Ill-posedness for the nonlinear Schrödinger equation with quadratic non-linearity in low dimensions

Tsukasa Iwabuchi; Takayoshi Ogawa

doi:10.1090/s0002-9947-2014-06000-5

Ill-posedness for the nonlinear Schrödinger equation with quadratic non-linearity in low dimensions

Tsukasa Iwabuchi, Takayoshi Ogawa

SCI - Mathematics

Research output: Contribution to journal › Article › peer-review

23 Citations (Scopus)

Abstract

We consider the ill-posedness issue for the nonlinear Schrödinger equation with a quadratic nonlinearity. We refine the Bejenaru-Tao result by constructing an example in the following sense. There exist a sequence of time T_N → 0 and solution u_N(t) such that u_N(T_N)→∞ in the Besov space (σ > 2) for one space dimension. We also construct a similar ill-posed sequence of solutions in two space dimensions in the scaling critical Sobolev space H⁻¹(ℝ²). We systematically utilize the modulation space for one dimension and the scaled modulation space for two dimensions.

Original language	English
Pages (from-to)	2613-2630
Number of pages	18
Journal	Transactions of the American Mathematical Society
Volume	367
Issue number	4
DOIs	https://doi.org/10.1090/s0002-9947-2014-06000-5
Publication status	Published - 2015 Apr 1

ASJC Scopus subject areas

Mathematics(all)
Applied Mathematics

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abstract = "We consider the ill-posedness issue for the nonlinear Schr{\"o}dinger equation with a quadratic nonlinearity. We refine the Bejenaru-Tao result by constructing an example in the following sense. There exist a sequence of time TN → 0 and solution uN(t) such that uN(TN)→∞ in the Besov space (σ > 2) for one space dimension. We also construct a similar ill-posed sequence of solutions in two space dimensions in the scaling critical Sobolev space H−1(ℝ2). We systematically utilize the modulation space for one dimension and the scaled modulation space for two dimensions.",

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N2 - We consider the ill-posedness issue for the nonlinear Schrödinger equation with a quadratic nonlinearity. We refine the Bejenaru-Tao result by constructing an example in the following sense. There exist a sequence of time TN → 0 and solution uN(t) such that uN(TN)→∞ in the Besov space (σ > 2) for one space dimension. We also construct a similar ill-posed sequence of solutions in two space dimensions in the scaling critical Sobolev space H−1(ℝ2). We systematically utilize the modulation space for one dimension and the scaled modulation space for two dimensions.

AB - We consider the ill-posedness issue for the nonlinear Schrödinger equation with a quadratic nonlinearity. We refine the Bejenaru-Tao result by constructing an example in the following sense. There exist a sequence of time TN → 0 and solution uN(t) such that uN(TN)→∞ in the Besov space (σ > 2) for one space dimension. We also construct a similar ill-posed sequence of solutions in two space dimensions in the scaling critical Sobolev space H−1(ℝ2). We systematically utilize the modulation space for one dimension and the scaled modulation space for two dimensions.

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Ill-posedness for the nonlinear Schrödinger equation with quadratic non-linearity in low dimensions

Abstract

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