arXiv subjects
Publications and source records attributed to Laurent Thomann.
We consider the nonlinear Schrödinger equation on a compact manifold near an elliptic periodic geodesic. Using a geometric optics construction, we construct quasimodes to a nonlinear stationary problem which are highly localized near the periodic geodesic. We show the nonlinear Schrödinger evolution of such a quasimode remains localized near the geodesic, at least for short times.
In this paper we construct a Gibbs measure for the derivative Schrödinger equation on the circle. The construction uses some renormalisations of Gaussian series and Wiener chaos estimates, ideas which have already been used by the second author in a work on the Benjamin-Ono equation.
In this article, we first present the construction of Gibbs measures associated to nonlinear Schrödinger equations with harmonic potential. Then we show that the corresponding Cauchy problem is globally well-posed for rough initial conditions in a statistical set (the support of the measures). Finally, we prove that the Gibbs measures are indeed invariant by the flow of the equation. As a byproduct of our analysis, we give a global well-posedness and scattering result for the $L^2$ critical and super-critical NLS (without harmonic potential).
We prove the equivalence between the smoothing effect for a Schrödinger operator and the decay of the associate spectral projectors. We give two applications to the Schrödinger operator in dimension one.
In this paper we consider the Schrödinger equation with power-like nonlinearity and confining potential or without potential. This equation is known to be well-posed with data in a Sobolev space $\H^{s}$ if $s$ is large enough and strongly ill-posed is $s$ is below some critical threshold $s_{c}$. Here we use the randomisation method of the inital conditions, introduced by N. Burq-N. Tzvetkov and we are able to show that the equation admits strong solutions for data in $\H^{s}$ for some $s<s_{c}$. In the appendix we prove the equivalence between the smoothing effect for a Schrödinger operator with confining potential and the decay of the associate spectral projectors.
In this paper we consider a Schrodinger equation on the circle with a quadratic nonlinearity. Thanks to an explicit computation of the first Picard iterate, we give a precision on the dynamic of the solution, whose existence was proved by C. E. Kenig, G. Ponce and L. Vega. We also show that the equation is well-posed in a space based on Lp norms in frequencies.
In this paper we consider supercritical nonlinear Schrödinger equations in an analytic Riemannian manifold $(M^d,g)$, 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.
Using variational methods, we construct approximate solutions for the Gross-Pitaevski equation which concentrate on circles in $\R^3$. These solutions will help to show that the $L^2$ flow is unstable for the usual topology and for the projective distance.
In this paper we are interested in constructing WKB approximations for the non linear cubic Schrödinger equation on a Riemannian surface which has a stable geodesic. These approximate solutions will lead to some instability properties of the equation.