A study on the resonant multi-soliton waves and the soliton molecule of the (3+1)-dimensional Kudryashov–Sinelshchikov equation

Abstract

In this paper, a reduced (3+1)-dimensional nonlinear wave equation describing the dynamic of liquid with gas bubble, namely Kudryashov-Sinelshchikov (KS) equation is carefully investigated. After applying the simplified linear superposition principle (LSP) and the velocity resonance (VR) the fact that there has no existence of soliton molecule to KS equation is formally proofed. Simultaneously, two new resonant multi-soliton solutions constructed with distinct physical structures are also generated. