Boson Stars as Solitary Waves

Abstract

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