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On nonlocal variational problems with lack of compactness related to nonlinear optics
, 2010
"... Abstract. We give a simple proof of existence of solutions of the dispersion management and diffraction management equations for zero average dispersion, respectively diffraction. These solutions are found as maximizers of nonlinear and nonlocal variational problems which are invariant under a lar ..."
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Abstract. We give a simple proof of existence of solutions of the dispersion management and diffraction management equations for zero average dispersion, respectively diffraction. These solutions are found as maximizers of nonlinear and nonlocal variational problems which are invariant under a large noncompact group. Our proof of existence of maximizer is rather direct and avoids the use of Lions ’ concentration compactness argument or Ekeland’s variational principle. 1.
SUPEREXPONENTIAL DECAY OF DIFFRACTION MANAGED SOLITONS
"... Abstract. This is the second part of a series of papers where we develop rigorous decay estimates for breather solutions of an averaged version of the nonlinear Schrödinger equation. In this part we study the diffraction managed discrete nonlinear Schrödinger equation, an equation which describes c ..."
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Abstract. This is the second part of a series of papers where we develop rigorous decay estimates for breather solutions of an averaged version of the nonlinear Schrödinger equation. In this part we study the diffraction managed discrete nonlinear Schrödinger equation, an equation which describes coupled waveguide arrays with periodic diffraction management geometries. We show that, for vanishing average diffraction, all solutions of the nonlinear and nonlocal diffraction management equation decay superexponentially. As a byproduct of our method, we also have a simple proof of existence of diffraction managed solitons in the case of vanishing average diffraction. 1.
Dispersive estimates for the Schrödinger equation

, 2010
"... In this document we explore the issue of L1 → L ∞ estimates for the solution operator of the linear Schrödinger equation, iut −∆u+ V u = 0 u(x, 0) = f(x) ∈ S(Rn). We focus particularly on the five and seven dimensional cases. We prove that the solution operator precomposed with projection onto t ..."
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In this document we explore the issue of L1 → L ∞ estimates for the solution operator of the linear Schrödinger equation, iut −∆u+ V u = 0 u(x, 0) = f(x) ∈ S(Rn). We focus particularly on the five and seven dimensional cases. We prove that the solution operator precomposed with projection onto the absolutely continuous spectrum of H = − ∆ + V satisfies the following estimate ‖eitHPac(H)‖L1→L ∞. t−n2 under certain conditions on the potential V. Specifically, in Chapter 1 we prove the dispersive estimate is satisfied with optimal assumptions on smoothness, that is V ∈ C n−32 (Rn) for n = 5, 7 assuming that zero is regular, V (x) . 〈x〉−β and ∇jV (x) . 〈x〉−α, 1 ≤ j ≤ n−32 for some β> 3n+52 and α> 3, 8 in dimensions five and seven respectively. In Chapter 2 we show that for the five dimensional result one only needs that V (x) . 〈x〉−4 − in addition to the assumptions on the derivative and regularity of the potential. This more than cuts in half the required decay rate in the first chapter. Finally in Chapter 3 we consider a problem involving the nonlinear Schrödinger equation. In particular, we consider the following equation that arises in fiber optic communication systems,
On the continuous resonant equation for NLS: I. Deterministic analysis. Preprint: arXiv:1501.03760
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Wellposedness and averaging of NLS with timeperiodic dispersion management
 Adv. Diff. Eq
, 2013
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ANALYTICITY OF EXTREMALS TO THE AIRY STRICHARTZ INEQUALITY
"... Abstract. We prove that there exists an extremal function to the Airy Strichartz inequality ‖e −t∂3 xf ‖ L8 t,x (R×R) ≤ C‖f‖L2 (R), by using the linear profile decomposition. Furthermore we show that, if f is an extremal, then f is exponentially decaying in the Fourier space and so f can be extende ..."
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Abstract. We prove that there exists an extremal function to the Airy Strichartz inequality ‖e −t∂3 xf ‖ L8 t,x (R×R) ≤ C‖f‖L2 (R), by using the linear profile decomposition. Furthermore we show that, if f is an extremal, then f is exponentially decaying in the Fourier space and so f can be extended to be an entire function on the complex domain. 1.