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

PDF (Electronic version of an article published as [Analysis and Applications, Volume, Issue, Year, Pages] [10.1142/S0219530521500147] © [copyright World Scientific Publishing Company] [https://www.worldscientific.com/worldscinet/aa]) - Accepted Version Download (404kB)

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
Schools and Departments:	School of Mathematical and Physical Sciences > Mathematics
SWORD Depositor:	Mx Elements Account
Depositing User:	Mx Elements Account
Date Deposited:	23 Jul 2021 15:24
Last Modified:	27 Jul 2022 01:00
URI:	http://sro.sussex.ac.uk/id/eprint/100717

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