Singularly perturbed elliptic systems

We study the existence and concentration behavior of positive solutions for a class of Hamiltonian systems (two coupled nonlinear stationary Schrödinger equations). Combining the Legendre–Fenchel transformation with mountain pass theorem, we prove the existence of a family of positive solutions concentrating at a point in the limit, where related functionals realize their minimum energy. In some cases, the location of the concentration point is given explicitly in terms of the potential functions of the stationary Schrödinger equations.     

Localizing estimates of the support of solutions of some nonlinear Schrödinger equations – The stationary case

The main goal of this paper is to study the nature of the support of the solution of suitable nonlinear Schrödinger equations, mainly the compactness of the support and its spatial localization. This question touches the very foundations underlying the derivation of the Schrödinger equation, since it is well-known a solution of a linear Schrödinger equation perturbed by a regular potential never vanishes on a set of positive measure. A fact, which reflects the impossibility of locating the particle. Here we shall prove that if the perturbation involves suitable singular nonlinear terms then the support of the solution is a compact set, and so any estimate on its spatial localization implies very rich information on places not accessible by the particle. Our results are obtained by the application of certain energy methods which connect the compactness of the support with the local vanishing of a suitable “energy function” which satisfies a nonlinear differential inequality with an exponent less than one. The results improve and extend a previous short presentation by the authors published in 2006.     

Kato-type estimates for NLS equation in a scalar field and unique solvability of NKGS system in energy space

We consider Schrödinger equation in R²⁺¹${\mathbb{R}}^{2+1}$ with nonlinear scalar potential. The potentials are time-independent or determined as solutions to inhomogeneous wave equations. By constructing a modified propagator, we derive Kato-type smoothing estimates for the nonlinear Schrödinger (NLS) equation. With the help of these results, we prove the unique solvability of the nonlinear Klein–Gordon–Schrödinger (NKGS) system for all time in the energy space.     

On the role of quadratic oscillations in nonlinear Schrödinger equations

We consider a nonlinear semi-classical Schrödinger equation for which it is known that quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. If the initial data is an energy bounded sequence, we prove that the nonlinear term has an effect at leading order only if the initial data have quadratic oscillations; the proof relies on a linearizability condition (which can be expressed in terms of Wigner measures). When the initial data is a sum of such quadratic oscillations, we prove that the associate solution is the superposition of the nonlinear evolution of each of them, up to a small remainder term. In an appendix, we transpose those results to the case of the nonlinear Schrödinger equation with harmonic potential.     

Inhomogeneous Neumann initial–boundary value problem for the nonlinear Schrödinger equation

We consider the inhomogeneous Neumann initial–boundary value problem for the nonlinear Schrödinger equation, formulated on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial–boundary value problem and the asymptotic behavior of solutions for large time.     

Criticality of viscous Hamilton–Jacobi equations and stochastic ergodic control

This paper deals with nonlinear additive eigenvalue problems for viscous Hamilton–Jacobi equations which appear in stochastic ergodic control. Certain qualitative properties of principal eigenvalues and associated eigenfunctions are studied. Such analysis plays a key role in studying the recurrence and transience of feedback diffusions for the corresponding stochastic control problems. Our results can be regarded as a nonlinear extension of the criticality theory for Schrödinger operators with decaying potentials.     

Instabilities for supercritical Schrödinger equations in analytic manifolds

In this paper we consider supercritical nonlinear Schrödinger equations in an analytic Riemannian manifold (Md,g), where the metric g is analytic. Using an analytic WKB method, we are able to construct an Ansatz for the semiclassical equation for times independent of the small parameter. These approximate solutions will help to show two different types of instabilities. The first is in the energy space, and the second is an immediate loss of regularity in higher Sobolev norms.     

Multiple existence of sign changing solutions for coupled nonlinear Schrödinger equations

In this paper, we consider the multiple existence of sign changing solutions of coupled nonlinear Schrödinger equations

(P)     

Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations

In this article we study global-in-time Strichartz estimates for the Schrödinger evolution corresponding to long-range perturbations of the Euclidean Laplacian. This is a natural continuation of a recent article [D. Tataru, Parametrices and dispersive estimates for Schrödinger operators with variable coefficients, Amer. J. Math. 130 (2008) 571-634] of the third author, where it is proved that local smoothing estimates imply Strichartz estimates. By [D. Tataru, Parametrices and dispersive estimates for Schrödinger operators with variable coefficients, Amer. J. Math. 130 (2008) 571-634] the local smoothing estimates are known to hold for small perturbations of the Laplacian. Here we consider the case of large perturbations in three increasingly favorable scenarios: (i) without non-trapping assumptions we prove estimates outside a compact set modulo a lower order spatially localized error term, (ii) with non-trapping assumptions we prove global estimates modulo a lower order spatially localized error term, and (iii) for time independent operators with no resonance or eigenvalue at the bottom of the spectrum we prove global estimates for the projection onto the continuous spectrum.     

