## Abstract

The effect of inhomogeneity of nonlinear medium is discussed concerning the stability of standing waves e ^{i ω t} φ_{ω}(x) for a nonlinear Schrödinger equation with an inhomogeneous nonlinearity V (x)|u| ^{p - 1} u, where V (x) is proportional to the electron density. Here, ω > 0 and φ_{ω}(x) is a ground state of the stationary problem. When V (x) behaves like |x|^{-b} at infinity, where 0 < b < 2, we show that e ^{i ω t} φ _{ω}(x) is stable for p < 1 | (4 - 2b)/n and sufficiently small ω > 0. The main point of this paper is to analyze the linearized operator at standing wave solution for the case of V (x) - |x|^{-b} . Then, this analysis yields a stability result for the case of more general, inhomogeneous V (x) by a certain perturbation method.

Original language | English |
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Pages (from-to) | 1157-1177 |

Number of pages | 21 |

Journal | Annales Henri Poincare |

Volume | 6 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2005 Dec |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Nuclear and High Energy Physics
- Mathematical Physics