The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities

By Terence Tao, Monica Visan, Xiaoyi Zhang. 2005

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We undertake a comprehensive study of the nonlinear Schrödinger equation where u(t, x) is a complex-valued function in spacetime , λ1 and λ2 are nonzero real constants, and . We address questions related to local and global well-posedness, finite time blowup, and asymptotic behaviour. Scattering is considered both in the energy space H 1(ℝ n ) and in the pseudoconformal space Σ := {f ∈ H 1(ℝ n ); xf ∈ L 2(ℝ n )}. Of particular interest is the case when both nonlinearities are defocusing and correspond to the -critical, respectively -critical NLS, that is, λ1, λ2 > 0 and , . The results at the endpoint are conditional on a conjectured global existence and spacetime estimate for the -critical nonlinear Schrödinger equation, which has been verified in dimensions n ≥ 2 for radial data in Tao et al. (Tao et al. to appear a,b) and Killip et al. (preprint). As an off-shoot of our analysis, we also obtain a new, simpler proof of scattering in for solutions to the nonlinear Schrödinger equation with , which was first obtained by Ginibre and Velo (1985).

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/36a9087e3e956f0592eb10e80e08e0d105e9eb96

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Citations in the corpus

Almost sure global well-posedness for the energy-critical defocusing nonlinear wave equation on $\mathbb{R}^d$, $d=4$ and $5$ (2014, neutral): “erministic energy-critical NLW and NLS. Moreover, perturbation theory has been previously used to prove global well-posedness for various other equations in the deterministic setting. See for example [22, 37, 63, 38]. In [63], Tao, Vişan, 1 The argument for obtaining almost sure global existence whe”
The cubic nonlinear Schr\"odinger equation in two dimensions with radial data (2007, background): “To this one may add stability of well-posedness under perturbations of the equation; see [69].”
Semilinear Schrödinger flows on hyperbolic spaces: scattering in H1 (2008, background): “1 (b) is in sharp contrast with its Euclidean analogue: on Euclidean spaces scattering to linear solutions is only known for exponents σ ∈ (2/d, 2/(d− 2)), see [23, 35, 20, 19, 43], and also for σ = 2/(d− 2), see [11, 21, 40, 37, 45].”
Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1 (2010, background): “Deﬁne the function The stability lemma from [91] implies that A ( m ) is a continuous function of m , which implies that { m : A ( m ) = ∞ } is a closed set.”
Normalized ground states for the NLS equation with combined nonlinearities (2018, neutral): “Differently to what happen in [49], we don’t make any assumption on the weighted mass current of the initial datum in order to prove finite time blow-up.”
Scattering threshold for the focusing nonlinear Klein–Gordon equation (2010, background): “Note that it is not a big trouble in the Schrödinger case (see [47]), because the same Strichartz estimate is used both for higher and lower powers.”
Normalized solutions for a Schrödinger equation with critical growth in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{documen (2021, neutral): “For the reader interested in normalized solutions for the Schrödinger equations, we would also like to refer [9,11,12,17,25,29,31,34,38,39,41] and references therein.”
Orbital stability of ground states for a Sobolev critical Schrödinger equation (2020, background): “Concerning the case of NLS with mixed nonlinearities let us quote [1, 2, 15,16,26,29,30,36].”