Symmetry breaking in the periodic Thomas--Fermi--Dirac--von Weizs{\"a}cker model

We consider the Thomas--Fermi--Dirac--von~Weizs{\"a}cker model for a system composed of infinitely many nuclei placed on a periodic lattice and electrons with a periodic density. We prove that if the Dirac constant is small enough, the electrons have the same periodicity as the nuclei. On the other hand if the Dirac constant is large enough, the 2-periodic electronic minimizer is not 1-periodic, hence symmetry breaking occurs. We analyze in detail the behavior of the electrons when the Dirac constant tends to infinity and show that the electrons all concentrate around exactly one of the 8 nuclei of the unit cell of size 2, which is the explanation of the breaking of symmetry. Zooming at this point, the electronic density solves an effective nonlinear Schr\"odinger equation in the whole space with nonlinearity $u^{7/3}-u^{4/3}$. Our results rely on the analysis of this nonlinear equation, in particular on the uniqueness and non-degeneracy of positive solutions

On uniqueness and non-degeneracy of anisotropic polarons

We study the anisotropic Choquard--Pekar equation which de-scribes a polaron in an anisotropic medium. We prove the uniqueness and non-degeneracy of minimizers in a weakly anisotropic medium. In addition, for a wide range of anisotropic media, we derive the symmetry properties of minimizers and prove that the kernel of the associated linearized operator is reduced, apart from three functions coming from the translation invariance, to the kernel on the subspace of functions that are even in each of the three principal directions of the medium

Results on the spectral stability of standing wave solutions of the Soler model in 1-D

We study the spectral stability of the nonlinear Dirac operator in dimension 1+1, restricting our attention to nonlinearities of the form $f(\langle\psi,\sigma_3\psi\rangle_{\mathbb{C}^2}) \sigma_3$. We obtain bounds on eigenvalues for the linearized operator around standing wave solutions of the form $e^{-i\omega t} \phi_0$. For the case of power nonlinearities $f(s)= |s|^p$, $p>0$, we obtain a range of frequencies $\omega$ such that the linearized operator has no unstable eigenvalues on the axes of the complex plane. As a crucial part of the proofs, we obtain a detailed description of the spectra of the self-adjoint blocks in the linearized operator. In particular, we show that the condition $\langle\phi_0,\sigma_3\phi_0\rangle_{\mathbb{C}^2} > 0$ characterizes groundstates analogously to the Schr\"{o}dinger case.Comment: 46 pages, 3 figure

Sugarcane (Saccharum X officinarum): A Reference Study for the Regulation of Genetically Modified Cultivars in Brazil

Global interest in sugarcane has increased significantly in recent years due to its economic impact on sustainable energy production. Sugarcane breeding and better agronomic practices have contributed to a huge increase in sugarcane yield in the last 30 years. Additional increases in sugarcane yield are expected to result from the use of biotechnology tools in the near future. Genetically modified (GM) sugarcane that incorporates genes to increase resistance to biotic and abiotic stresses could play a major role in achieving this goal. However, to bring GM sugarcane to the market, it is necessary to follow a regulatory process that will evaluate the environmental and health impacts of this crop. The regulatory review process is usually accomplished through a comparison of the biology and composition of the GM cultivar and a non-GM counterpart. This review intends to provide information on non-GM sugarcane biology, genetics, breeding, agronomic management, processing, products and byproducts, as well as the current technologies used to develop GM sugarcane, with the aim of assisting regulators in the decision-making process regarding the commercial release of GM sugarcane cultivars

Symétrie et brisure de symétrie pour certains problèmes non linéaires

This thesis is devoted to the mathematical study of two quantum systems described by nonlinear models: the anisotropic polaron and the electrons in a periodic crystal. We first prove the existence of minimizers, and then discuss the question of uniqueness for both problems. In the first part, we show the uniqueness and nondegeneracy of the minimizer for the polaron, described by the Choquard--Pekar anisotropic equation, assuming that the dielectric matrix of the medium is almost isotropic. In the strong anisotropic setting, we leave the question of uniqueness open but identify the symmetry that can possibly be degenerate. In the second part, we study the electrons of a crystal in the periodic Thomas--Fermi--Dirac--Von~Weizsäcker model, varying the parameter in front of the Dirac term. We show uniqueness and nondegeneracy of the minimizer when this parameter is small enough et prove the occurrence of symmetry breaking when it is large.Cette thèse est consacrée à l'étude mathématique de deux systèmes quantiques décrits par des modèles non linéaires : le polaron anisotrope et les électrons d'un cristal périodique. Après avoir prouvé l'existence de minimiseurs, nous nous intéressons à la question de l'unicité pour chacun des deux modèles. Dans une première partie, nous montrons l'unicité du minimiseur et sa non-dégénérescence pour le polaron décrit par l'équation de Choquard--Pekar anisotrope, sous la condition que la matrice diélectrique du milieu est presque isotrope. Dans le cas d'une forte anisotropie, nous laissons la question de l'unicité en suspens mais caractérisons précisément les symétries pouvant être dégénérées. Dans une seconde partie, nous étudions les électrons d'un cristal dans le modèle de Thomas--Fermi--Dirac--Von~Weizsäcker périodique, en faisant varier le paramètre devant le terme de Dirac. Nous montrons l'unicité et la non-dégénérescence du minimiseur lorsque ce paramètre est suffisamment petit et mettons en évidence une brisure de symétrie lorsque celui-ci est grand

