Fröhlich, Jürg and Jonsson, B. Lars G. and Lenzmann, Enno.
(2007)
* Boson stars as solitary waves.*
Communications in Mathematical Physics, 274 (1).
pp. 1-30.

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Official URL: http://edoc.unibas.ch/49587/

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## Abstract

We study the nonlinear equation

i \partial_t \psi = (\sqrt{-\Delta + m^2} - m)\psi - ( |x|^{-1} \ast |\psi|^2 ) \psi \quad {\rm on}\,\mathbb{R}^3

which is known to describe the dynamics of pseudo-relativistic boson stars in the mean-field limit. For positive mass parameters, m > 0, we prove existence of travelling solitary waves, \psi(t,x) = e^{{i}{t}\mu} \varphi_{v}(x - vt) , for some \mu \in {\mathbb{R}} and with speed |v| < 1, where c = 1 corresponds to the speed of light in our units. Due to the lack of Lorentz covariance, such travelling solitary waves cannot be obtained by applying a Lorentz boost to a solitary wave at rest (with v = 0). To overcome this difficulty, we introduce and study an appropriate variational problem that yields the functions \varphi_v \in {\bf H}^{1/2}({\mathbb{R}}^3) as minimizers, which we call boosted ground states. Our existence proof makes extensive use of concentration-compactness-type arguments.

In addition to their existence, we prove orbital stability of travelling solitary waves \psi(t, x) = e^{{i}{t}\mu}\varphi_v(x - vt) and pointwise exponential decay of \varphi_v(x) in x.

i \partial_t \psi = (\sqrt{-\Delta + m^2} - m)\psi - ( |x|^{-1} \ast |\psi|^2 ) \psi \quad {\rm on}\,\mathbb{R}^3

which is known to describe the dynamics of pseudo-relativistic boson stars in the mean-field limit. For positive mass parameters, m > 0, we prove existence of travelling solitary waves, \psi(t,x) = e^{{i}{t}\mu} \varphi_{v}(x - vt) , for some \mu \in {\mathbb{R}} and with speed |v| < 1, where c = 1 corresponds to the speed of light in our units. Due to the lack of Lorentz covariance, such travelling solitary waves cannot be obtained by applying a Lorentz boost to a solitary wave at rest (with v = 0). To overcome this difficulty, we introduce and study an appropriate variational problem that yields the functions \varphi_v \in {\bf H}^{1/2}({\mathbb{R}}^3) as minimizers, which we call boosted ground states. Our existence proof makes extensive use of concentration-compactness-type arguments.

In addition to their existence, we prove orbital stability of travelling solitary waves \psi(t, x) = e^{{i}{t}\mu}\varphi_v(x - vt) and pointwise exponential decay of \varphi_v(x) in x.

Faculties and Departments: | 05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Lenzmann) |
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UniBasel Contributors: | Lenzmann, Enno |

Item Type: | Article, refereed |

Article Subtype: | Research Article |

Publisher: | Springer |

ISSN: | 0010-3616 |

e-ISSN: | 1432-0916 |

Note: | Publication type according to Uni Basel Research Database: Journal article |

Identification Number: | |

Last Modified: | 28 Nov 2017 08:37 |

Deposited On: | 28 Nov 2017 08:37 |

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