Coupled nonlinear Schrödinger systems with potentials

Coupled nonlinear Schrödinger systems describe some physical phenomena such as the propagation in birefringent optical fibers, Kerr-like photorefractive media in optics and Bose-Einstein condensates. In this paper, we study the existence of concentrating solutions of a singularly perturbed coupled nonlinear Schrödinger system, in presence of potentials. We show how the location of the concentration points depends strictly on the potentials.     

Multiple solutions for quasilinear elliptic equations with a finite potential well

We consider multiplicity of solutions for a class of quasilinear problems which has received considerable attention in the past, including the so called Modified Nonlinear Schrödinger Equations. By combining a new variational approach via q-Laplacian regularization and the compactness arguments from [4] we establish infinitely many bound state solutions for the quasilinear Schrödinger type equations, extending the earlier work of [4] for semilinear equations.     

On the blow up phenomenon of the critical nonlinear Schrödinger equation

In this paper we consider the blow up phenomenon of critical nonlinear Schrödinger equations in dimension 1D and 2D. We define the minimal mass as the L² norm necessary to ignite a wave collapse and we stress its role in the blow up mechanism. Asymptotic compactness properties and L²-concentration are proved. The proof relies on linear and nonlinear profile decompositions.     

Variational iteration method for solving cubic nonlinear Schrödinger equation

The variational iteration method is applied to solve the cubic nonlinear Schrödinger (CNLS) equation in one and two space variables. In both cases, we will reduce the CNLS equation to a coupled system of nonlinear equations. Numerical experiments are made to verify the efficiency of the method. Comparison with the theoretical solution shows that the variational iteration method is of high accuracy.     

The complexity of wave propagation of the nonlinear Schrödinger equation with weak periodic external field

A nonlinear Schrödinger equation with external field is obtained for nonlinear optics in a non-homogeneous medium. On the basis of the Melnikov function and KAM theory, the complexity of the wave propagation of the equation is studied in the case of weak periodic external field.     

The inverse scattering problem for Schrödinger and Klein–Gordon equations with a nonlocal nonlinearity

We study the inverse scattering problem for the nonlinear Schrödinger equation and for the nonlinear Klein–Gordon equation with the generalized Hartree type nonlinearity. We reconstruct the nonlinearity from knowledge of the scattering operator, which improves the known results.     

Solitary waves and a stability analysis of an equation of short and long dispersive waves

A universal model for the interaction of long nonlinear waves and packets of short waves with long linear carrier waves is given by a system in which an equation of Korteweg–de Vries (KdV) type is coupled to an equation of nonlinear Schrödinger (NLS) type. The system has solutions of steady form in which one component is like a solitary-wave solution of the KdV equation and the other component is like a ground-state solution of the NLS equation. We study the stability of solitary-wave solutions to an equation of short and long waves by using variational methods based on the use of energy–momentum functionals and the techniques of convexity type. We use the concentration compactness method to prove the existence of solitary waves. We prove that the stability of solitary waves is determined by the convexity or concavity of a function of the wave speed.     

Weighted energy decay for 3D Klein-Gordon equation

We obtain a dispersive long-time decay in weighted energy norms for solutions of the 3D Klein-Gordon equation with generic potential. The decay extends the results obtained by Jensen and Kato for the 3D Schrödinger equation. For the proof we modify the spectral approach of Jensen and Kato to make it applicable to relativistic equations.     

Ideas in the theory of random media

These lectures discuss the ideas of localization, intermittency, and random fluctuations in the theory of random media. These ideas are compared and contrasted with the older approach based on averaging. Within this framework, the topics discussed include: Anderson localization, turbulent diffusion and flows, periodic Schrödinger operators and averaging theory, longwave oscillations of elastic random media, stochastic differential equations, the spectral theory of Hamiltonians with (an infinite sequence of) wells, random Schrödinger operators, electrons in a random homogeneous field, influence of localization effects on the propagation of elastic waves, the Lyapunov spectrum (Lyapunov exponents), the Furstenberg and Oseledec theorems for ann-tuple of identically distributed unimodular matrices and their relation with the spectral theory of random Schrödinger or string operators, Rossby waves, averaging on random Schrödinger operators, percolation mechanisms, the moments method in the theory of sequences of random variables, the evolution of a magnetic field in the turbulent flow of a conducting fluid or plasma (the so-called kinematical dynamo problem), heat transmission in a randomly flowing fluid.     

Persistence of solutions to higher order nonlinear Schrödinger equation

Applying an Abstract Interpolation Lemma, we showed persistence of solutions of the initial value problem to higher order nonlinear Schrödinger equation, also called Airy-Schrödinger equation, in weighted Sobolev spaces X2,θ, for θ∈[0,1].     

Interaction of modulated pulses in the nonlinear Schrödinger equation with periodic potential

We consider a cubic nonlinear Schrödinger equation with periodic potential. In a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several Bloch bands is studied. The notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic interaction of these pulses.