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Solitary waves and excited states for Boson stars

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posted on 2023-06-10, 00:26 authored by Michael MelgaardMichael Melgaard, F D Y Zongo
We study the nonlinear, nonlocal, time-dependent partial differential equation i \pd_{t} \vf = (\sqrt{ -\D +m^{2} } -m) \vf -\left( \frac{1}{|x|} \ast | \vf |^{2} \right) \vf \mbox{ on } \R^{3} , which is known to describe the dynamics of quasi-relativistic boson stars in the mean-field limit. For positive mass parameter $m>0$ we establish existence of infinitely many (corresponding to distinct energies $\l_{k}$) travelling solitary waves, $\vf_{k}(x,t) = e^{i \l_{k} t } \f_{k}(x-vt)$, with speed $|v| <1$, where $c=1$ corresponds to the speed of light in our choice of units. These travelling solitary waves cannot be obtained by applying a Lorentz boost to a solitary wave at rest (with $v=0$) because Lorentz covariance fails. Instead we study a suitable variational problem for which the functions $\f_{k} \in \Hb^{1/2}(\R^{3})$ arise as solutions (called boosted excited states) to a Choquard type equation in $\R^{3}$, where the negative Laplacian is replaced by the pseudo-differential operator $\sqrt{ -\D +m^{2} } -m$ and an additional term $i (v \cdot
abla)$ enters. Moreover, we give a new proof for existence of boosted ground states. The results are based on perturbation methods in critical point theory.

History

Publication status

  • Published

File Version

  • Accepted version

Journal

Analysis and Applications

ISSN

0219-5305

Publisher

World Scientific

Page range

1-18

Department affiliated with

  • Mathematics Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2021-07-23

First Open Access (FOA) Date

2022-07-27

First Compliant Deposit (FCD) Date

2021-07-22

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