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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 consider the Degasperis-Procesi equation with the Coriolis effect. We demonstrate a rigorous justification of the solution of the classical Degasperis-Procesi equation as an approximation to that of the Degasperis-Procesi equation with the Coriolis effect in a long time scale when the effect of the Earth's rotation vanishes. Also, nonexistence of the classical Camassa-Holm type peaked solitary waves is investigated. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of peaked traveling wave solution of the generalized μ-Novikov equation with nonlocal cubic and quadratic nonlinearities. The equation is a μ-version of a linear combination of the Novikov equation and Camassa-Hom equation. It is found that the equation admits single peaked traveling wave solutions.

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]

In this paper, we study the existence of peaked traveling wave solution of the generalized μ-Novikov equation with nonlocal cubic and quadratic nonlinearities. The equation is a μ-version of a linear combination of the Novikov equation and Camassa-Hom equation. It is found that the equation admits single peaked traveling wave solutions. [ABSTRACT FROM AUTHOR]

Considered herein is a generalized periodic Novikov equation with cubic nonlinearity which is called the -Novikov equation. It is shown that the equation admits a single peaked soliton with a similar character of the -Camassa-Holm equation. [ABSTRACT FROM AUTHOR]

We study the Korteweg-de Vries equation posed on the quarter plane with asymptotically t-periodic boundary data for large t > 0. We derive an expression for the Dirichlet to Neumann map to all orders in the perturbative expansion of a small ϵ > 0 in the case of the asymptotically periodic boundary data. More precisely, we show that if the unknown Neumann boundary data are asymptotically periodic for large t in the sense that u x ( 0 , t ) and u x x ( 0 , t ) tend to periodic functions g ˜ 1 ( t ) and g ˜ 2 ( t ) for large t, respectively, then the periodic functions g ˜ 1 ( t ) and g ˜ 2 ( t ) can be characterized in terms of the given asymptotically periodic Dirichlet boundary datum u ( 0 , t ). Moreover, we determine effectively the Fourier coefficients of the functions g ˜ 1 ( t ) and g ˜ 2 ( t ) by solving a certain recursive algebraic equations. [ABSTRACT FROM AUTHOR]