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In this study, we establish the existence and nonexistence of smooth and peaked solitary wave solutions (or periodic) to the Degasperis-Procesi-Kadomtsev-Petviashvili (DP-KP) equation with a weak transverse effect. We have also shown that DP-KP equation possesses periodic shock waves similar to that of the Degasperis-Procesi equation.

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. [ABSTRACT FROM AUTHOR]