Boson stars as solitary waves

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.