The higher-order lump, breather and hybrid solutions for the generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation in fluid mechanics

Abstract

Under investigation in this paper is the (2 + 1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation, which can be utilized to describe certain nonlinear phenomena in fluid mechanics. We obtain the higher-order lump, breather and hybrid solutions, and analyze the effects of the constant coefficients $$h_1$$ , $$h_2$$ , $$h_4$$ and $$h_5$$ in that equation on those solutions, since the higher-order lump solutions are generalized via the long-wave limit method, and since the higher-order breather solutions and hybrid solutions composed of the solitons, breathers and lumps are derived. With the help of the analytic and graphic analysis, we get the following: (1) amplitudes of the humps and valleys of the first-order lumps are related to $$h_1$$ , $$h_2$$ , $$h_4$$ and $$h_5$$ , proportional to $$h_4$$ while inversely proportional to $$h_2$$ . Velocities of the first-order lumps are proportional to $$h_4$$ . The second-order lumps describe the interaction between the two first-order lumps, which is elastic since those lumps keep their shapes, velocities and amplitudes unchanged after the interaction. Effects of $$h_2$$ and $$h_4$$ on the second-order lumps are graphically illustrated. (2) Amplitudes of the first-order breathers are proportional to $$h_2$$ . Interaction between the breather waves is graphically presented. Effects of $$h_2$$ and $$h_1$$ on the amplitudes and shapes of the second-order breathers are graphically discussed. (3) Elastic interactions are graphically illustrated, between the first-order breathers and one solitons, the first-order lumps and one solitons, as well as the first-order breathers and first-order lumps. Also graphically illustrated, amplitudes of all those three kinds of hybrid solutions are inversely proportional to $$h_2$$ , and velocity of the one soliton is positively correlated to $$h_4$$ .