Boson Stars as Solitary Waves

We study the nonlinear equation $i \partial_t \psi = (\sqrt{-\Delta + m^2} - m) \psi - ( |x|^{-1} \ast |\psi|^2 ) \psi$ on $\RR^3$, 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, $\psi(t,x) = e^{i t \mu} \sol_{v}(x-vt)$, 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 $\sol_v \in \Hhalf(\RR^3)$ 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 $\psi(t,x) = e^{it \mu} \sol_v(x-vt)$ and pointwise exponential decay of $\sol_v(x)$ in $x$.