On decaying properties of nonlinear Schr\"odinger equations

In this paper we discuss quantitative (pointwise) decay estimates for solutions to the 3D cubic defocusing Nonlinear Schr\"odinger equation with various initial data, deterministic and random. We show that nonlinear solutions enjoy the same decay rate as the linear ones. The regularity assumption on the initial data is much lower than in previous results (see \cite{fan2021decay} and the references therein) and moreover we quantify the decay, which is another novelty of this work. Furthermore, we show that the (physical) randomization of the initial data can be used to replace the $L^1$-data assumption (see \cite{fan2022note} for the necessity of the $L^1$-data assumption). At last, we note that this method can be also applied to derive decay estimates for other nonlinear dispersive equations.Comment: 24 pages. Comments are welcome

Scattering of the three-dimensional cubic nonlinear Schrödinger equation with partial harmonic potentials

In this paper, we consider the following three dimensional defocusing cubic nonlinear Schrödinger equation (NLS) with partial harmonic potential \begin{equation} \left\{\begin{array}{l} i\partial_tu + \left(\Delta_{\mathbb{R}^3}-x^2\right)u = |u|^2u, \\ u|_{t=0} = u_0 \\ \end{array}\right. \tag{NLS} \end{equation} Out main result shows that the solution $u$ scatters for any given initial data $u_0$ with finite mass and energy. The main new ingredient in our approach is to approxmate (NLS) in the large-scale case by a relevant dispersive continuous resonant (DCR) system. The proof of global well-posedness and scattering of the new (DCR) system is greatly inspired by the fundamental works of Dodson [29, 31, 32] in his study of scattering for the mass-critical nonlinear Schrödinger equation. The analysis of (DCR) system allows us to utilize the additional regularity of the smooth nonlinear profile so that the celebrated concentration-compactness/rigidity argument of Kenig and Merle [61,62] applies

Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrodinger equation

We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This behavior is quantified by the growth of higher Sobolev norms: given any δ[much less-than]1,K [much greater-than] 1, s > 1, we construct smooth initial data u 0 with ||u0||Hs , so that the corresponding time evolution u satisfies u(T)Hs[greater than]K at some time T. This growth occurs despite the Hamiltonian’s bound on ||u(t)||H1 and despite the conservation of the quantity ||u(t)||L2. The proof contains two arguments which may be of interest beyond the particular result described above. The first is a construction of the solution’s frequency support that simplifies the system of ODE’s describing each Fourier mode’s evolution. The second is a construction of solutions to these simpler systems of ODE’s which begin near one invariant manifold and ricochet from arbitrarily small neighborhoods of an arbitrarily large number of other invariant manifolds. The techniques used here are related to but are distinct from those traditionally used to prove Arnold Diffusion in perturbations of Hamiltonian systems

Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation

We consider the cubic fourth order nonlinear Schr\"odinger equation on the circle. In particular, we prove that the mean-zero Gaussian measures on Sobolev spaces $H^s(\mathbb{T})$, $s > \frac34$, are quasi-invariant under the flow.Comment: 41 pages. To appear in Probab. Theory Related Field

