 05417 Juerg Froehlich, B. Lars G. Jonsson, Enno Lenzmann
 Boson Stars as Solitary Waves
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Dec 14, 05

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Abstract. 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 pseudorelativistic boson stars in the meanfield limit. For positive mass parameters, $m > 0$, we prove existence of travelling solitary waves, $\psi(t,x) = e^{i t \mu} \sol_{v}(xvt)$, 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 $\sol_v \in H^{1/2}(\RR^3)$ as minimizers, which we call boosted ground states. Our existence proof makes extensive use of concentrationcompactnesstype arguments. In addition to their existence, we prove orbital stability of travelling solitary waves $\psi(t,x) = e^{it \mu} \sol_v(xvt)$ and pointwise exponential decay of $\sol_v(x)$ in $x$.
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