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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.

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