Skip to main content
SpringerLink
Log in

Zero-Dispersion Limit for the Benjamin–Ono Equation on the Torus with Bell Shaped Initial Data

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

Abstract

We consider the zero-dispersion limit for the Benjamin–Ono equation on the torus. We prove that when the initial data is bell shaped, the zero-dispersion limit exists in the weak sense and is uniform on every compact time interval. Moreover, the limit is equal to the signed sum of branches for the multivalued solution of the inviscid Burgers equation obtained by the method of characteristics. This result is similar to the one obtained by Miller and Xu for the Benjamin–Ono equation on the real line for decaying and positive initial data. We also establish some precise asymptotics of the spectral data with initial data \(u_0(x)=-\beta \cos (x)\), \(\beta >0\), justifying our approximation method, which is analogous to the work of Miller and Wetzel concerning a family of rational potentials for the Benjamin–Ono equation on the real line.

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
Fig. 3

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Data availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Benjamin, T.B.: Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29(3), 559–592 (1967). https://doi.org/10.1017/S002211206700103X

    Article  ADS  MATH  Google Scholar 

  2. Ono, H.: Algebraic solitary waves in stratified fluids. J. Phys. Soc. Jpn. 39(4), 1082–1091 (1975). https://doi.org/10.1143/JPSJ.39.1082

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Miller, P.D., Xu, Z.: On the zero-dispersion limit of the Benjamin–Ono Cauchy problem for positive initial data. Commun. Pure Appl. Math. 64(2), 205–270 (2011). https://doi.org/10.1002/cpa.20345

    Article  MathSciNet  MATH  Google Scholar 

  4. Matsuno, Y.: Nonlinear modulation of periodic waves in the small dispersion limit of the Benjamin–Ono equation. Phys. Rev. E 58(6), 7934 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  5. Moll, A.: Finite gap conditions and small dispersion asymptotics for the classical periodic Benjamin–Ono equation. Quart. Appl. Math. 78, 671–702 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  6. Moll, A.: Exact Bohr–Sommerfeld conditions for the quantum periodic Benjamin–Ono equation. SIGMA 15(098), 1–27 (2019)

    MathSciNet  MATH  Google Scholar 

  7. Gérard, P., Kappeler, T.: On the integrability of the Benjamin–Ono equation on the torus. Commun. Pure Appl. Math. (2020). https://doi.org/10.1002/cpa.21896

    Article  MATH  Google Scholar 

  8. Gérard, P., Kappeler, T., Topalov, P.: Sharp well-posedness results of the Benjamin–Ono equation in \(H^s({\mathbb{T}},{\mathbb{R}})\) and qualitative properties of its solution. Acta Math. (to appear). arxiv:2004.04857 [math.AP] (2020)

  9. Gérard, P., Kappeler, T., Topalov, P.: On the spectrum of the Lax operator of the Benjamin–Ono equation on the torus. J. Funct. Anal. 279(12), 108762 (2020). https://doi.org/10.1016/j.jfa.2020.108762

    Article  MathSciNet  MATH  Google Scholar 

  10. Gérard, P., Kappeler, T., Topalov, P.: On the analytic Birkhoff normal form of the Benjamin–Ono equation and applications. Nonlinearity (to appear). arxiv:2103.07981 (2021)

  11. Gérard, P., Kappeler, T., Topalov, P.: On the analyticity of the nonlinear Fourier transform of the Benjamin–Ono equation on \({\mathbb{T}}\) (2021)

  12. Gérard, P.: A nonlinear Fourier transform for the Benjamin–Ono equation on the torus and applications. In: Séminaire Laurent Schwartz—EDP et Applications, pp. 1–19. https://doi.org/10.5802/slsedp.138 (2019-2020)

  13. Miller, P.D., Xu, Z.: The Benjamin–Ono hierarchy with asymptotically reflectionless initial data in the zero-dispersion limit. Commun. Math. Sci. 10(1), 117–130 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fokas, A., Ablowitz, M.: The inverse scattering transform for the Benjamin–Ono equation—a pivot to multidimensional problems. Stud. Appl. Math. 68(1), 1–10 (1983). https://doi.org/10.1002/sapm19836811

    Article  MathSciNet  MATH  Google Scholar 

  15. Coifman, R.R., Wickerhauser, M.V.: The scattering transform for the Benjamin–Ono equation. Inverse Prob. 6(5), 825 (1990). https://doi.org/10.1088/0266-5611/6/5/011

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Kaup, D., Matsuno, Y.: The inverse scattering transform for the Benjamin–Ono equation. Stud. Appl. Math. 101(1), 73–98 (1998). https://doi.org/10.1111/1467-9590.00086

    Article  MathSciNet  MATH  Google Scholar 

  17. Wu, Y.: Simplicity and finiteness of discrete spectrum of the Benjamin–Ono scattering operator. SIAM J. Math. Anal. 48(2), 1348–1367 (2016). https://doi.org/10.1137/15M1030649

