Boson Stars as Solitary Waves

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 pseudo-relativistic boson stars in the mean-field limit. For positive mass parameters, $m>0$, we prove existence of travelling solitary waves, $ψ(t,x) = e^{i t μ} \sol_{v}(x-vt)$, 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 $\sol_v \in \Hhalf(\RR^3)$ 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 $ψ(t,x) = e^{it μ} \sol_v(x-vt)$ and pointwise exponential decay of $\sol_v(x)$ in $x$.

30 pages

References in corpus (4)

• Well-Posedness for Semi-Relativistic Hartree Equations of Critical Type
• Blow-Up for Nonlinear Wave Equations describing Boson Stars
• Effective Dynamics for Boson Stars
• Mean Field Dynamics of Boson Stars

Cited by in corpus (48)

• A guide to the Choquard equation
• Well-Posedness for Semi-Relativistic Hartree Equations of Critical Type
• Uniqueness of Ground States for Pseudo-Relativistic Hartree Equations
• Blow-Up for Nonlinear Wave Equations describing Boson Stars
• Numerical study of fractional Nonlinear Schrödinger equations
• Nondispersive solutions to the L2-critical half-wave equation
• Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator
• Effective Dynamics for Boson Stars
• Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation
• On Traveling Solitary Waves and Absence of Small Data Scattering for Nonlinear Half-Wave Equations
• Existence and mass concentration of pseudo-relativistic Hartree equation
• On Blow-up Profile of Ground States of Boson Stars with External Potential
• Stable ground states for the relativistic gravitational Vlasov-Poisson system
• Fractional Schrödinger-Poisson systems with a general subcritical or critical nonlinearity
• On instability of standing waves for the mass-supercritical fractional nonlinear Schrödinger equation
• On fractional Schrödinger equations with Hartree type nonlinearities
• Nondegeneracy and Stability of Antiperiodic Bound States for Fractional Nonlinear Schrödinger Equations
• Ground states for semi-relativistic Schrödinger-Poisson-Slater energies
• On fractional Choquard equations
• The Lieb-Yau Conjecture for Ground States of Pseudo-Relativistic Boson Stars
• Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
• Blowup dynamics for Mass Critical Half-wave equation in 3D
• Many-Body Blow-Up Profile of Boson Stars with External Potentials
• Optimal convergence rate of nonrelativistic limit for the nonlinear pseudo-relativistic equations
• Nonrelativistic limit of standing waves for pseudo-relativistic nonlinear Schrödinger equations
• A constrained minimization problem related to two coupled pseudo-relativistic Hartree equations
• Asymptotic decay of solutions for sublinear fractional Choquard equations
• Long time dynamics for semirelativistic NLS and Half Wave in arbitrary dimension
• Uniqueness of ground states for the L^2-critical boson star equation
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• Ground states of a coupled pseudo-relativistic Hartree system: existence and concentration behavior
• Normalized ground states for the fractional nonlinear Schrödinger equations
• Nonlocal Pertubations of Fractional Choquard Equation
• On a generalized pseudorelativistic Schrödinger equation with supercritical growth
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• Soliton dynamics for the generalized Choquard equation
• Semiclassical analysis for pseudo-relativistic Hartree equations
• Nondispersive solutions to the mass critical half-wave equation in two dimensions
• On small travelling waves to the mass critical fractional NLS
• Ground states for pseudo-relativistic Hartree equations of critical type
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• Convergence of states for polaron models in the classical limit
• On the stability of 2D dipolar Bose-Einstein condensates
• Boson Stars with Long-range Perturbations
• Direct methods for pseudo-relativistic Schrödinger operators
• On the existence of multiple normalized solutions for a class of fractional Choquard equations with mixed nonlinearities
• The fractional in time Schrödinger equation with a Hartree perturbation
• Ground states for the pseudo-relativistic Hartree equation with external potential