Sur certains systèmes hamiltoniens liés à l’équation de Szegő cubique

The main purpose of this Ph.D. thesis is to study the long time behavior of solutionsto some Hamiltonian PDEs, i∂_t u=X_H (u), including global existence, growth of high Sobolev norms, scattering and long time approximation by resonant dynamics.In this context, at first we consider the Szegő equation on the circle S1 perturbed bya linear potential, i∂_t u=∏ |u|² u+α∫ u,α∈R, (α-Szegő) where ∏ is the projector onto the non-negative frequencies. For α=0, it turns out tobe the cubic Szegő equation, which was recently introduced by Gérard and Grellier as amathematical toy model of a non-linear totally non dispersive equation.We study the global well-posedness, the integrability and the dynamics of the singularvalues of the related Hankel operators of the α –Szegő equation. Moreover, we establishthe following properties for this equation on a class of invariant submanifolds, with anarbitrary large dimension. For α0, there exist trajectories on which everySobolev norm of regularity s>½ , exponentially tends to infinity in time.Second, we study the wave-guide Schrödinger equation posed on the spatial domain(x,y)∈R×T ，i∂_t U+∂_xx U-|D_y |U=|U|² U,(WS)Adapting an idea by Hani–Pausader–Tzvetkov–Visciglia, we establish a modified scattering theory between small solutions to this equation and small solutions to the cubic Szegő equation. Combining this scattering theory with a recent result by Gérard–Grellier, we infer existence of global solutions to (WS) which are unbounded in the space L_x^2 H_y^s (R×T) for every s>½ .Cette thèse est principalement consacrée à l’étude du comportement en temps long de solutions de certaines équations aux dérivées partielles hamiltoniennes, du type i∂_t u=X_H (u), en particulier l’existence globale, la croissance des normes de Sobolev, la diffusion et l’approximation par la dynamique résonante.Dans ce contexte, nous considérons d’abord une perturbation de l’équation de Szegő cubique par un potentiel linéaire, i∂_t u=∏ |u|² u+α∫ u,α∈R, (α-Szegő) où ∏▒ désigne le projecteur de Szegő sur les fréquences positives. Pour α=0, cette équation est l’équation de Szegő cubique, étudiée récemment par Gérard et Grellier comme modèle mathématique d’équation non linéaire et non dispersive. Pour l’équation (α–Szegő), nous établissons le caractère bien posé et la complète intégrabilité, et étudions la dynamique des valeurs singulières des opérateurs de Hankel associés. En outre, nous montrons les propriétés suivantes pour cette équation, sur une classe de sous–variétés invariantes de dimensions finies arbitrairement grandes : si α0, il existe des trajectoires le long desquelles toutes les normes de Sobolev de régularité plus grande que ½ tendent exponentiellement vers l’infini en temps.Dans une seconde partie, nous étudions un système mixte Schrödinger–ondes sur le cylinder (x,y)∈R×T , i∂_t U+∂_xx U-|D_y |U=|U|² U,(WS)En adaptant une idée de Hani–Pausader–Tzvetkov–Visciglia, nous établissons une théorie du scattering modifiée reliant les petites solutions de cette équation et les petites solutions de l’équation de Szegő cubique. En combinant cette théorie du scattering avec un résultat récent de Gérard–Grellier, nous en déduisons l’existence de solutions globales de (WS) qui sont non bornées dans l’espace L_x² H_y^s (R×T) pour tout s>½

