# Solitary waves and excited states for Boson stars

Melgaard, M and Zongo, F D Y (2021) Solitary waves and excited states for Boson stars. Analysis and Applications. pp. 1-18. ISSN 0219-5305

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## Abstract

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 $i (v \cdot \nabla)$ enters. Moreover, we give a new proof for existence of boosted ground states. The results are based on perturbation methods in critical point theory.

Item Type: Article School of Mathematical and Physical Sciences > Mathematics Mx Elements Account Mx Elements Account 23 Jul 2021 15:24 27 Jul 2022 01:00 http://sro.sussex.ac.uk/id/eprint/100717