On the one class of hyperbolic systems.
On the real analyticity of the scattering operator for the Hartree equation
We study the real analyticity of the scattering operator for the Hartree equation . To this end, we exploit interior and exterior cut-off in time and space, together with a compactness argument to overcome difficulties which arise from absence of good properties as for the Klein-Gordon equation, such as the finite speed of propagation and ideal time decay estimate. Additionally, the method in this paper allows us to simplify the proof of analyticity of the scattering operator for the nonlinear...
On the rigidity of minimal mass solutions to the focusing mass-critical NLS for rough initial data.
On the scattering in Gevrey classes for the subcritical Hartree and Schrödinger equations
On the Virasoro structure of symmetry algebras of nonlinear partial differential equations.
On Threshold Eigenvalues and Resonances for the Linearized NLS Equation
We prove the instability of threshold resonances and eigenvalues of the linearized NLS operator. We compute the asymptotic approximations of the eigenvalues appearing from the endpoint singularities in terms of the perturbations applied to the original NLS equation. Our method involves such techniques as the Birman-Schwinger principle and the Feshbach map.
On two-dimensional finite-gap potential Schrödinger and Dirac operators with singular spectral curves.
Optical vortices in dispersive nonlinear Kerr-type media.
Optimal feedback control of Ginzburg-Landau equation for superconductivity via differential inclusion
Slightly below the transition temperatures, the behavior of superconducting materials is governed by the Ginzburg-Landau (GL) equation which characterizes the dynamical interaction of the density of superconducting electron pairs and the exited electromagnetic potential. In this paper, an optimal control problem of the strength of external magnetic field for one-dimensional thin film superconductors with respect to a convex criterion functional is considered. It is formulated as a nonlinear coefficient...
Orbital stability of standing waves for a class of Schrödinger equations with unbounded potential.
Periodic and solitary wave solutions of two component Zakharov-Yajima-Oikawa system, using Madelung's approach.
Periodic solutions of nonlinear Schrödinger equations
Persistence of solutions to nonlinear evolution equations in weighted Sobolev spaces.
Perturbations of the harmonic map equation
We consider perturbations of the harmonic map equation in the case where the source and target manifolds are closed riemannian manifolds and the latter is in addition of nonpositive sectional curvature. For any semilinear and, under some extra conditions, quasilinear perturbation, the space of classical solutions within a homotopy class is proved to be compact. For generic perturbations the set of solutions is finite and we present a count of this set. An important ingredient for our analysis is...
Plane wave stability of some conservative schemes for the cubic Schrödinger equation
The plane wave stability properties of the conservative schemes of Besse [SIAM J. Numer. Anal.42 (2004) 934–952] and Fei et al. [Appl. Math. Comput.71 (1995) 165–177] for the cubic Schrödinger equation are analysed. Although the two methods possess many of the same conservation properties, we show that their stability behaviour is very different. An energy preserving generalisation of the Fei method with improved stability is presented.
Positive solutions for nonlinear Schrödinger equations with deepening potential well
Problèmes d’évolution liés à l’énergie de Ginzburg-Landau
Propagation et réflexion des singularités pour l'équation de Schrödinger non linéaire
Nous construisons un calcul paradifférentiel adapté à l'équation de Schrödinger qui nous permet de montrer un théorème de propagation des singularités pour l'équation de Schrödinger non linéaire en adaptant la méthode de Bony. Nous construisons également la version tangentielle du calcul précédent qui nous permet de montrer un théorème de réflexion transverse des singularités pour l'équation de Schrödinger non linéaire. Nous utilisons alors ce théorème pour calculer l'opérateur...
Propagation of uniform Gevrey regularity of solutions to evolution equations
We investigate the propagation of the uniform spatial Gevrey , σ ≥ 1, regularity for t → +∞ of solutions to evolution equations like generalizations of the Euler equation and the semilinear Schrödinger equation with polynomial nonlinearities. The proofs are based on direct iterative arguments and nonlinear Gevrey estimates.
-deformed bi-local fields. II.