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59
Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation
 Math. Research Letters
"... Abstract. We prove an “almost conservation law ” to obtain globalintime wellposedness for the cubic, defocussing nonlinear Schrödinger equation in Hs (Rn) when n = 2, 3 and s> 4 5, , respectively. 7 6 1. ..."
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Cited by 88 (29 self)
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Abstract. We prove an “almost conservation law ” to obtain globalintime wellposedness for the cubic, defocussing nonlinear Schrödinger equation in Hs (Rn) when n = 2, 3 and s> 4 5, , respectively. 7 6 1.
GLOBAL EXISTENCE AND SCATTERING FOR ROUGH SOLUTIONS OF A NONLINEAR SCHRÖDINGER EQUATION ON R³
, 2003
"... We prove global existence and scattering for the defocusing, cubic nonlinear Schrödinger equation in Hs (R3) for s> 4. The main new estimate in the argument is a Morawetztype inequality for the solution φ. 5 This estimate bounds ‖φ(x, t)‖L4 x,t (R3×R), whereas the wellknown Morawetztype estima ..."
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Cited by 69 (16 self)
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We prove global existence and scattering for the defocusing, cubic nonlinear Schrödinger equation in Hs (R3) for s> 4. The main new estimate in the argument is a Morawetztype inequality for the solution φ. 5 This estimate bounds ‖φ(x, t)‖L4 x,t (R3×R), whereas the wellknown Morawetztype estimate of LinStrauss controls
Resonant decompositions and the Imethod for cubic nonlinear Schrödinger
 on R 2 . Disc. Cont. Dynam. Systems A
"... Abstract. The initial value problem for the cubic defocusing nonlinear Schrödinger equation i∂tu + ∆u = u  2 u on the plane is shown to be globally wellposed for initial data in H s (R 2) provided s> 1/2. The proof relies upon an almost conserved quantity constructed using multilinear correcti ..."
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Cited by 28 (4 self)
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Abstract. The initial value problem for the cubic defocusing nonlinear Schrödinger equation i∂tu + ∆u = u  2 u on the plane is shown to be globally wellposed for initial data in H s (R 2) provided s> 1/2. The proof relies upon an almost conserved quantity constructed using multilinear correction terms. The main new difficulty is to control the contribution of resonant interactions to these correction terms. The resonant interactions are significant due to the multidimensional setting of the problem and some orthogonality issues which arise. 1.
Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm, Discrete Contin
 Dyn. Syst
"... Abstract. We study the longtime behaviour of the focusing cubic NLS on R in the Sobolev norms H s for 0 < s < 1. We obtain polynomial growthtype upper bounds on the H s norms, and also limit any orbital H s instability of the ground state to polynomial growth at worst; this is a partial anal ..."
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Cited by 14 (3 self)
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Abstract. We study the longtime behaviour of the focusing cubic NLS on R in the Sobolev norms H s for 0 < s < 1. We obtain polynomial growthtype upper bounds on the H s norms, and also limit any orbital H s instability of the ground state to polynomial growth at worst; this is a partial analogue of the H 1 orbital stability result of Weinstein [27], [26]. In the sequel to this paper we generalize this result to other nonlinear Schrödinger equations. Our arguments are based on the “Imethod ” from earlier papers [9][15] which pushes down from the energy norm, as well as an “upsidedown Imethod ” which pushes up from the L 2 norm. 1.
Global wellposedness and scattering for a class of nonlinear Schrödinger equations below the energy space
, 2006
"... We prove global wellposedness and scattering for the nonlinear Schrödinger equation with powertype nonlinearity iut + ∆u = u  p 4 4 u, < p < n n−2, u(0, x) = u0(x) ∈ Hs (Rn), n ≥ 3, below the energy space, i.e., for s < 1. In [14], J. Colliander, M. Keel, G. Staffilani, H. Takaoka, ..."
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Cited by 13 (0 self)
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We prove global wellposedness and scattering for the nonlinear Schrödinger equation with powertype nonlinearity iut + ∆u = u  p 4 4 u, < p < n n−2, u(0, x) = u0(x) ∈ Hs (Rn), n ≥ 3, below the energy space, i.e., for s < 1. In [14], J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao established polynomial growth of the H s xnorm of the solution, and hence global wellposedness for initial data in H s x, provided s is sufficiently close to 1. However, their bounds are insufficient to yield scattering. In this paper, we use the a priori interaction Morawetz inequality to show that scattering holds in H s (R n) whenever s is larger than some value 0 < s0(n, p) < 1.
