Eur. Phys. J. Plus (2017) 132: 255
https://doi.org/10.1140/epjp/i2017-11515-8

Regular Article

Integrability, solitons, periodic and travelling waves of a generalized (3+1)-dimensional variable-coefficient nonlinear-wave equation in liquid with gas bubbles

Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, 100191, Beijing, China

^* e-mail: gaoyt163@163.com

Received: 23 December 2016
Accepted: 24 April 2017
Published online: 9 June 2017

Abstract

Under investigation in this paper is a generalized (3+1)-dimensional varible-coefficient nonlinear-wave equation, which has been presented for nonlinear waves in liquid with gas bubbles. The bilinear form, Bäcklund transformation, Lax pair and infinitely-many conservation laws are obtained via the binary Bell polynomials. One-, two- and three-soliton solutions are generated by virtue of the Hirota method. Travelling-wave solutions are derived with the aid of the polynomial expansion method. The one-periodic wave solutions are constructed by the Hirota-Riemann method. Discussions among the soliton, periodic- and travelling-wave solutions are presented: I) the soliton velocities are related to the variable coefficients, while the soliton amplitudes are unaffected; II) the interaction between the solitons is elastic; III) there are three cases of the travelling-wave solutions, i.e., the triangle-type periodical, bell-type and soliton-type travelling-wave solutions, while we notice that bell-type travelling-wave solutions can be converted into one-soliton solutions via taking suitable parameters; IV) the one-periodic waves approach to the solitary waves under some conditions and can be viewed as a superposition of overlapping solitary waves, placed one period apart.