The cubic nonlinear Schrödinger equation in two dimensions with radial data

Killip, Rowan; Tao, Terence; Vişan, Monica

doi:10.4171/jems/180

Cited by 201 publications

(351 citation statements)

References 77 publications

(162 reference statements)

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“…Remark 1.24. A closely related result in the spherically symmetric case was established in [Killip et al 2008b, Corollary 1.12], in which it was shown that any blowup of a spherically symmetric Strichartz class solution in two dimensions must concentrate an amount of mass at least equal to the ground state M(Q); the same result in higher dimensions follows by the same argument together with the results in [Killip et al 2007]. Indeed, we will use the results in [Killip et al 2007] to establish the spherically symmetric case of this theorem.…”

Section: Definition 115 (Semi-strong and Semi-strichartz Solutions)mentioning

confidence: 99%

Global existence and uniqueness results for weak solutions of the focusing mass-critical nonlinear Schrödinger equation

Tao

2009

APDE

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We consider the focusing mass-critical NLS iu t + u = −|u| 4/d u in high dimensions d ≥ 4, with initial data u(0) = u 0 having finite massIt is well known that this problem admits unique (but not global) strong solutions in the Strichartz class, and also admits global (but not unique) weak solutions in L ∞ t L 2 x . In this paper we introduce an intermediate class of solution, which we call a semi-Strichartz class solution, for which one does have global existence and uniqueness in dimensions d ≥ 4. In dimensions d ≥ 5 and assuming spherical symmetry, we also show the equivalence of the Strichartz class and the strong solution class (and also of the semi-Strichartz class and the semi-strong solution class), thus establishing unconditional uniqueness results in the strong and semi-strong classes. With these assumptions we also characterise these solutions in terms of the continuity properties of the mass function t → M(u(t)).

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Section: Definition 115 (Semi-strong and Semi-strichartz Solutions)mentioning

confidence: 99%

Global existence and uniqueness results for weak solutions of the focusing mass-critical nonlinear Schrödinger equation

Tao

2009

APDE

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“…The scattering for radially symmetric solutions with L 2 initial data was recently established in [20] for 2d, and in [22] it Q(x), or a blow up solution which is a pseudoconformal image of e it Q(x) (a "self-similar solution"), or a globally defined solution with quadratically decaying in time L 4/d+2 norm which implies scattering as t → ±∞.…”

Section: Furthermore This Nls Equation Enjoys Several Invariances Imentioning

confidence: 99%

Threshold solutions for the focusing 3D cubic Schrödinger equation

Duyckaerts¹,

Roudenko²

2010

Rev. Mat. Iberoam.

143

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We study the focusing 3d cubic NLS equation with H 1 data at the mass-energy threshold, namely, when. In earlier works of Holmer-Roudenko and Duyckaerts-Holmer-Roudenko, the behavior of solutions (i.e., scattering and blow up in finite time) was classified when. In this paper, we first exhibit 3 special solutions: e it Q and Q ± , where Q is the ground state, Q ± exponentially approach the ground state solution in the positive time direction, Q + has finite time blow up and Q − scatters in the negative time direction. Secondly, we classify solutions at this threshold and obtain that up toḢ 1/2 symmetries, they behave exactly as the above three special solutions, or scatter and blow up in both time directions as the solutions below the mass-energy threshold. These results are obtained by studying the spectral properties of the linearized Schrödinger operator in this mass-supercritical case, establishing relevant modulational stability and careful analysis of the exponentially decaying solutions to the linearized equation.

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“…global existence) for critical equations in the defocusing case, or in focusing cases in which the solution is "smaller" in mass or energy than that of the ground state. See [28], [81], [30] About the author Terence Tao is professor and James and Carol Collins Chair at the Department of Mathematics at the University of California, Los Angeles. He was a recipient of the Fields Medal in 2006 and is a Fellow of the National Academy of Sciences.…”

Section: Further Developmentsmentioning

confidence: 99%

Why are solitons stable?

Tao

2008

Bull. Amer. Math. Soc.

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126

119

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Abstract. The theory of linear dispersive equations predicts that waves should spread out and disperse over time. However, it is a remarkable phenomenon, observed both in theory and practice, that once nonlinear effects are taken into account, solitary wave or soliton solutions can be created, which can be stable enough to persist indefinitely. The construction of such solutions can be relatively straightforward, but the fact that they are stable requires some significant amounts of analysis to establish, in part due to symmetries in the equation (such as translation invariance) which create degeneracy in the stability analysis. The theory is particularly difficult in the critical case in which the nonlinearity is at exactly the right power to potentially allow for a self-similar blowup. In this article we survey some of the highlights of this theory, from the more classical orbital stability analysis of Weinstein and Grillakis-Shatah-Strauss, to the more recent asymptotic stability and blowup analysis of Martel-Merle and Merle-Raphael, as well as current developments in using this theory to rigorously demonstrate controlled blowup for several key equations.

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The cubic nonlinear Schrödinger equation in two dimensions with radial data

Cited by 201 publications

References 77 publications

Global existence and uniqueness results for weak solutions of the focusing mass-critical nonlinear Schrödinger equation

Global existence and uniqueness results for weak solutions of the focusing mass-critical nonlinear Schrödinger equation

Threshold solutions for the focusing 3D cubic Schrödinger equation

Why are solitons stable?

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