Abstract
We study the nonlinear equation
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.
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Communicated by H.-T. Yau
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Fröhlich, J., Jonsson, B.L.G. & Lenzmann, E. Boson Stars as Solitary Waves. Commun. Math. Phys. 274, 1–30 (2007). https://doi.org/10.1007/s00220-007-0272-9
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DOI: https://doi.org/10.1007/s00220-007-0272-9