Boson stars as solitary waves

• Published in: Communications in mathematical physics Vol. 274; no. 1; pp. 1 - 30
• Main Authors: FRÖHLICH, Jürg, JONSSON, B. Lars G, LENZMANN, Enno
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer 01.08.2007
• Subjects: collapse; Exact sciences and technology; Mathematical methods in physics; Other topics in mathematical methods in physics; Physics; quantum-mechanics; Variational problem; Nonlinear equations; Travelling waves; Mathematical physics; Numerical stability; Exponential decay; Concentration compactness
• ISSN: 0010-3616; 1432-0916; 1432-0916
• DOI: 10.1007/s00220-007-0272-9

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.