We study the Whitham equations, which describe the semiclassical limit of the defocusing nonlinear Schrödinger equation. The limit is governed by a pair of hyperbolic equations of two independent variables for a short time starting from the initial time. After this hyperbolic solution breaks down, t
On blowup for the pseudo-conformally invariant nonlinear Schrödinger equation II
✍ Scribed by Hayato Nawa; Masayoshi Tsutsumi
- Publisher
- John Wiley and Sons
- Year
- 1998
- Tongue
- English
- Weight
- 96 KB
- Volume
- 51
- Category
- Article
- ISSN
- 0010-3640
No coin nor oath required. For personal study only.
✦ Synopsis
We study the δ-measure-like blowup of solutions to the pseudo-conformally invariant nonlinear Schrödinger equation
For N = 1 or N ≥ 2 and u0 radially symmetric, we prove that if the blowup solution u(t) satisfies |u(t, x)| 2 dx u0 2 δ0(dx) in the sense of measures as t ↑ Tm (i.e., weakly * in B , which is the dual of
where Tm > 0 is the maximal existence time and δ0 is the Dirac measure at 0 ∈ R N , then u0 must satisfy
and we have lim t→Tm |x|u(t) L 2 (R N ) = 0 .
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