### Abstract

We consider the one-dimensional focusing nonlinear Schrödinger equation (NLS) with a delta potential and even initial data. The problem is equivalent to the solution of the initial/boundary problem for NLS on a half-line with Robin boundary conditions at the origin. We follow the method of Bikbaev and Tarasov which utilizes a Bäcklund transformation to extend the solution on the half-line to a solution of the NLS equation on the whole line. We study the asymptotic stability of the stationary 1-soliton solution of the equation under perturbation by applying the nonlinear steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou. Our work strengthens, and extends, the earlier work on the problem by Holmer and Zworski.

Original language | English (US) |
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Pages (from-to) | 143-156 |

Number of pages | 14 |

Journal | Letters in Mathematical Physics |

Volume | 96 |

Issue number | 1-3 |

DOIs | |

State | Published - Jun 1 2011 |

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### Keywords

- Riemann-Hilbert problem
- initial-boundary value problem
- long-time behavior
- nonlinear Schrödinger equation
- nonlinear steepest descent

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics