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Pré-Publication, Document De Travail Année : 2020

Almost sure scattering for the one dimensional nonlinear Schrödinger equation

Résumé

We consider the one-dimensional nonlinear Schrödinger equation with a nonlinearity of degree $p>1$. We exhibit measures on the space of initial data for which we describe the non trivial evolution by the linear Schr\"odinger flow and we show that their nonlinear evolution is absolutely continuous with respect to this linear evolution. We deduce from this precise description the global well-posedness of the equation for $p>1$ and scattering for $p>3$. To the best of our knowledge, it is the first occurence where the description of quasi-invariant measures allows to get quantitative asymptotics (here scattering properties) for the nonlinear evolution.
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Dates et versions

hal-03081927 , version 1 (18-12-2020)
hal-03081927 , version 2 (11-12-2021)
hal-03081927 , version 3 (04-04-2024)

Identifiants

  • HAL Id : hal-03081927 , version 3

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Nicolas Burq, Laurent Thomann. Almost sure scattering for the one dimensional nonlinear Schrödinger equation. 2020. ⟨hal-03081927v3⟩
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