Bilinear form and solutions of a (3+1)-dimensional generalized nonlinear evolution equation for the shallow-water waves

Abstract

A (3+1)-dimensional generalized nonlinear evolution equation for the shallow-water waves is investigated. Bilinear form is derived and semi-rational solutions are constructed via the Kadomtsev–Petviashvili hierarchy reduction. Interactions between the lumps and solitons are analyzed. For the first-order semi-rational solutions, we observe that (1) the lump and the soliton fuse into the soliton; (2) the lump arises from the soliton and then separates from the soliton; (3) the first-order semi-rational solutions on the y−z plane possess a line profile and the wave shape changes with t varying. For the multi-semi-rational solutions, we find that on the x−y plane, the two lumps fuse into the two solitons, and on the x−z and y−z planes, the two lumps emerge on and then split from the two solitons. For the higher-order semi-rational solutions, we observe three kinds of interaction phenomena: (1) The two lumps fuse into the soliton; (2) The two lumps arise from the soliton and then separate from the soliton; (3) The two lumps which propagate towards each other fuse into one lump, and then that one splits into two other lumps. Influences of the coefficients in the original equation on the semi-rational solutions are also revealed.

Keywords:

• Shallow-water waves
• semi-rational solutions
• interactions
• (3+1)-dimensional generalized nonlinear evolution equation
• Kadomtsev–Petviashvili hierarchy reduction

2010 Mathematics Subject Classification:

• nonlinear partial differential equation

Acknowledgments

The authors express sincere thanks to the members of our discussion group for their valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Via the transformations $u(x,t)\to U(X,T),x\to \sqrt{3}X$ and $t\to 6\sqrt{3}T$, the equation $2u_t+u_{xxx}-2u{u}_x=0$ has been transformed to the KdV equation [33,34].

2 When $h_1=3,h_2=-2,h_3=-1,h_4=h_5=2$, Equation (2(2) $h_1{u}_{xz}+(h_2{u}_t+h_3{u}_{xxx}+h_{4}u{u}_x{)}_y+h_5(u_x{\mathrm{\partial }}_x^{-1}{u}_y{)}_x=0,$(2) ) can be reduced to Equation (1(1) $3u_{xz}-(2u_t+u_{xxx}-2u{u}_x{)}_y+2(u_x{\mathrm{\partial }}_x^{-1}{u}_y{)}_x=0,$(1) ).

Additional information

Funding

This work has been supported by the National Natural Science Foundation of China under Grant Nos. 11272023, 11772017, and by the Fundamental Research Funds for the Central Universities [grant number 50100002016105010].