the case of two-dimensional Coulomb gas, our result proves with a method that is simply energy-comparison based that the points in weighted Fekete sets are equidistributed. In addition, we also showed that the energy is equidistributed.

#### The nonlinear Schrödinger equation in fiber optics

##### with Stephan de Bièvre, Guillaume Dujardin, Matteo Conforti, Arnaud Mussot, Alexandre Kudlinski and Stefano Trillo

As part of LabEx CEMPI, mathematicians of the Laboratoire Paul Painlevé and of the Inria team MEPHYSTO started a collaboration with physicists of PhLAM (laboratoire de physique des lasers de l’Université Lille 1) on modelling, analysis and the numerical simulation of wave propagation in the optical fibers made and tested at PhLAM.

The propagation of a wave packet in an optical fiber is described by a nonlinear Schrodinger equation where the coefficients of the dispersion of the group velocity and the nonlinearity may vary along the direction of propagation. Our goal is to understand the effects of a periodic variation of these two factors on the propagation of a wave packet and give an interpretation of the experiences of our colleagues physicists from the mathematical point of view and through numerical simulations.

### Book chapter

#### Orbital stability: analysis meets geometry

##### with Stephan de Bièvre and François Genoud

We present an introduction to the orbital stability of relative equilibria of Hamiltonian dynamical systems on (finite and infinite dimensional) Banach spaces. A convenient formulation of the theory of Hamiltonian dynamics with symmetry and the corresponding momentum maps is proposed that allows us to highlight the interplay between (symplectic) geometry and (functional) analysis in the proofs of orbital stability of relative equilibria via the so-called energy-momentum method. The theory is illustrated with examples from finite dimensional systems, as well as from Hamiltonian PDE's, such as solitons, standing and plane waves for the nonlinear Schrödinger equation, for the wave equation, and for the Manakov system.

### Ph.D. Thesis

#### A mathematical study of nonlinear models from relativistic quantum physics

My Ph.D. thesis is devoted to the mathematical study of
some nonlinear models from relativistic quantum physics. The
goal was to have a theoretical understanding of these models
which are little studied from a rigorous mathematical point of
view because of the difficulties related to the presence of the
relativistic effects. Nevertheless, the relativistic models are
often used by chemists and physicists ; this is why a
theoretical analysis seems necessary. The aim was to give a
rigorous mathematical framework for these models; to show the
existence of solutions and to provide a good definition of the ground
state. The works contained in my thesis provide interesting results in this direction.

All models considered are nonlinear models and they are characterized
by the presence of the Dirac operator $H_0 =-i\sum_ {k =
1}^3\alpha_k\partial_k + \beta m$, which is a first-order
differential operator whose spectrum contains a negative
unbounded interval. As a consequence, the associated energy
functionals are strongly indefinite.
On this study, I used some methods of nonlinear analysis and variational calculus adapted to these difficulties.

More precisely, in the first part, I proved by a perturbation method the existence of solutions of the coupled Einstein–Dirac–Maxwell equations for a static, spherically symmetric system of two fermions in a singlet spinor state and for a weak electromagnetic coupling. In the second part, I studied a relativistic mean-field model that describes the behavior of nucleons in the atomic nucleus. I provided a condition that ensures the existence of a ground state solution of the relativistic mean-field equations in a static case; in particular, I related the existence of critical points of a strongly indefinite energy functional to strict concentration-compactness inequalities.

My Ph.D. thesis (in French) is available here.