Boson stars as solitary waves

We study the nonlinear equation i theta t psi = (root-Delta+m(2) - m) psi - (vertical bar x vertical bar(-1) * vertical bar psi vertical bar(2)) psi on R(3), which is known to describe the dynamics of pseudo-relativistic boson stars in the meanfield limit. For positive mass parameters, m > 0, we prove existence of travelling solitary waves, psi(t, x) = ei t mu phi(v)(x - vt), for some mu is an element of R and with speed vertical bar v vertical bar < 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 phi(v) H(1/2)(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(v)(x - vt) and pointwise exponential decay of phi(v)(x) in x.