Symmetry and symmetry breaking for some nonlinear problems

Cette thèse est consacrée à l'étude mathématique de deux systèmes quantiques décrits par des modèles non linéaires : le polaron anisotrope et les électrons d'un cristal périodique. Après avoir prouvé l'existence de minimiseurs, nous nous intéressons à la question de l'unicité pour chacun des deux modèles. Dans une première partie, nous montrons l'unicité du minimiseur et sa non-dégénérescence pour le polaron décrit par l'équation de Choquard--Pekar anisotrope, sous la condition que la matrice diélectrique du milieu est presque isotrope. Dans le cas d'une forte anisotropie, nous laissons la question de l'unicité en suspens mais caractérisons précisément les symétries pouvant être dégénérées. Dans une seconde partie, nous étudions les électrons d'un cristal dans le modèle de Thomas--Fermi--Dirac--Von~Weizsäcker périodique, en faisant varier le paramètre devant le terme de Dirac. Nous montrons l'unicité et la non-dégénérescence du minimiseur lorsque ce paramètre est suffisamment petit et mettons en évidence une brisure de symétrie lorsque celui-ci est grand.This thesis is devoted to the mathematical study of two quantum systems described by nonlinear models: the anisotropic polaron and the electrons in a periodic crystal. We first prove the existence of minimizers, and then discuss the question of uniqueness for both problems. In the first part, we show the uniqueness and nondegeneracy of the minimizer for the polaron, described by the Choquard--Pekar anisotropic equation, assuming that the dielectric matrix of the medium is almost isotropic. In the strong anisotropic setting, we leave the question of uniqueness open but identify the symmetry that can possibly be degenerate. In the second part, we study the electrons of a crystal in the periodic Thomas--Fermi--Dirac--Von~Weizsäcker model, varying the parameter in front of the Dirac term. We show uniqueness and nondegeneracy of the minimizer when this parameter is small enough et prove the occurrence of symmetry breaking when it is large

On uniqueness and non-degeneracy of anisotropic polarons

We study the anisotropic Choquard–Pekar equation which de-scribes a polaron in an anisotropic medium. We prove the uniqueness and non-degeneracy of minimizers in a weakly anisotropic medium. In addition, for a wide range of anisotropic media, we derive the symmetry properties of minimizers and prove that the kernel of the associated linearized operator is reduced, apart from three functions coming from the translation invariance, to the kernel on the subspace of functions that are even in each of the three principal directions of the medium.nonnonouirechercheInternationa

On one-dimensional Bose gases with two- and (critical) attractive three-body interactions

We consider a one-dimensional, trapped, focusing Bose gas where $N$ bosons interact with each other via both a two-body interaction potential of the form $aN^{\alpha-1} U(N^\alpha(x-y))$ and an attractive three-body interaction potential of the form $-bN^{2\beta-2} W(N^\beta(x-y,x-z))$, where $a\in\mathbb{R}$, $b,\alpha>0$, $0 0$ is fixed. Finally, we analyze the behavior of the many-body ground states when the convergence $b_N \nearrow \mathfrak{b}$ is "faster" than the slow enough convergence $0<a_N \searrow 0$

On one-dimensional Bose gases with two- and (critical) attractive three-body interactions

We consider a one-dimensional, trapped, focusing Bose gas where $N$ bosons interact with each other via both a two-body interaction potential of the form $aN^{\alpha-1} U(N^\alpha(x-y))$ and an attractive three-body interaction potential of the form $-bN^{2\beta-2} W(N^\beta(x-y,x-z))$, where $a\in\mathbb{R}$, $b,\alpha>0$, $0 0$ is fixed. Finally, we analyze the behavior of the many-body ground states when the convergence $b_N \nearrow \mathfrak{b}$ is "faster" than the slow enough convergence $0<a_N \searrow 0$

On one-dimensional Bose gases with two- and (critical) attractive three-body interactions

We consider a one-dimensional, trapped, focusing Bose gas where $N$ bosons interact with each other via both a two-body interaction potential of the form $aN^{\alpha-1} U(N^\alpha(x-y))$ and an attractive three-body interaction potential of the form $-bN^{2\beta-2} W(N^\beta(x-y,x-z))$, where $a\in\mathbb{R}$, $b,\alpha>0$, $0 0$ is fixed. Finally, we analyze the behavior of the many-body ground states when the convergence $b_N \nearrow \mathfrak{b}$ is "faster" than the slow enough convergence $0<a_N \searrow 0$