### Abstract

We consider the Cauchy problem for the cubic nonlinear Schrödinger equation in one space dimension (1) {iu_{t} + 1/2u_{xx} + u^{-3} = 0, t ∈ R, x ∈ R, u(0, x) = u_{0}(x), x ∈ R.} Cubic type nonlinearities in one space dimension heuristically appear to be critical for large time. We study the global existence and large time asymptotic behavior of solutions to the Cauchy problem (1). We prove that if the initial data u_{0} ∈ H^{1,0} ∩ H^{0,1} are small and such that sup_{|ξ| ≤ 1} | arg ℱu_{0}(ξ) - πn/2 | < π/8 for some n ∈ Z, and inf_{|ξ| ≤ 1} |ℱu_{0}(ξ)| > 0, then the solution has an additional logarithmic time-decay in the short range region |x| ≤ √t. In the far region |x| > √t the asymptotics have a quasilinear character.

Original language | English |
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Pages (from-to) | 1065-1085 |

Number of pages | 21 |

Journal | Canadian Journal of Mathematics |

Volume | 54 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2002 Oct |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Canadian Journal of Mathematics*,

*54*(5), 1065-1085. https://doi.org/10.4153/CJM-2002-039-3