Hybrid exact solutions of the (3 + 1)-dimensional variable-coefficient nonlinear wave equation in liquid with gas bubbles

Abstract

In this paper, we study some exact solutions of the (3 + 1)-dimensional variable-coefficient nonlinear wave equation in liquid with gas bubbles. Based on a given transformation and the bilinear form, together with symbolic computations, we obtain the multi-soliton solutions and periodic solutions including X-periodic, Y-periodic and 2-periodic wave solutions. By using of the long wave limit method, we obtain lump solutions and lump-periodic solutions that lumps moving on a background with different periodic waves. The properties of the solutions are analyzed graphically.

Keywords

• Variable-coefficient nonlinear wave equation
• Bilinear form
• Periodic solution
• Lump solution
• Lump-periodic solution