    Article  MathSciNet  MATH  Google Scholar 

  18. Wu, Y.: Jost solutions and the direct scattering problem of the Benjamin–Ono equation. SIAM J. Math. Anal. 49(6), 5158–5206 (2017). https://doi.org/10.1137/17M1124528

    Article  MathSciNet  MATH  Google Scholar 

  19. Sun, R.: Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds. Commun. Math. Phys. 383, 1051–1092 (2021). https://doi.org/10.1007/s00220-021-03996-1

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Miller, P.D., Wetzel, A.N.: Direct scattering for the Benjamin–Ono equation with rational initial data. Stud. Appl. Math. 137(1), 53–69 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Miller, P.D., Wetzel, A.N.: The scattering transform for the Benjamin–Ono equation in the small-dispersion limit. Physica D 333, 185–199 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Lax, P.D., David Levermore, C.: The small dispersion limit of the Korteweg-de Vries equation. Commun. Pure Appl. Math. 36(3), 253–290571593809829 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  23. Venakides, S.: The zero dispersion of the Korteweg–de Vries equation for initial potentials with non-trivial reflection coefficient. Commun. Pure Appl. Math. 38(2), 125–155 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  24. Venakides, S.: The Korteweg–de Vries Equation with Small Dispersion: Higher Order Lax-Levermore Theory, pp. 255–262. Springer, Berlin (1991)

    MATH  Google Scholar 

  25. Deift, P., Venakides, S., Zhou, X.: New results in small dispersion KdV by an extension of the steepest descent method for Riemann–Hilbert problems. Int. Math. Res. Not. 1997(6), 285–299 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Claeys, T., Grava, T.: Universality of the break-up profile for the KdV equation in the small dispersion limit using the Riemann–Hilbert approach. Commun. Math. Phys. 286(3), 979–1009 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Claeys, T., Grava, T.: Painlevé II asymptotics near the leading edge of the oscillatory zone for the Korteweg–de Vries equation in the small-dispersion limit. Commun. Pure Appl. Math. 63(2), 203–232 (2010)

    Article  MATH  Google Scholar 

  28. Claeys, T., Grava, T.: Solitonic asymptotics for the Korteweg–de Vries equation in the small dispersion limit. SIAM J. Math. Anal. 42(5), 2132–2154 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Venakides, S.: The zero dispersion limit of the Korteweg–de Vries equation with periodic initial data. Trans. Am. Math. Soc. 189–226 (1987)

  30. Deng, G., Biondini, G., Trillo, S.: Small dispersion limit of the Korteweg–de Vries equation with periodic initial conditions and analytical description of the Zabusky–Kruskal experiment. Physica D 333, 137–147 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Zabusky, N.J., Kruskal, M.D.: Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15(6), 240 (1965)

    Article  ADS  MATH  Google Scholar 

  32. Wong, R.: Asymptotic approximations of integrals. Corrected reprint of the 1989 original edn. Classics Appl. Math., vol. 34. SIAM, Philadelphia, PA (2001). https://doi.org/10.1137/1.9780898719260

  33. Gassot, L.: Lax eigenvalues in the zero-dispersion limit for the Benjamin–Ono equation on the torus. arXiv preprint arXiv:2301.03919 (2023)

  34. Gassot, L.: The third order Benjamin–Ono equation on the torus: well-posedness, traveling waves and stability. Ann. Inst. Henri Poincaré C, Anal. Non linéaire 38(3), 815–840 (2021). https://doi.org/10.1016/j.anihpc.2020.09.004

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Lorentz, G., Zeller, K.: Degree of approximation by monotone polynomials I. J. Approx. Theory 1(4), 501–504 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  36. Dzyubenko, G., Pleshakov, M.: Comonotone approximation of periodic functions. Math. Notes 83(1), 180–189 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Hörmander, L.: The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. Springer, Berlin (2015)

    MATH  Google Scholar 

  38. Faraut, J.: Calcul Intégral. EDP sciences, France (2021)

    Google Scholar 

Download references

Acknowledgements

The author is grateful to Patrick Gérard for relevant advice on this problem, in particular, for providing notes on Proposition 21 about the cosine function. The author would also like to warmly thank the reviewer for precise and constructive remarks about this work. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 while the author was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the “Hamiltonian Methods in Dispersive and Wave Evolution Equations” program.

Author information

Authors and Affiliations

  1. ICERM, Brown University, 121 South Main Street, Providence, 02903, RI, USA

    Louise Gassot

  2. Univ Rennes, CNRS, IRMAR - UMR 6625, 35000, Rennes, France

    Louise Gassot

Authors
  1. Louise Gassot
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Louise Gassot.

Additional information

Communicated by A. Ionescu.

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gassot, L. Zero-Dispersion Limit for the Benjamin–Ono Equation on the Torus with Bell Shaped Initial Data. Commun. Math. Phys. 401, 2793–2843 (2023). https://doi.org/10.1007/s00220-023-04701-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-023-04701-0

Search

Navigation