arXiv subjects

Simona Rota Nodari

Publications and source records attributed to Simona Rota Nodari.

At least 19 recordsLinked to original sources

arXiv2026

A PDE approach to Benamou--Brenier formula for the Schrödinger problem

We studied the Benamou--Brenier formulation of the Schrödinger problem, focusing on a gap between theoretical results and applications, that often involve measures with unbounded support. While the existing proof in the literature relies on the compactness of the marginals' supports to ensure the necessary regularity of the Schrödinger potentials, we extend the validity of the Benamou--Brenier formula to the larger class of sub-Gaussian probability measures. Exploiting fine estimates on the Hessian of the potentials and the entropic interpolation, we provide an almost self-contained proof that establishes the existence of a velocity field with the appropriate polynomial growth that ensures the right integrability. This result justifies the use of the dynamic formulation in more general settings, such as Gaussian and mixture-of-Gaussians models, important also for the applications.

math.OC↗

arXiv2025

Existence and non-existence of minimizers for a multi-particle model with concave nonlinearity

We study a multi--particle model including a kinetic energy and a non linear local self-interaction, both in the bosonic and fermionic cases. In both cases, we prove that the model is well-posed if the number of particles is large enough. In particular, we show that there is a nonlinearity for which the model with $N=2$ particles is well-posed, while the model with $N=1$ is not.

math.AP↗

arXiv2025

A numerical study of stability for solitary waves of a quasi-linear Schr{ö}dinger equation

We discuss the (in)stability of solitary waves for a quasi-linear Schr{ö}dinger equation. The equation contains a quasi-linear term, responsible for a saturation effect, as well as a power nonlinearity. For different exponents of the nonlinearity, we determine analytically the asymptotic behavior of the $L^2$-mass of the solution as a function of the frequency close to the critical frequencies, which leads to natural conjectures concerning their stability. Depending on the exponent and the dimension, we expect all solitary waves to be stable, or the emergence of both a stable and an unstable branch of solutions. We investigate our conjectures numerically, and find compatible results both for the mass-energy relation and the dynamics. We observe that perturbations of solitary waves on the unstable branch may converge dynamically to the stable solution of a similar mass, or disperse. More general initial conditions show a similar behavior.

math.AP↗

arXiv2024

Standing wave solutions of a quasilinear Schrödinger equation in the small frequency limit

This article is concerned with the quasilinear Schrödinger equation \[ Δu-ωu+|u|^{p-1}u+δΔ(|u|^2)u=0, \] where $δ>0$, $N=2$ and $p>1$ or $N\ge3$ and $1 0$, our main results establish the asymptotic behavior of $u_ω$ in the limit $ω\to 0^+$. Three different regimes arise, termed 'subcritical', 'critical' and 'supercritical', corresponding respectively (when $N\ge3$) to $1 0$, $M(ω)$ is increasing if $1<p\le 1+\frac4N$ and decreasing if $1+\frac4N< p\le\frac{N+2}{N-2}$. In the supercritical case, the monotonicity of $M(ω)$ depends on the dimension, except in the regime $p\ge 3+\frac4N$, where $M(ω)$ is always decreasing close to $ω=0$. The crucial role played by $M(ω)$ for the orbital stability of the standing wave $e^{iωt}u_ω$, and for the uniqueness of normalized ground states, is discussed in the introduction.

math.AP↗

arXiv2023

A simple testbed for stability analysis of quantum dissipative systems

We study a two-state quantum system with a non linearity intended to describe interactions with a complex environment, arising through a non local coupling term. We study the stability of particular solutions, obtained as constrained extrema of the energy functional of the system. The simplicity of the model allows us to justify a complete stability analysis. This is the opportunity to review in details the techniques to investigate the stability issue. We also bring out the limitations of perturbative approaches based on simpler asymptotic models.

