Existence et stabilité de solitons, multi-solitons et solutions explosives dans quelques équations dispersives non linéaires
Résumé
The goal of this thesis is to present my research work following my
PhD. My PhD thesis was devoted to questions related to the
existence and stability of standing waves solutions to nonlinear
dispersive PDE like the nonlinear Schr\"odinger and Klein-Gordon
equations. After my PhD, my interest shifted toward more elaborate
solutions of dispersive PDE, in particular the
multi-solitons. Standing waves are still present in my work, not as
the main object of study, but as building blocks for the analysis of
more complicated nonlinear objects.
The first chapter of this document is devoted to a general
presentation of the context of my work.
The second chapter is devoted to
basic facts concerning nonlinear Schr\"odinger equations and serves as
a framework setting for many of the other works presented in this
document.
The third chapter is devoted to the presentation of my works on
multi-solitons. We start by presenting two existence results, one for
excited states multi-solitons of nonlinear Schr\"odinger equations and the other for multi-solitons of
Klein-Gordon equations based on stable solitons. We then show the
existence of infinite trains of solitons in nonlinear Schr\"odinger
equations. We conclude this chapter by a stability result for the
multi-solitons of the derivative nonlinear Schr\"odinger equation.
The fourth chapter is devoted to the presentation of my results on
blow-up and stability in different contexts. We start with a result on
the existence of minimal mass blowing up solutions for a Schr\"odinger
equation with double power nonlinearity. Then we study the Cauchy
problem and the stationary states of a singularly perturbed
Gross-Pitaevskii equation. Next we investigate, using a variety of
techniques, the stability of space periodic standing waves of one
dimensional cubic nonlinear Schr\"odinger equations. We continue with
considerations on nonlinear Schr\"odinger systems and we conclude with
results on stability of standing waves for semi-classical equations.
The fifth and last chapter is devoted to two works in progress. The
first one concerns the Manakov system: we exhibit a new family of
standing waves and study their existence and stability. The second one
concerns the excited states of nonlinear Schr\"odinger equations: we
obtain the excited states by constructing numerical schemes inspired from their
variational characterizations.
Origine : Fichiers produits par l'(les) auteur(s)
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