Skip to main content
SpringerLink
Log in

The Existence and Uniqueness of Global Admissible Conservative Weak Solution for the Periodic Single-Cycle Pulse Equation

  • Published:
Journal of Mathematical Fluid Mechanics Aims and scope Submit manuscript

Abstract

This paper is devoted to studying the existence and uniqueness of global admissible conservative weak solution for the periodic single-cycle pulse equation without any additional assumptions. Firstly, introducing a new set of variables, we transform the single-cycle pulse equation into an equivalent semilinear system. Using the standard ordinary differential equation theory, the global solution of the semilinear system is studied. Secondly, returning to the original coordinates, we get a global admissible conservative weak solution for the periodic single-cycle pulse equation. Finally, choosing some vital test functions which are different from [Bressan (Discrete Contin. Dyn. Syst 35:25-42, 2015), Brunelli (Phys. Lett. A 353:475-478, 2006)], we find a equation to single out a unique characteristic curve through each initial point. Moreover, the uniqueness of global admissible conservative weak solution is obtained.

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., Chen, G., Zhang, Q.: Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics. Discrete Contin. Dyn. Syst. 35, 25–42 (2015)

    Article  MathSciNet  Google Scholar 

  2. Bressan, A.: Constantin a: global solutions of the Hunter-Saxton equation. SIAM J. Math. Anal. 37, 996–1026 (2005)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Brunelli, J.C.: The bi-Hamiltonian structure of the short pulse equation. Phys. Lett. A 353, 475–478 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  5. Cheng, Q., Zhang, D., Wei, G., Hu, S., Wang, S.: Basis of Real Variable Function and Functional Analysis (in Chinese). Higher Education Press, China (2010)

  6. Constantin, A.: On the scattering problem for the Camassa-Holm equation. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457, 953–970 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  7. Constantin, A., Escher, J.: Global existence and blow-up for a shallow water equation. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 26(4), 303–328 (1998)

    MathSciNet  MATH  Google Scholar 

  8. Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998)

    Article  MathSciNet  Google Scholar 

  9. Constantin, A., Escher, J.: Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation. Comm. Pure Appl. Math. 51, 475–504 (1998)

    Article  MathSciNet  Google Scholar 

  10. Constantin, A., Gerdjikov, V.S., Ivanov, R.I.: Inverse scattering transform for the Camassa-Holm equation. Inverse Probl. 22, 2197–2207 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  11. Constantin, A., Molinet, L.: Global weak solutions for a shallow water equation. Comm. Math. Phys. 211, 45–61 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  12. Dai, H., Pavlov, M.: Transformations for the Camassa-Holm equation, its high-frequency limit and the Sinh-Gordon equation. J. Phys. Soc. Japan 67, 3655–3657 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  13. Danchin, R.: A note on well-posedness for Camassa-Holm equation. J. Differ. Equ. 192, 429–444 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  14. Edwards R.E.: Fourier Series. A Modern Introduction. Vol. 1, volume 64 of Graduate Texts in Mathematics. Springer-Verlag, New York-Berlin (1979)

  15. Fuchssteiner B., Fokas A. S.: Symplectic structures, their Backlund transformations and hereditary symmetries. Phys. D 4, 47–66 (1981/82)

  16. Grunert, K., Holden, H., Raynaud, X.: Lipschitz metric for the periodic Camassa-Holm equation. J. Differ. Equ. 250, 1460–1492 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  17. Guo, Z., Liu, X., Molinet, L., Yin, Z.: Ill-posedness of the Camassa-Holm and related equations in the critical space. J. Differ. Equ. 266, 1698–1707 (2019)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Hone, A.N.W., Novikov, V., Wang, J.: Generalizations of the short pulse equation. Lett. Math. Phys. 108, 927–947 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  20. Hunter, J.K., Saxton, R.: Dynamics of director fields. SIAM J. Appl. Math. 51, 1498–1521 (1991)

    Article  MathSciNet  Google Scholar 

  21. Li, J., Zhang, K.: Global existence of dissipative solutions to the Hunter-Saxton equation via vanishing viscosity. J. Differ. Equ. 250, 1427–1447 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  22. Li, M., Yin, Z.: Global existence and local well-posedness of the single-cycle pulse equation. J. Math. Phys. 58, 101515 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  23. Liu, Y., Pelinovsky, D., Sakovich, A.: Wave breaking in the short-pulse equation. Dyn. Partial Differ. Equ. 6, 291–310 (2009)

    Article  MathSciNet  Google Scholar 

  24. Parkes, E.J.: A note on loop-soliton solutions of the short-pulse equation. Phys. Lett. A 374, 4321–4323 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  25. Pelinovsky, D., Sakovich, A.: Global well-posedness of the short-pulse and sine-Gordon equations in energy space. Comm. Partial Differ. Equ. 35, 613–629 (2010)

    Article  MathSciNet  Google Scholar 

  26. Sakovich, A., Sakovich, S.: The short pulse equation is integrable. J. Phys. Soc. Japan 74, 33–40 (2005)

    Article  Google Scholar 

  27. Sakovich, A., Sakovich, S.: Solitary wave solutions of the short pulse equation. J. Phys. A 39, L361–L367 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  28. Sakovich, S.: Transformation and integrability of a generalized short pulse equation. Commun. Nonlinear Sci. Numer. Simul. 39, 21–28 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  29. Schäfer, T., Wayne, C.E.: Propagation of ultra-short optical pulses in cubic nonlinear media. Phys. D 196, 90–105 (2004)

    Article  MathSciNet  Google Scholar 

  30. Schmeisser, H.J., Triebel, H.: Topics in Fourier Analysis and Function Spaces. A Wiley-Interscience Publication, John Wiley & Sons Ltd, Chichester (1987)

  31. Xin, Z., Zhang, P.: On the weak solutions to a shallow water equation. Comm. Pure Appl. Math. 53, 1411–1433 (2000)

    Article  MathSciNet  Google Scholar 

  32. Xin, Z., Zhang, P.: On the uniqueness and large time behavior of the weak solutions to a shallow water equation. Comm. Partial Differ. Equ. 27, 1815–1844 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Guo was partially supported by the Guangdong Basic and Applied Basic Research Foundation (No. 2020A1515111092) and Research Fund of Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic Technology (No. 2020B1212030010). Yin was partially supported by NNSFC (No. 11671407), Guangdong Special Support Program (No. 8-2015) and the key project of NSF of Guangdong Province (No. 2016A030311004).

Author information

Authors and Affiliations

  1. School of Mathematics and Big Data, Foshan University, Foshan, 528000, China

    Yingying Guo

  2. Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

    Zhaoyang Yin

Authors
  1. Yingying Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhaoyang Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yingying Guo.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by A. Constantin.

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, Y., Yin, Z. The Existence and Uniqueness of Global Admissible Conservative Weak Solution for the Periodic Single-Cycle Pulse Equation. J. Math. Fluid Mech. 24, 57 (2022). https://doi.org/10.1007/s00021-022-00691-6

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00021-022-00691-6

Keywords

Mathematics Subject Classification

Search

Navigation