math.AP↗

arXiv2023

Plane wave stability analysis of Hartree and quantum dissipative systems

We investigate the stability of plane wave solutions of equations describing quantum particles interacting with a complex environment. The models take the form of PDE systems with a non local (in space or in space and time) self-consistent potential; such a coupling lead to challenging issues compared to the usual non linear Schr{ö}dinger equations. The analysis relies on the identification of suitable Hamiltonian structures and Lyapounov functionals. We point out analogies and differences between the original model, involving a coupling with a wave equation, and its asymptotic counterpart obtained in the large wave speed regime. In particular, while the analogies provide interesting intuitions, our analysis shows that it is illusory to obtain results on the former based on a perturbative analysis from the latter.

math.AP↗

arXiv2021

The Dirac-Klein-Gordon system in the strong coupling limit

We study the Dirac equation coupled to scalar and vector Klein-Gordon fields in the limit of strong coupling and large masses of the fields. We prove convergence of the solutions to those of a cubic non-linear Dirac equation, given that the initial spinors coincide. This shows that in this parameter regime, which is relevant to the relativistic mean-field theory of nuclei, the retarded interaction is well approximated by an instantaneous, local self-interaction. We generalize this result to a many-body Dirac-Fock equation on the space of Hilbert-Schmidt operators.

math.AP↗

arXiv2021

On a nonlinear Schr{ö}dinger equation for nucleons in one space dimension

We study a 1D nonlinear Schr{ö}dinger equation appearing in the description of a particle inside an atomic nucleus. For various nonlinearities, the ground states are discussed and given in explicit form. Their stability is studied numerically via the time evolution of perturbed ground states. In the time evolution of general localized initial data, they are shown to appear in the long time behaviour of certain cases.

math.AP↗

arXiv2020

The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications

In this paper we first prove a general result about the uniqueness and non-degeneracy of positive radial solutions to equations of the form $Δu+g(u)=0$. Our result applies in particular to the double power non-linearity where $g(u)=u^q-u^p-μu$ for $p>q>1$ and $μ>0$, which we discuss with more details. In this case, the non-degeneracy of the unique solution $u_μ$ allows us to derive its behavior in the two limits $μ\to0$ and $μ\toμ_*$ where $μ_*$ is the threshold of existence. This gives the uniqueness of energy minimizers at fixed mass in certain regimes. We also make a conjecture about the variations of the $L^2$ mass of $u_μ$ in terms of $μ$, which we illustrate with numerical simulations. If valid, this conjecture would imply the uniqueness of energy minimizers in all cases and also give some important information about the orbital stability of $u_μ$.

math.AP↗

arXiv2018

Orbital stability via the energy-momentum method: the case of higher dimensional symmetry groups

We consider the orbital stability of relative equilibria of Hamiltonian dynamical systems on Banach spaces, in the presence of a multi-dimensional invariance group for the dynamics. We prove a persistence result for such relative equilibria, present a generalization of the Vakhitov-Kolokolov slope condition to this higher dimensional setting, and show how it allows to prove the local coercivity of the Lyapunov function, which in turn implies orbital stability. The method is applied to study the orbital stability of relative equilibria of nonlinear Schr{ö}dinger and Manakov equations. We provide a comparison of our approach to the one by Grillakis-Shatah-Strauss.

math.AP↗

arXiv2016

Equidistribution of jellium energy for Coulomb and Riesz Interactions

For general dimension $d$ we prove the equidistribution of energy at the micro-scale in $\mathbb R^d$, for the optimal point configurations appearing in Coulomb gases at zero temperature. At the microscopic scale, i.e. after blow-up at the scale corresponding to the interparticle distance, in the case of Coulomb gases we show that the energy concentration is precisely determined by the macroscopic density of points, independently of the scale. This uses the "jellium energy" which was previously shown to control the next-order term in the large particle number asymptotics of the minimum energy. As a corollary, we obtain sharp error bounds on the discrepancy between the number of points and its expected average of optimal point configurations for Coulomb gases, extending previous results valid only for $2$-dimensional log-gases. For Riesz gases with interaction potentials $g(x)=|x|^{-s}, s\in]\min\{0,d-2\},d[$ and one-dimensional log-gases, we prove the same equidistribution result under an extra hypothesis on the decay of the localized energy, which we conjecture to hold for minimizing configurations. In this case we use the Caffarelli-Silvestre description of the non-local fractional Laplacians in $\mathbb R^d$ to localize the problem.

