Solitons and periodic waves for the (2 + 1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics

Abstract

Fluid mechanics has the applications in a wide range of disciplines, such as oceanography, astrophysics, meteorology, and biomedical engineering. Under investigation in this paper is the ($$2+1$$)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics. Via the Pfaffian technique and certain constraint on the real constant $$\alpha $$, the Nth-order Pfaffian solutions are derived. One- and two-soliton solutions are obtained via the Nth-order Pfaffian solutions. Based on the Hirota–Riemann method, one- and two-periodic wave solutions are constructed. With the help of the analytic and graphic analysis, we notice that: (1) of the one soliton, amplitude is irrelevant to $$\gamma $$, a real constant coefficient in the equation, velocity along the x direction is independent of $$\gamma $$, while velocity along the y direction is proportional to $$\gamma $$; (2) one soliton keeps its amplitude and velocity invariant during the propagation and total amplitude of the two solitons in the interaction region is lower than that of any soliton; (3) one-periodic wave can be viewed as a superposition of the overlapping solitary waves, placed one period apart; (4) periodic behaviors for the two-periodic wave exist along the x and y directions, respectively; (5) under certain limiting conditions, one-periodic wave solutions approach to the one-soliton solutions and two-periodic wave solutions approach to the two-soliton solutions.