Strichartz estimates and the nonlinear Schrödinger-type equations

Cette thèse est consacrée à l'étude des aspects linéaires et non-linéaires des équations de type Schrödinger [ i partial_t u + |nabla|^sigma u = F, quad |nabla| = sqrt {-Delta}, quad sigma in (0, infty).] Quand $sigma = 2$, il s'agit de l'équation de Schrödinger bien connue dans de nombreux contextes physiques tels que la mécanique quantique, l'optique non-linéaire, la théorie des champs quantiques et la théorie de Hartree-Fock. Quand $sigma in (0,2) backslash {1}$, c'est l'équation Schrödinger fractionnaire, qui a été découverte par Laskin (voir par exemple cite{Laskin2000} et cite{Laskin2002}) en lien avec l'extension de l'intégrale de Feynman, des chemins quantiques de type brownien à ceux de Lévy. Cette équation apparaît également dans des modèles de vagues (voir par exemple cite{IonescuPusateri} et cite{Nguyen}). Quand $sigma = 1$, c'est l'équation des demi-ondes qui apparaît dans des modèles de vagues (voir cite{IonescuPusateri}) et dans l'effondrement gravitationnel (voir cite{ElgartSchlein}, cite{FrohlichLenzmann}). Quand $sigma = 4$, c'est l'équation Schrödinger du quatrième ordre ou biharmonique introduite par Karpman cite{Karpman} et par Karpman-Shagalov cite{KarpmanShagalov} pour prendre en compte le rôle de la dispersion du quatrième ordre dans la propagation d'un faisceau laser intense dans un milieu massif avec non-linéarité de Kerr. Cette thèse est divisée en deux parties. La première partie étudie les estimations de Strichartz pour des équations de type Schrödinger sur des variétés comprenant l'espace plat euclidien, les variétés compactes sans bord et les variétés asymptotiquement euclidiennes. Ces estimations de Strichartz sont utiles pour l'étude de l'équations dispersives non-linéaire à régularité basse. La seconde partie concerne l'étude des aspects non-linéaires tels que les caractères localement puis globalement bien posés sous l'espace d'énergie, ainsi que l'explosion de solutions peu régulières pour des équations non-linéaires de type Schrödinger. Dans le Chapitre 1, nous discutons des estimations de Strichartz pour les équations de type Schrödinger avec $sigma in (0, infty)$ sur l'espace euclidien $mathbb{R}^d$. Dans le Chapitre 2, nous prouvons des estimations de Strichartz pour les équations de type Schrödinger avec $sigma in (0, infty) backslash {1}$ sur $mathhbb{R}^d$ équipé d'une métrique lisse bornée $g$. Au Chapitre 3, nous utilisons les estimations de Strichartz prouvées au Chapitre 2 pour montrer les estimations de Strichartz pour les équations de type Schrödinger avec $sigma in (0, infty) backslash {1 }$ sur les variétés compactes sans bord. Au Chapitre 4, nous montrons des estimations de Strichartz globales pour les équations de type Schrödinger avec $sigma in (0, infty) backslash {1}$ sur les variétés asymptotiquement euclidiennes sous la condition de non-capture. Dans le Chapitre 5, nous utilisons les estimations de Strichartz données au Chapitre 1 (entre autres) pour étudier le caractère localement bien posé des équations non-linéaires de type Schrödinger avec la non-linéarité de type puissance et $sigma in (0, infty)$ posées sur $mathbb{R}^d$. Dans le Chapitre 6, nous étudions le le caractère globalement bien posé de l'équation de Schrödinger non-linéaire du quatrième ordre $sigma = 4$ défocalisante et $L^2$ critique, en considérant séparément deux cas $d = 4$ et $d geq 5$ qui correspondent respectivement à la non-linéarité algébrique et non-algébrique. Dans le Chapitre 7, nous étudions l'explosion des solutions peu régulières de l'équation de Schrödinger non-linéaire du quatrième ordre focalisante $L^2$ critique. Comme au Chapitre 6, nous considérons aussi séparément deux cas $d = 4$ et $d geq 5$.This dissertation is devoted to the study of linear and nonlinear aspects of the Schrödinger-type equations [ i partial_t u + |nabla|^sigma u = F, quad |nabla| = sqrt {-Delta}, quad sigma in (0, infty).] When $sigma = 2$, it is the well-known Schrödinger equation arising in many physical contexts such as quantum mechanics, nonlinear optics, quantum field theory and Hartree-Fock theory. When $sigma in (0,2) backslash {1}$, it is the fractional Schrödinger equation, which was discovered by Laskin (see e.g. cite{Laskin2000} and cite{Laskin2002}) owing to the extension of the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths. This equation also appears in the water waves model (see e.g. cite{IonescuPusateri} and cite{Nguyen}). When $sigma = 1$, it is the half-wave equation which arises in water waves model (see cite{IonescuPusateri}) and in gravitational collapse (see cite{ElgartSchlein}, cite{FrohlichLenzmann}). When $sigma =4$, it is the fourth-order or biharmonic Schrödinger equation introduced by Karpman cite {Karpman} and by Karpman-Shagalov cite{KarpmanShagalov} taking into account the role of small fourth-order dispersion term in the propagation of intense laser beam in a bulk medium with Kerr nonlinearity. This thesis is divided into two parts. The first part studies Strichartz estimates for Schrödinger-type equations on manifolds including the flat Euclidean space, compact manifolds without boundary and asymptotically Euclidean manifolds. These Strichartz estimates are known to be useful in the study of nonlinear dispersive equation at low regularity. The second part concerns the study of nonlinear aspects such as local well-posedness, global well-posedness below the energy space and blowup of rough solutions for nonlinear Schrödinger-type equations. In Chapter 1, we discuss Strichartz estimates for Schrödinger-type equations with $sigma in (0, infty)$ on the Euclidean space $R^d$. In Chapter 2, we derive Strichartz estimates for Schrödinger-type equations with $sigma in (0, infty) backslash {1}$ on $R^d$ equipped with a smooth bounded metric $g$.In Chapter 3, we make use of Strichartz estimates proved in Chapter 2 to show Strichartz estimates for Schrödinger-type equations with $sigma in (0, infty) backslash {1}$ on compact manifolds without boundary. In Chapter 4, we prove global in time Strichartz estimates for Schrödinger-type equations with $sigma in (0, infty) backslash {1}$ on asymptotically Euclidean manifolds under the non-trapping condition. In Chapter 5, we use Strichartz estimates given in Chapter 1 (among other things) to study the local well-posedness of the power-type nonlinear Schrödinger-type equations with $sigma in (0, infty)$ posed on $R^d$. In Chapter 6, we study the global well-posedness for the defocusing mass-critical nonlinear fourth-order Schrödinger equation $sigma =4$ below the energy space. We will consider separately two cases $d=4$ and $d geq 5$ which respectively correspond to the algebraic and non-algebraic nonlinearity. In Chapter 7, we study the blowup of rough solutions to the focusing mass-critical nonlinear fourth-order Schrödinger equation. As in Chapter 6, we also consider separately two cases $d=4$ and $d geq 5$

