Boson Stars as Solitary Waves

By Juerg Froehlich, B. Jonsson, E. Lenzmann. 2005

212 citations recorded by the source corpus.

AbstractWe study the nonlinear equation $$i \partial_t \psi = (\sqrt{-\Delta + m^2} - m)\psi - ( |x|^{-1} \ast |\psi|^2 ) \psi \quad {\rm on}\,\mathbb{R}^3$$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 $$\varphi_v \in {\bf H}^{1/2}({\mathbb{R}}^3)$$ 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.

Subfields: Mathematical Physics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/26bc1749d070684ed503176d8d412fa9a321b795

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In Mathematical Physics

In Nuclear and High Energy Physics

E. Lenzmann

In Statistical and Nonlinear Physics

Citations in the corpus

Uniqueness of ground states for pseudorelativistic Hartree equations (2008, methodology): “tion that the dispersive nonlinear PDE (1.7) exhibits a rich variety of phenomena, such as stable and unstable traveling solitary waves, as well as finite-time blowup solutions indicating the “gravitational collapse” of a boson star; see [ FJL07b , FJL07a, FL07].”
On fractional Choquard equations (2014, background): “2(R3),u>0, and in [13] it is shown that the dynamical evolution of boson stars is described by the nonlinear evolution equation i∂tψ= p −∆+m2ψ−(K2 ∗|ψ|2)ψ (m≥0) for a ﬁeld ψ: [0,T) ×R3 →C(see also [18,19,21]). The square root of the Laplacian also appears in the semi-relativistic Schr¨odinger-Poi”
Fractional Schrödinger–Poisson Systems with a General Subcritical or Critical Nonlinearity (2015, unknown): “1 [22] R.L. Frank, E. Lenzmann, L. Silvestre, Uniqueness of radial solutions for the fractional laplacian, Comm. Pure.”
Blowup for nonlinear wave equations describing boson stars (2005, background): “me. Remark. In contrast to this result, we mention that travelling ground state solitary waves are orbitally stable whenever the mass parameter is positive, i.e., m > 0 holds in (2.1). We refer to [3] for this result. 3 Proof of Main Results 3.1 Proof of Theorem 2.2 The proof of Theorem 2.2 is or”
Well-posedness for Semi-relativistic Hartree Equations of Critical Type (2005, background): “Equation (1) arises, for instance, as an effective description of boson stars, see, e. g., [4, 5 , 12], where F(u) is a focusing Hartree nonlinearity given by”