Skip to main content
SpringerLink
Log in

Justification of the NLS Approximation for the Euler–Poisson Equation

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

Abstract

The nonlinear Schrödinger (NLS) equation can be derived as a formal approximation equation describing the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the ion Euler–Poisson equation. In this paper, we rigorously justify such approximation by giving error estimates in Sobolev norms between exact solutions of the ion Euler–Poisson system and the formal approximation obtained via the NLS equation. The justification consists of several difficulties such as the resonances and loss of regularity, due to the quasilinearity of the problem. These difficulties are overcome by introducing normal form transformation and cutoff functions and carefully constructed energy functional of the equation.

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

Similar content being viewed by others

References

  1. Alazard, T., Delort, J.M.: Sobolev estimates for two dimensional gravity water waves. Astérisque 374, viii\(+\)241 (2015)

  2. Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM Studies in Applied Mathematics, Vol. 4. SIAM (1981)

  3. Delort, J.M.: Long-time Sobolev stability for small solutions of quasi-linear Klein–Gordon equations on the circle. Trans. Am. Math. Soc. 361(08), 4299–4365 (2009)

    Article  MathSciNet  Google Scholar 

  4. Düll, W.P.: Validity of the Korteweg–de Vries approximation for the two-dimensional water wave problem in the arc length formulation. Commun. Pure Appl. Math. 65(3), 381–429 (2012)

    Article  MathSciNet  Google Scholar 

  5. Düll, W.P.: Justification of the nonlinear Schrödinger approximation for a quasilinear Klein–Gordon equation. Commun. Math. Phys. 355, 1189–1207 (2017)

    Article  ADS  Google Scholar 

  6. Delort, J.M., Szeftel, J.: Long-time existence for small data nonlinear Klein–Gordon equations on tori and spheres. Int. Math. Res. Not. 37(37), 1897–1966 (2004)

    Article  MathSciNet  Google Scholar 

  7. Düll, W.P., Schneider, G., Wayne, C.E.: Justification of the Nonlinear Schrödinger equation for the evolution of gravity driven 2D surface water waves in a canal of finite depth. Arch. Ration. Mech. Anal. 220(2), 543–602 (2016)

    Article  MathSciNet  Google Scholar 

  8. Guo, Y.: Smooth irrotational flows in the large to the Euler–Poisson system in \(R^{3+1}\). Commun. Math. Phys. 195(2), 249–265 (1998)

    Article  ADS  Google Scholar 

  9. Guo, Y., Han, L., Zhang, J.: Absence of shocks for one dimensional Euler–Poisson system. Arch. Ration. Mech. Anal. 223, 1057–1121 (2017)

    Article  MathSciNet  Google Scholar 

  10. Germain, P., Masmoudi, N.: Global existence for the Euler–Maxwell system. Ann. Sci. Ecole Norm. Sup. 47(4), 469–503 (2014)

    Article  MathSciNet  Google Scholar 

  11. Guo, Y., Pausader, B.: Global smooth ion dynamics in the Euler–Poisson system. Commun. Math. Phys. 303(1), 89–125 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  12. Guo, Y., Pu, X.: KdV limit of the Euler–Poisson system. Arch. Ration. Mech. Anal. 211, 673–710 (2014)

    Article  MathSciNet  Google Scholar 

  13. Han-Kwan, D.: From Vlasov–Poisson to Korteweg–de Vries and Zakharov–Kuznetsov. Commun. Math. Phys. 324(3), 961–993 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  14. Hunter, J., Ifrim, M., Tataru, D., et al.: Long time solutions for a Burgers–Hilbert equation via a modified energy method. Proc. Am. Math. Soc. 143(8), 3407–3412 (2015)

    Article  MathSciNet  Google Scholar 

  15. Hunter, J., Ifrim, M., Tataru, D., et al.: Two dimensional water waves in holomorphic coordinates. Commu. Math. Phys. 346(2), 483–552 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  16. Ionescu, A.D., Pausader, B.: The Euler–Poisson system in 2D: global stability of the constant equilibrium solution. Int. Math. Res. Not. 2013, 761–826 (2013)

    Article  MathSciNet  Google Scholar 

  17. Ionescu, A.D., Pusateri, F.: Global solutions for the gravity water waves system in 2d. Invent. Math. 199(3), 653–804 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  18. Ionescu, A.D., Pusateri, F.: Global analysis of a model for capillary water waves in two dimensions. Commun. Pure Appl. Math. 69(11), 2015–2071 (2016)

    Article  MathSciNet  Google Scholar 

  19. Ionescu, A.D., Pusateri, F.: Global regularity for 2d water waves with surface tension, arXiv:1408.4428 (2017)

  20. Ifrim, M., Tataru, D.: The lifespan of small data solutions in two dimensional capillary water waves. Arch. Ration. Mech. Anal. 225(3), 1279–1346 (2017)

