Skip to main content
Log in

Boson Stars as Solitary Waves

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramowitz, M., Stegun, I.A.: ed.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. New York: Dover Publications Inc., 1992, Reprint of the 1972 edition

  2. Cazenave T. and Lions P.-L. (1982). Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85: 549–561

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. Elgart A. and Schlein B. (2006). Mean field dynamics of Boson Stars. Comm. Pure Appl. Math. 60(4): 500–545

    Article  MathSciNet  Google Scholar 

  4. Fröhlich J., Jonsson B.L.G. and Lenzmann E. (2007). Effective dynamics for Boson stars. Nonlinearity 32: 1031–1075

    Article  ADS  Google Scholar 

  5. Fröhlich, J., Lenzmann, E.: Blow-up for nonlinear wave equations describing Boson Stars. http://arxiv.org/list/math-ph/0511003, 2006 to appear in Comm. Pure Appl. Math.

  6. Hislop, P.D.: Exponential decay of two-body eigenfunctions: A review. In: Proceedings of the Symposium on Mathematical Physics and Quantum Field Theory (Berkeley, CA, 1999), Volume 4 of Electron. J. Differ. Equ. Conf., pages 265–288 (electronic), San Marcos, TX, 2000. Southwest Texas State Univ.

  7. Lenzmann, E.: Well-posedness for semi-relativistic Hartree equations of critical type. http://arXiv.org/list/math.AP/0505456, 2005, to appear in Mathematical Physics, Analysis, and Geometry

  8. Lieb E.H. (1977). Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57: 93–105

    ADS  MathSciNet  Google Scholar 

  9. Lieb, E.H., Loss, M.: Analysis, Volume 14 of Graduate Studies in Mathematics. A Providence, RI: Amer. Math. Soc., second edition, 2001

  10. Lieb E.H. and Thirring W. (1984). Gravitational collapse in quantum mechanics with relativistic kinetic energy. Ann. Physics 155(2): 494–512

    Article  ADS  MathSciNet  Google Scholar 

  11. Lieb E.H. and Yau H.-T. (1987). The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112: 147–174

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. Lions P.-L. (1984). The concentration-compactness principle in the calculus of variations. The locally compact case, Part I. Ann. Inst. Henri Poincaré, 1(2): 109–145

    MATH  Google Scholar 

  13. Stein E.M. (1993). Harmonic Analysis. Princeton University Press, Princeton, NJ

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jürg Fröhlich.

Additional information

Communicated by H.-T. Yau

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fröhlich, J., Jonsson, B.L.G. & Lenzmann, E. Boson Stars as Solitary Waves. Commun. Math. Phys. 274, 1–30 (2007). https://doi.org/10.1007/s00220-007-0272-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-007-0272-9

Keywords

Navigation