N-Lump Solutions to a (3+1)-Dimensional Variable-Coefficient Generalized Nonlinear Wave Equation in a Liquid with Gas Bubbles

In this paper, the (3+1)-dimensional variable-coefficient generalized nonlinear wave equation arising in a liquid with gas bubbles is studied. The Hirota bilinear technique and binary bell polynomials are considered. Some new analytic solutions containing one-lump soliton, two-lump soliton, and three-lump soliton, and also 1-breather and 1-lump, the interaction 1-bell-shaped soliton with 1-lump or 1-breather, and other soliton solutions to the mentioned equation are obtained and analyzed. To create the solutions of Maple symbolic package is used. Using suitable mathematical assumptions and Hirota’s bilinear technique, the new types of M-soliton and N-soliton solutions are derived and established in view of the hyperbolic, trigonometric, and rational functions of the governing equation. To achieve this, an illustrative example of the nonlinear wave equation in a liquid with gas bubbles is provided to demonstrate the feasibility and reliability of the procedure used in this study. The trajectory solutions of the N-soliton are shown explicitly and graphically (2D, density, and 3D graphs). The effect of parameters on the behavior of acquired solutions for N=3,4,5 orders are also discussed. By comparing the proposed method with the other existing methods, the results show that the execution of this method is concise, simple, and straightforward. The results are useful for obtaining and explaining some new soliton phenomena.