Boson Stars as Solitary Waves

We study the nonlinear equation i∂tψ = (√ −∆+m2 −m ) ψ − (|x| ∗ |ψ|)ψ on R, 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, ψ(t, x) = eφv(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 φv ∈ H(R) 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 ψ(t, x) = eφv(x − vt) and pointwise exponential decay of φv(x) in x.