## Abstract

We study the initial-value problem for the quadratic nonlinear Schrödinger equation iu_{t} + 1/2u_{xx}= δ_{x}ū^{2}, x ∈ ℝ, t > 1, u(1, x) = u_{1}(x), x ∈ ℝ. For small initial data u_{1} ∈ H^{2,2} we prove that there exists a unique global solution u ∈ C([1, ∞); H^{2,2}) of this Cauchy problem. Moreover we show that the large time asymptotic behavior of the solution is defined in the region ΙxΙ ≤ C√t by the self-similar solution 1/√tMS(x/ √t) such that the total mass 1/√t ∫_{ℝ} MS(x/√t)dx = ∫_{ℝ} u_{1}(x)dx, and in the far region ΙxΙ > √t the asymptotic behavior of solutions has rapidly oscillating structure similar to that of the cubic nonlinear Schrödinger equations.

Original language | English |
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Pages (from-to) | 1-38 |

Number of pages | 38 |

Journal | Electronic Journal of Differential Equations |

Volume | 2008 |

Publication status | Published - 2008 Feb 1 |

Externally published | Yes |

## Keywords

- Large time asymptotic
- Nonlinear schrodinger equation
- Self-similar solutions

## ASJC Scopus subject areas

- Analysis