Periodic peakons to a generalized μ-Camassa–Holm–Novikov equation

Abstract

In this paper, we study the existence of periodic peaked solitary waves to a generalized μ-Camassa–Holm–Novikov equation with nonlocal cubic and quadratic nonlinearities. The equation is a μ-version of a linear combination of the Camassa–Holm, modified Camassa–Holm, and Novikov equations. It is shown that the proposed equation admits a single peakons. It is natural extension of the previous results obtained in [Khesin B, Lenells J, Misiolek G. Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms. Math Ann. 2008;342:617–656; Moon B. The existence of the single-peaked traveling waves to the μ-Novikov equation. Appl Anal. 2018;97:1540–1548; Qu CZ, Fu T, Liu Y. Well-posedness, wave breaking and peakons for a modified μ-Camassa–Holm equation. J Funct Anal. 2014;266(2):433–477.] for the μ-Camassa–Holm, modified μ-Camassa–Holm, and μ-Novikov equations, respectively.

Keywords:

• Generalized μ-Camassa–Holm–Novikov equation
• Peakons

2010 Mathematics Subject Classifications:

• 35B30
• 35G25

Acknowledgments

The work of Hwang was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT)(2020R1F1A1A01067944). The work of Moon was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2020R1F1A1A01048468).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work of Hwang was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) [grant number Geunbo Hwang/2020R1F1A1A01067944]. The work of Moon was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) [grant number Byungsoo Moon/2020R1F1A1A01048468].