Skip to main content
SpringerLink
Log in

On Self-Similar Solutions to a Kinetic Equation Arising in Weak Turbulence Theory for the Nonlinear Schrödinger Equation

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We construct a family of self-similar solutions with fat tails to a quadratic kinetic equation. This equation describes the long time behaviour of weak solutions with finite mass to the weak turbulence equation associated to the nonlinear Schrödinger equation. The solutions that we construct have finite mass, but infinite energy. In Kierkels and Velázquez (J Stat Phys 159:668–712, 2015) self-similar solutions with finite mass and energy were constructed. Here we prove upper and lower exponential bounds on the tails of these solutions.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)

    MATH  Google Scholar 

  2. Duistermaat, J.J., Kolk, J.A.C.: Distributions: Theory and Applications. Birkhäuser, Berlin (2010)

    Book  MATH  Google Scholar 

  3. Düring, G., Josserand, C., Rica, S.: Weak turbulence for a vibrating plate: can one hear a Kolmogorov spectrum? Phys. Rev. Lett. 97, 025503 (2006)

    Article  ADS  Google Scholar 

  4. Dyachenko, S., Newell, A.C., Pushkarev, A., Zakharov, V.E.: Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation. Physica D 57, 96–160 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Escobedo, M., Mischler, S., Ricard, M.R.: On self-similarity and stationary problem for fragmentation and coagulation models. Ann. Inst. H. Poincaré Anal. Non Linéaire 22, 99–125 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Escobedo, M., Velázquez, J.J.L.: On the theory of weak turbulence for the nonlinear Schrödinger equation, Memoirs AMS 238(1124) (2015)

  7. Fournier, N., Laurençot, P.: Existence of self-similar solutions to Smoluchowski’s Coagulation Equation. Commun. Math. Phys. 256, 589–609 (2005)

    Article  ADS  MATH  Google Scholar 

  8. Hasselmann, K.: On the non-linear energy transfer in a gravity-wave spectrum. Part 1. General Theory. J. Fluid Mech. 12, 481–500 (1962)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Hasselmann, K.: On the non-linear energy transfer in a gravity-wave spectrum. Part 2. Conservation theorems, wave-particle analogy, irreversibility. J. Fluid Mech. 15, 273–281 (1963)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Kierkels, A.H.M., Velázquez, J.J.L.: On the transfer of energy towards infinity in the theory of weak turbulence for the nonlinear Schrödinger equation. J. Stat. Phys. 159, 668–712 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Lukacs, E.: Characteristic Functions, 2nd edn. C. Griffin & Co. Ltd., High Wycombe (1970)

    MATH  Google Scholar 

  12. Menon, G., Pego, R.L.: Approach to self-similarity in Smoluchowski’s coagulation equations. Commun. Pure Appl. Math. 57, 1197–1232 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Nazarenko, S.: Wave turbulence. Lecture Notes in Physics 825. Springer, New York (2011)

    Google Scholar 

  14. Niethammer, B., Velázquez, J.J.L.: Self-similar solutions with fat tails for Smoluchowski’s coagulation equation with locally bounded kernels. Commun. Math. Phys. 318, 505–532 (2013)

    Article  ADS  MATH  Google Scholar 

  15. Niethammer, B., Velázquez, J.J.L.: Exponential tail behavior of self-similar solutions to Smoluchowski’s coagulation equation. Commun. PDE 39, 2314–2350 (2014)

    Article  MATH  Google Scholar 

  16. Niethammer, B., Throm, S., Velázquez, J.J.L.: Self-similar solutions with fat tails for Smoluchowski’s coagulation equation with singular kernels. Ann. Inst. H. Poincaré Anal. Non Linéaire (2015). doi:10.1016/j.anihpc.2015.04.002

  17. Peierls, R.: Zur kinetischen Theorie der Wärmeleitung in Kristallen. Ann. Phys. 395, 1055–1101 (1929)

    Article  MATH  Google Scholar 

  18. Pomeau, Y.: Asymptotic time behaviour of nonlinear classical field equations. Nonlinearity 5, 707–720 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Simon, J.: Sobolev, Besov and Nikolskii fractional spaces: imbeddings and comparisons for vector valued spaces on an interval. Ann. Mat. Pura Appl. 157, 117–148 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  20. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Pub. Co., New York (1978)

    MATH  Google Scholar 

  21. Triebel, H.: Theory of Function Spaces. Birkhäuser, Berlin (1983)

    Book  MATH  Google Scholar 

  22. Zakharov, V.E.: Weak-turbulence spectrum in a plasma without a magnetic field. Zh. Eksp. Teor. Fiz. 51, 688–696 (1967) [Sov. Phys. JEPT 24, 455–459 (1967)]

  23. Zakharov, V.E., Filonenko, N.N.: Weak turbulence of capillary waves. Zh. Prikl. Mekh. Tekh. Fiz. 8(5), 62–67 (1967) [J. Appl. Mech. Tech. Phys. 8(5), 37–40 (1967)]

  24. Zakharov, V.E., L’vov, V.S., Falkovich, G.: Kolmogorov Spectra of Turbulence I. Wave Turbulence. Springer, New York (1992)

Download references

Acknowledgments

The authors acknowledge support through the CRC 1060 The mathematics of emergent effects at the University of Bonn, that is funded through the German Science Foundation (DFG).

Author information

Authors and Affiliations

  1. Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    A. H. M. Kierkels & J. J. L. Velázquez

Authors
  1. A. H. M. Kierkels
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. J. L. Velázquez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. H. M. Kierkels.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kierkels, A.H.M., Velázquez, J.J.L. On Self-Similar Solutions to a Kinetic Equation Arising in Weak Turbulence Theory for the Nonlinear Schrödinger Equation. J Stat Phys 163, 1350–1393 (2016). https://doi.org/10.1007/s10955-016-1505-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-016-1505-0

Keywords

Mathematics Subject Classification

Search

Navigation