Skip to main content
SpringerLink
Log in

On the isospectral problem of the Camassa–Holm equation

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

In this paper, we study an isospectral problem of a weighted Sturm–Liouville equation with the Dirichlet boundary condition, which lies at the basis of the integrability of the Camassa–Holm equation. We will choose the general setting of the so-called measure differential equations to solve the optimization problem on eigenvalues. It should be noticed that our technique in this paper can be used to deal with other self-adjoint boundary conditions.

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. Bressan, A., Constantin, A.: Global conservative solutions of the Camassa–Holm equation. Arch. Ration. Mech. Anal. 183, 215–239 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Camassa, R., Holm, D., Hyman, J.: A new integrable shallow water equation. Adv. Appl. Mech. 31, 1–33 (1994)

    Article  MATH  Google Scholar 

  4. Carter, M., van Brunt, B.: The Lebesgue-Stieltjes Integral: A Practical Introduction. Springer, New York (2000)

    Book  MATH  Google Scholar 

  5. Chu, J., Meng, G., Zhang, M.: Continuity and minimization of spectrum related with the periodic Camassa–Holm equation. J. Differential Equations 265, 1678–1695 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. Constantin, A.: On the spectral problem for the periodic Camassa–Holm equation. J. Math. Anal. Appl. 210, 215–230 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Constantin, A.: On the Cauchy problem for the periodic Camassa–Holm equation. J. Differential Equations 141, 218–235 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Constantin, A.: Quasi-periodicity with respect to time of spatially periodic finite-gap solutions of the Camassa–Holm equation. Bull. Sci. Math. 122, 487–494 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Constantin, A.: On the inverse spectral problem for the Camassa–Holm equation. J. Funct. Anal. 155, 352–363 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Constantin, A., McKean, H.P.: A shallow water equation on the circle. Comm. Pure Appl. Math. 52, 949–982 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Eckhardt, J.: The inverse spectral transform for the conservative Camassa–Holm flow with decaying initial data. Arch. Ration. Mech. Anal. 224, 21–52 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Eckhardt, J., Kostenko, A.: An isospectral problem for global conservative multi-peakon solutions of the Camassa–Holm equation. Comm. Math. Phys. 329, 893–918 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Eckhardt, J., Teschl, G.: On the isospectral problem of the dispersionless Camassa–Holm equation. Adv. Math. 235, 469–495 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fu, Y., Liu, Y., Qu, C.: On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations. J. Funct. Anal. 262, 3125–3158 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Halas, Z., Tvrdý, M.: Continuous dependence of solutions of generalized linear differential equations on a parameter. Funct. Differential Equations 16, 299–313 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Holden, H., Raynaud, X.: Periodic conservative solutions of the Camassa–Holm equation. Ann. Inst. Fourier (Grenoble) 58, 945–988 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. McKean, H.P.: Breakdown of the Camassa–Holm equation. Comm. Pure Appl. Math. 56, 998–1015 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Meng, G.: Extremal problems for eigenvalues of measure differential equations. Proc. Amer. Math. Soc. 143, 1991–2002 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Meng, G., Yan, P.: Optimal lower bound for the first eigenvalue of the fourth order equation. J. Differential Equations 261, 3149–3168 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Meng, G., Zhang, M.: Dependence of solutions and eigenvalues of measure differential equations on measures. J. Differential Equations 254, 2196–2232 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mingarelli, A.B.: Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions. Lecture Notes in Mathematics, vol. 989. Springer, New York (1983)

  22. Qi, J., Chen, S.: Extremal norms of the potentials recovered from inverse Dirichlet problems. Inverse Probl. 32, 035007, 13 pp. (2016)

  23. Schwabik, Š: Generalized Ordinary Differential Equations. World Scientific, Singapore (1992)

    Book  MATH  Google Scholar 

  24. Yan, P., Zhang, M.: Continuity in weak topology and extremal problems of eigenvalues of the \(p\)-Laplacian. Trans. Amer. Math. Soc. 363, 2003–2028 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhang, M.: Minimization of the zeroth Neumann eigenvalues with integrable potentials. Ann. Inst. H. Poincaré C Anal. Non Linéaire 29, 501–523 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhu, H., Shi, Y.: Dependence of eigenvalues on the boundary conditions of Sturm-Liouville problems with one singular endpoint. J. Differential Equations 263, 5582–5609 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The third author is supported by the National Natural Science Foundation of China (Grant Nos. 12071456 and 12271509) and the Fundamental Research Funds for the Central Universities.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Yaru Dou & Gang Meng

  2. The High School Affiliated to Renmin University of China, Beijing, 100080, China

    Jieyu Han

Authors
  1. Yaru Dou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jieyu Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gang Meng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gang Meng.

Additional information

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

Dou, Y., Han, J. & Meng, G. On the isospectral problem of the Camassa–Holm equation. Arch. Math. 121, 67–76 (2023). https://doi.org/10.1007/s00013-023-01870-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-023-01870-1

Keywords

Mathematics Subject Classification

Search

Navigation