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Boson stars as solitary waves. (English) Zbl 1126.35064

Summary: We study the nonlinear equation \[ i \partial_t \psi = (\sqrt{-\Delta + m^2} - m)\psi - (|x|^{-1} \ast |\psi|^2) \psi \quad \text{on }\mathbb{R}^3 \] 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 \(\varphi_v \in {\mathbf H}^{1/2}({\mathbb{R}}^3)\) 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 \(x\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
85A30 Hydrodynamic and hydromagnetic problems in astronomy and astrophysics
35Q51 Soliton equations
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
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