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SHARP GLOBAL WELLPOSEDNESS FOR KDV AND MODIFIED KDV ON R AND T
"... 1.1. GWP below the conservation law 707 1.2. The operator I and almost conserved quantities 708 ..."
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1.1. GWP below the conservation law 707 1.2. The operator I and almost conserved quantities 708
GLOBAL WELLPOSEDNESS FOR THE L 2CRITICAL NONLINEAR SCHRÖDINGER EQUATION IN HIGHER DIMENSIONS
, 2006
"... Abstract. The initial value problem for the L 2 critical semilinear Schrödinger equation in R n, n ≥ 3 is considered. We show that the problem is globally well posed in H s (R n) when 1> s> √ 7−1 3 for n = 3, and when 1> s> −(n−2)+ (n−2) 2 +8(n−2) for n ≥ 4. We ..."
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Abstract. The initial value problem for the L 2 critical semilinear Schrödinger equation in R n, n ≥ 3 is considered. We show that the problem is globally well posed in H s (R n) when 1> s> √ 7−1 3 for n = 3, and when 1> s> −(n−2)+ (n−2) 2 +8(n−2) for n ≥ 4. We
Global rough solutions to the critical generalized KdV
, 908
"... We prove that the initial value problem (IVP) for the critical generalized KdV equation ut +uxxx+(u 5)x = 0 on the real line is globally wellposed in H s (R) in s> 3/5 with the appropriate smallness assumption on the initial data. 1 ..."
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We prove that the initial value problem (IVP) for the critical generalized KdV equation ut +uxxx+(u 5)x = 0 on the real line is globally wellposed in H s (R) in s> 3/5 with the appropriate smallness assumption on the initial data. 1
THE LOW REGULARITY GLOBAL SOLUTIONS FOR THE CRITICAL GENERALIZED KDV EQUATION
, 908
"... Abstract. In this paper, we establish the global wellposedness for the critical generalized KdV equation with the low regularity data. To be precise, we show that a unique and global solution exists for initial data in the Sobolev space Hs`R ´ with s> 1. Of course, we require that the mass is st ..."
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Abstract. In this paper, we establish the global wellposedness for the critical generalized KdV equation with the low regularity data. To be precise, we show that a unique and global solution exists for initial data in the Sobolev space Hs`R ´ with s> 1. Of course, we require that the mass is strictly less than 2 that of the ground state in the focusing case. This follows from “Imethod”, which was introduced by Colliander, Keel, Staffilani, Takaoka and Tao, and improves the result in [20]. 1.
Global wellposedness for Schrödinger equation with derivative in H 2
 R), J. Diff. Eq
, 2011
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THE SEMICLASSICAL MODIFIED NONLINEAR SCHRÖDINGER EQUATION I: MODULATION THEORY AND SPECTRAL ANALYSIS
, 2007
"... Abstract. We study an integrable modification of the focusing nonlinear Schrödinger equation from the point of view of semiclassical asymptotics. In particular, (i) we establish several important consequences of the mixedtype limiting quasilinear system including the existence of maps that embed th ..."
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Abstract. We study an integrable modification of the focusing nonlinear Schrödinger equation from the point of view of semiclassical asymptotics. In particular, (i) we establish several important consequences of the mixedtype limiting quasilinear system including the existence of maps that embed the limiting forms of both the focusing and defocusing nonlinear Schrödinger equations into the framework of a single limiting system for the modified equation, (ii) we obtain bounds for the location of discrete spectrum for the associated spectral problem that are particularly suited to the semiclassical limit and that generalize known results for the spectrum of the nonselfadjoint ZakharovShabat spectral problem, and (iii) we present a multiparameter family of initial data for which we solve the associated spectral problem in terms of special functions for all values of the semiclassical scaling parameter. We view our results as part of a broader project to analyze the semiclassical limit of the modified nonlinear Schrödinger equation via the noncommutative steepest descent procedure of Deift and Zhou, and we also present a selfcontained development of a RiemannHilbert problem of inverse scattering that differs from those given in the literature and that is welladapted to semiclassical asymptotics. 1.
Ground states for the higher order dispersion managed NLS equation in the absence of average dispersion
 J. Differential Equations
"... Abstract The problem of existence of ground states in higher order dispersion managed NLS equation is considered. The ground states are stationary solutions to dispersive equations with nonlocal nonlinearity which arise as averaging approximations in the context of strong dispersion management in o ..."
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Abstract The problem of existence of ground states in higher order dispersion managed NLS equation is considered. The ground states are stationary solutions to dispersive equations with nonlocal nonlinearity which arise as averaging approximations in the context of strong dispersion management in optical communications. The main result of this note states that the averaged equation possesses ground state solutions in the practically and conceptually important case of the vanishing residual dispersions.
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.
MASS CONCENTRATION PHENOMENON FOR THE QUINTIC NONLINEAR SCHRÖDINGER EQUATION IN 1D
, 2006
"... Abstract. We consider the L 2critical quintic focusing nonlinear Schrödinger equation (NLS) on R. It is well known that H 1 solutions of the aforementioned equation blowup in finite time. In higher dimensions, for H 1 spherically symmetric blowup solutions of the L 2critical focusing NLS, there ..."
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Abstract. We consider the L 2critical quintic focusing nonlinear Schrödinger equation (NLS) on R. It is well known that H 1 solutions of the aforementioned equation blowup in finite time. In higher dimensions, for H 1 spherically symmetric blowup solutions of the L 2critical focusing NLS, there is a minimal amount of concentration of the L 2norm (the mass of the ground state) at the origin. In this paper we prove the existence of a similar phenomenon for the 1d case and rougher initial data, (u0 ∈ H s, s < 1), without any additional assumption. 1.