math-ph↗

arXiv2015

Orbital stability: analysis meets geometry

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 Schr{ö}dinger equation, for the wave equation, and for the Manakov system.

math.AP↗

arXiv2014

Uniqueness and non-degeneracy for a nuclear nonlinear Schrödinger equation

We prove the uniqueness and non-degeneracy of positive solutions to a cubic nonlinear Schrödinger (NLS) type equation that describes nucleons. The main difficulty stems from the fact that the mass depends on the solution itself. As an application, we construct solutions to the $σ$--$ω$ model, which consists of one Dirac equation coupled to two Klein-Gordon equations (one focusing and one defocusing).

math.AP↗

arXiv2014

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

We consider two related problems: the first is the minimization of the "Coulomb renormalized energy" of Sandier-Serfaty, which corresponds to the total Coulomb interaction of point charges in a uniform neutralizing background (or rather variants of it). The second corresponds to the minimization of the Hamiltonian of a two-dimensional "Coulomb gas" or "one-component plasma", a system of n point charges with Coulomb pair interaction, in a confining potential (minimizers of this energy also correspond to "weighted Fekete sets"). In both cases we investigate the microscopic structure of minimizers, i.e. at the scale corresponding to the interparticle distance. We show that in any large enough microscopic set, the value of the energy and the number of points are "rigid" and completely determined by the macroscopic density of points. In other words, points and energy are "equidistributed" in space (modulo appropriate scalings). The number of points in a ball is in particular known up to an error proportional to the radius of the ball. We also prove a result on the maximal and minimal distances between points. Our approach involves fully exploiting the minimality by reducing to minimization problems with fixed boundary conditions posed on smaller subsets.

math-ph↗

arXiv2012

Symmetric Excited States for a Mean-Field Model for a Nucleon

In this paper, we consider a stationary model for a nucleon interacting with the $ω$ and $σ$ mesons in the atomic nucleus. The model is relativistic, and we study it in a nuclear physics nonrelativistic limit. By a shooting method, we prove the existence of infinitely many solutions with a given angular momentum. These solutions are ordered by the number of nodes of each component.

math.AP↗

arXiv2012

Symmetric ground states for a stationary relativistic mean-field model for nucleons in the nonrelativistic limit

In this paper we consider a model for a nucleon interacting with the $ω$ and $σ$ mesons in the atomic nucleus. The model is relativistic, but we study it in the nuclear physics nonrelativistic limit, which is of a very different nature from the one of the atomic physics. Ground states with a given angular momentum are shown to exist for a large class of values for the coupling constants and the mesons' masses. Moreover, we show that, for a good choice of parameters, the very striking shapes of mesonic densities inside and outside the nucleus are well described by the solutions of our model.

math.AP↗

arXiv2012

Ground States for a Stationary Mean-Field Model for a Nucleon

In this paper we consider a variational problem related to a model for a nucleon interacting with the $ω$ and $σ$ mesons in the atomic nucleus. The model is relativistic, and we study it in a nuclear physics nonrelativistic limit, which is of a very different nature than the nonrelativistic limit in the atomic physics. Ground states are shown to exist for a large class of values for the parameters of the problem, which are determined by the values of some physical constants.

math.AP↗

arXiv2011

The relativistic mean-field equations of the atomic nucleus

In nuclear physics, the relativistic mean-field theory describes the nucleus as a system of Dirac nucleons which interact via meson fields. In a static case and without nonlinear self-coupling of the $σ$ meson, the relativistic mean-field equations become a system of Dirac equations where the potential is given by the meson and photon fields. The aim of this work is to prove the existence of solutions of these equations. We consider a minimization problem with constraints that involve negative spectral projectors and we apply the concentration-compactness lemma to find a minimizer of this problem. We show that this minimizer is a solution of the relativistic mean-field equations considered.

math-ph↗