Mathematical analysis of Bose mixtures and related models: ground state theory and effective dynamics.

This PhD thesis contains new results on the mathematical study of Bose-Einstein condensation and the main part of it is devoted to mixtures of condensates, i.e., systems composed of multiple bosonic species in interaction. We prove the validity of effective ground state theories for mixtures in the Gross-Pitaevskii and mean-field regime. We show that the ground state energy asymptotics, in the large-N limit, is captured by the minimum of a suitable one-body functional. Moreover, we prove that in the ground state all species exhibit Bose-Einstein condensation onto the minimizer of that functional. For mixtures in the mean-field regime, we provide a rigorous justification of Bogoliubov\u2019s theory. This is done by computing the contribution to the ground state energy which is due to excited particles. We also prove a norm approximation for the ground state vector, in the Fock space norm. From the time-dependent viewpoint, we prove for the first time the validity of the effective equations that were previously known due to heuristic physical arguments, and that are confirmed by robust experimental evidence. Our results show that, for mixtures in the Gross-Pitaevskii and mean-field regime, the effective dynamics is governed by a system of non-linear Schr\uf6dinger equations, one for each species of the mixture. In the final part of the thesis we present additional results on problems and models related to the study on mixtures. We were able to derive the effective dynamics for spinor- and pseudo-spinor condensates. The equations that we obtain are precisely those of modern experiments with ultra-cold spin bosons. We also show that the mean-field model provide a time-dependent control of condensation that is very accurate for the typical duration times of experiments. A further result is the global well-posedness in the energy space of the singular Hartree equation. Last, we present new remarks on the adaptation of known techniques that one needs in order to prove the derivation of the magnetic Gross-Pitaevskii equation

Dispersive shock waves and modulation theory

There is growing physical and mathematical interest in the hydrodynamics of dissipationless/dispersive media. Since G. B. Whitham’s seminal publication fifty years ago that ushered in the mathematical study of dispersive hydrodynamics, there has been a significant body of work in this area. However, there has been no comprehensive survey of the field of dispersive hydrodynamics. Utilizing Whitham’s averaging theory as the primary mathematical tool, we review the rich mathematical developments over the past fifty years with an emphasis on physical applications. The fundamental, large scale, coherent excitation in dispersive hydrodynamic systems is an expanding, oscillatory dispersive shock wave or DSW. Both the macroscopic and microscopic properties of DSWs are analyzed in detail within the context of the universal, integrable, and foundational models for uni-directional (Korteweg–de Vries equation) and bi-directional (Nonlinear Schrödinger equation) dispersive hydrodynamics. A DSW fitting procedure that does not rely upon integrable structure yet reveals important macroscopic DSW properties is described. DSW theory is then applied to a number of physical applications: superfluids, nonlinear optics, geophysics, and fluid dynamics. Finally, we survey some of the more recent developments including non-classical DSWs, DSW interactions, DSWs in perturbed and inhomogeneous environments, and two-dimensional, oblique DSWs

A Two-Soliton with Transient Turbulent Regime for the Cubic Half-Wave Equation on the Real Line

We consider the focusing cubic half-wave equation on the real line $$i \partial_t u + |D| u = |u|^2 u, \ \ \widehat{|D|u}(\xi)=|\xi|\hat{u}(\xi), \ \ (t,x)\in \Bbb R_+\times \Bbb R.$$ We construct an asymptotic global-in-time compact two-soliton solution with arbitrarily small $L^2$-norm which exhibits the following two regimes: (i) a transient turbulent regime characterized by a dramatic and explicit growth of its $H^1$-norm on a finite time interval, followed by (ii) a saturation regime in which the $H^1$-norm remains stationary large forever in time.ERC consolidator gran