Skip to main content
SpringerLink
Log in

A Justification of the Modulation Approximation to the 3D Full Water Wave Problem

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

Abstract

We consider modulational solutions to the 3D inviscid incompressible irrotational infinite depth water wave problem, neglecting surface tension. For such solutions, it is well known that one formally expects the modulation to be a profile traveling at group velocity and governed by a 2D hyperbolic cubic nonlinear Schrödinger equation. In this paper we justify this fact by providing rigorous error estimates in Sobolev spaces. We reproduce the multiscale calculation to derive an approximate wave packet-like solution to the evolution equations with mild quadratic nonlinearities constructed by Sijue Wu. Then we use the energy method along with the method of normal forms to provide suitable a priori bounds on the difference between the true and approximate 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

Similar content being viewed by others

References

  1. Ablowitz, M.J., Segur, H.: Solitons and the inverse scattering transform. In: SIAM Studies in Applied Mathematics, vol. 4. SIAM (2000)

  2. Bergh, J., Löfström, J.: Interpolation spaces: an introduction. In: Grundlehren der mathematischen Wissenschaften, vol. 223. Springer, Berlin (1976)

  3. Cazenave, T.: Semilinear Schrödinger equations. In: Courant Lecture Notes, vol. 10. American Mathematical Society, Providence

  4. Coifman R.R., David G., Meyer Y.: La solution des conjectures de Calderón. Adv. Math. 48, 144–148 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  5. Coifman R.R., McIntosh A., Meyer Y.: L’intégral de Cauchy Definit un Operateur Borne sur L 2 Pour Les Courbes Lipschitziennes. Ann. Math. 2nd Series 116(2), 361–387 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  6. Craig W., Schanz U., Sulem C.: The modulational regime of three-dimensional water waves and the davey-stewartson system. Annales de l’Institute Henri Poincare Analyse nonlineaire 14(5), 615–667 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Gallay T., Schneider G.: KP description of unidirectional long waves. The model case. Proc. Roy. Soc. Edinburgh Sect. A 131, 885–898 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Germain P., Masmoudi N., Shatah J.: Global solutions for the gravity water waves equation in dimension 3. Ann. Math. 175(2), 691–754 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ghidaglia J.M., Saut J.C.: On the initial value problem for the davey-stewartson systems. Nonlinearity 3, 475–506 (1990)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Gilbert, J.E., Murray, M.A.M.: Clifford algebras and Dirac operators in harmonic analysis. In: Cambridge Studies in Advanced Mathematics, vol. 26. Cambridge University, Cambridge (1991)

  11. Kirrmann P., Schneider G., Mielke A.: The validity of modulation equations for extended systems with cubic nonlinearities. Proc. Roy. Soc. Edinburgh Sect. A 122(1–2), 85–91 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lannes, D.: The water waves problem: mathematical analysis and asymptotics. In: Mathematical Surveys and Monographs, vol. 188. American Mathematical Society, Providence (2013)

  13. Lannes D., Rauch J.: Validity of nonlinear geometric optics with times growing logarithmically. Proc. Am. Math. Soc. 129(4), 1087–1096 (2000)

    Article  MathSciNet  Google Scholar 

  14. Abou Salem, W.K.: On the renormalization group approach to perturbation theory for PDEs. Ann. Henri Poincare 11, 1007–1021 (2010)

  15. Schneider G.: Validity and limitation of the Newell-Whitehead equation. Math. Nachr. 176, 249–263 (1995)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. Schneider G., Wayne C.E.: Justification of the NLS approximation for a quasilinear water wave model. J. Differ. Equ. 251(2), 238–269 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Shatah J.: Normal forms and quadratic nonlinear Klein-Gordon equations. Comm. Math. Phys. 38(5), 685–696 (1985)

    MATH  MathSciNet  Google Scholar 

  19. Stein E.M.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)

    MATH  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. Wu S.: Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Am. Math. Soc. 12(2), 445–495 (1999)

    Article  MATH  Google Scholar 

  22. Wu S.: Global wellposedness of the 3-d full water wave problem. Inventiones Mathematicae 182, 125–220 (2010)

    Google Scholar 

  23. Zakharov V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. Zhurnal Prikladnoi Mekhaniki i Teckhnicheskoi Fiziki 9(2), 86–94 (1968)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Duke University, Durham, NC, 27708, USA

    Nathan Totz

Authors
  1. Nathan Totz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nathan Totz.

Additional information

Communicated by N. Totz

The author would like to thank Sijue Wu for her discussions on her formulation of the 3D water wave problem, as well as her helpful comments and suggestions on the draft of this paper. The author was supported in part by NSF grant DMS-0800194.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Totz, N. A Justification of the Modulation Approximation to the 3D Full Water Wave Problem. Commun. Math. Phys. 335, 369–443 (2015). https://doi.org/10.1007/s00220-014-2259-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-014-2259-7

Keywords

Search

Navigation