    Article  MathSciNet  Google Scholar 

  21. Jang, J.: The two-dimensional Euler–Poisson system with spherical symmetry. J. Math. Phys. 53(2), 023701 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  22. Jang, J., Li, D., Zhang, X.: Smooth global solutions for the two-dimensional Euler–Poisson system. Forum Math. 26(3), 645–701 (2014)

    Article  MathSciNet  Google Scholar 

  23. Kalyakin, L.A.: Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium. Sb. Math. 60, 457–483 (1988)

    Article  MathSciNet  Google Scholar 

  24. Kirrmann, P., Schneider, G., Mielke, A.: The validity of modulation equations for extended systems with cubic nonlinearities. Proc. R. Soc. Edinb. Sect. A 122, 85–91 (1992)

    Article  MathSciNet  Google Scholar 

  25. Lannes, D., Linares, F., Saut, J.C.: The Cauchy Problem for the Euler–Poisson System and Derivation of the Zakharov–Kuznetsov Equation and Studies in Phase Space Analysis with Applications to PDEs, pp. 181–213. Springer, New York (2013)

    Book  Google Scholar 

  26. Liu, H., Pu, X.: Long wavelength limit for the quantum Euler–Poisson equation. SIAM J. Math. Anal. 48(4), 2345–2381 (2016)

    Article  MathSciNet  Google Scholar 

  27. Liu, H., Pu, X.: The QKP limit of the quantum Euler–Poisson equation, arXiv:1706.07313 (submitted)

  28. Li, D., Wu, Y.: The Cauchy problem for the two dimensional Euler–Poisson system. J. Eur. Math. Soc. 10, 2211–2266 (2014)

    Article  MathSciNet  Google Scholar 

  29. Pu, X.: Dispersive limit of the Euler–Poisson system in higher dimensions. SIAM J. Math. Anal. 45(2), 834–878 (2013)

    Article  MathSciNet  Google Scholar 

  30. Shatah, J.: Normal forms and quadratic nonlinear Klein–Gordon equations. Commun. Pure Appl. Math. 38(5), 685–696 (1985)

    Article  MathSciNet  Google Scholar 

  31. Schneider, G.: Approximation of the Korteweg–de Vries equation by the nonlinear Schrödinger eqution. J. Differ. Equ. 147(2), 333–354 (1998)

    Article  ADS  Google Scholar 

  32. Schneider, G.: Justification of modulation equations for hyperbolic systems via normal forms. NoDEA Nonlinear Differ. Equ. Appl. 5(1), 69–82 (1998)

    Article  MathSciNet  Google Scholar 

  33. Schneider, G.: Justification of the NLS approximation for the KdV equation using the Miura transformation. Adv. Math. Phys. 2011, 854719 (2011)

    Article  MathSciNet  Google Scholar 

  34. Shimizu, K., Ichikawa, Y.H.: Automodulation of ion oscillation modes in plasma. J. Phys. Soc. Jpn. 33(3), 789–792 (1972)

    Article  ADS  Google Scholar 

  35. Schneider, G., Wayne, C.E.: The long-wave limit for the water wave problem I. The case of zero surface tension. Commun. Pure Appl. Math. 53(12), 1475–1535 (2000)

    Article  MathSciNet  Google Scholar 

  36. Schneider, G., Wayne, C.E.: The rigorous approximation of long-wavelength capillary–gravity waves. Arch. Ration. Mech. Anal. 162(3), 247–285 (2002)

    Article  MathSciNet  Google Scholar 

  37. Schneider, G., Wayne, C.E.: Justification of the NLS approximation for a quasilinear water wave model. J. Differ. Equ. 251, 238–269 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  38. Totz, N.: A justification of the modulation approximation to the 3D full water wave problem. Commun. Math. Phys. 335, 369–443 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  39. Totz, N., Wu, S.: A rigorous justification of the modulation approximation to the 2D full water wave problem. Comm. Math. Phys. 310(3), 817–883 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  40. Wu, S.: Almost global wellposedness of the 2-D full water wave problem. Invent. Math. 177, 45–135 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  41. Zakharov, V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. Sov. Phys. J. Appl. Mech. Technol. Phys. 4, 190–194 (1968)

    ADS  Google Scholar 

Download references

Acknowledgements

The second author thanks Professor Yan Guo for his valuable discussions and suggestions on this paper. The authors thank the referees for their helpful comments of the manuscript.

Author information

Authors and Affiliations

  1. Faculty of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, 030006, People’s Republic of China

    Huimin Liu

  2. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, People’s Republic of China

    Xueke Pu

Authors
  1. Huimin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xueke Pu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xueke Pu.

Ethics declarations

Funding

This study was funded by the National Natural Science Foundation of China (Grant Number 11871172).

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by C. De Lellis.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by NSFC (11871172).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, H., Pu, X. Justification of the NLS Approximation for the Euler–Poisson Equation. Commun. Math. Phys. 371, 357–398 (2019). https://doi.org/10.1007/s00220-019-03576-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-019-03576-4

Search

Navigation