DECAY ESTIMATES AND SMOOTHNESS FOR SOLUTIONS OF THE DISPERSION MANAGED NONLINEAR SCHRÖDINGER EQUATION
"... Abstract. We study the decay and smoothness of solutions of the dispersion managed nonlinear Schrödinger equation in the critical case of zero residual dispersion. Using new xspace versions of bilinear Strichartz estimates, we show that the solutions are not only smooth, but also fast decaying. 1. ..."
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Cited by 12 (6 self)
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Abstract. We study the decay and smoothness of solutions of the dispersion managed nonlinear Schrödinger equation in the critical case of zero residual dispersion. Using new xspace versions of bilinear Strichartz estimates, we show that the solutions are not only smooth, but also fast decaying. 1.
Global wellposedness for a periodic nonlinear Schrödinger equation
 in 1D and 2D, arXiv: math.AP/0602660v1
, 2006
"... Abstract. The initial value problem for the L 2 critical semilinear Schrödinger equation with periodic boundary data is considered. We show that the problem is globally well posed in H s (T d), for s> 4/9 and s> 2/3 in 1D and 2D respectively, confirming in 2D a statement of Bourgain in [3]. We ..."
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Cited by 11 (2 self)
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Abstract. The initial value problem for the L 2 critical semilinear Schrödinger equation with periodic boundary data is considered. We show that the problem is globally well posed in H s (T d), for s> 4/9 and s> 2/3 in 1D and 2D respectively, confirming in 2D a statement of Bourgain in [3]. We use the “Imethod”. This method allows one to introduce a modification of the energy functional that is well defined for initial data below the H 1 (T d) threshold. The main ingredient in the proof is a “refinement ” of the Strichartz’s estimates that hold true for solutions defined on the rescaled space, T d λ = R d /λZ d, d = 1, 2.
Polynomial upper bounds for the instability of the nonlinear Schrödinger equation below the energy norm
 Commun. Pure Appl. Anal
"... Abstract. We continue the study (initiated in [18]) of the orbital stability of the ground state cylinder for focussing nonlinear Schrödinger equations in the H s (R n) norm for 1 − ε < s < 1, for small ε. In the L 2subcritical case we obtain a polynomial bound for the time required to move ..."
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Cited by 9 (3 self)
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Abstract. We continue the study (initiated in [18]) of the orbital stability of the ground state cylinder for focussing nonlinear Schrödinger equations in the H s (R n) norm for 1 − ε < s < 1, for small ε. In the L 2subcritical case we obtain a polynomial bound for the time required to move away from the ground state cylinder. If one is only in the H 1subcritical case then we cannot show this, but for defocussing equations we obtain global wellposedness and polynomial growth of H s norms for s sufficiently close to 1. 1.
ON THE CAUCHY PROBLEM FOR THE DERIVATIVE NONLINEAR SCHRÖDINGER EQUATION WITH PERIODIC BOUNDARY CONDITION
, 2005
"... ABSTRACT. It is shown that the Cauchy problem associated to the derivative nonlinear Schrödinger equation ∂tu − i∂2 xu = λ∂x(u  2u) is locally wellposed for initial data u(0) ∈ Hs (T), if s ≥ 1 and λ is real. The proof is based on an adaption of the gauge 2 transformation to periodic functions a ..."
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Cited by 9 (1 self)
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ABSTRACT. It is shown that the Cauchy problem associated to the derivative nonlinear Schrödinger equation ∂tu − i∂2 xu = λ∂x(u  2u) is locally wellposed for initial data u(0) ∈ Hs (T), if s ≥ 1 and λ is real. The proof is based on an adaption of the gauge 2 transformation to periodic functions and sharp multilinear estimates for the gauge equivalent equation in Fourier restriction norm spaces. By the use of a conservation law, the problem is shown to be globally wellposed for s ≥ 1 and data which is small in L2. 1.