Nonlinear fractional Schrödinger equations in one dimension
By A. Ionescu, F. Pusateri. 2012
216 citations recorded by the source corpus.
Abstract We consider the question of global existence of small, smooth, and localized solutions of a certain fractional semilinear cubic NLS in one dimension, i ∂ t u − Λ u = c 0 | u | 2 u + c 1 u 3 + c 2 u u ¯ 2 + c 3 u ¯ 3 , Λ = Λ ( ∂ x ) = | ∂ x | 1 2 , where c 0 ∈ R and c 1 , c 2 , c 3 ∈ C . This model is motivated by the two-dimensional water wave equation, which has a somewhat similar structure in the Eulerian formulation, in the case of irrotational flows. We show that one cannot expect linear scattering, even in this simplified model. More precisely, we identify a suitable nonlinear logarithmic correction, and prove global existence and modified scattering of solutions.
Subfields: Mathematical Physics, Modeling and Simulation
View Paper: https://www.semanticscholar.org/paper/3537c2722dadc2a3e19dd36d6e64931f5ee0de90
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In Mathematical Physics
- A. Ionescu
- F. Pusateri
- Michael Taylor
- Zihua Guo
- Carlos E. Kenig
- Luc Molinet
- Vittoria Pierfelice
- Jean-Philippe Anker
In Modeling and Simulation
Citations in the corpus
- Global solutions for the gravity water waves system in 2d (2013, methodology): “RU D. IONESCU AND FABIO PUSATERI The proof of (4.32) constitutes the heart of the proof for the decay in Eulerian coordinates, and is performed in section 6, using a construction similar to our paper [IPu12]. Using (4.31) and an estimate similar to (4.20), we can also obtain the following: Corollary”
- Asymptotic stability of solitons for mKdV (2015, unknown): “A similar approach was also successfully employed in the proofs of global regularity and modified scattering for 2d gravity [23, 24, 25] and capillary [26, 27] water waves, and in other higher dimensional dispersive models [29, 49].”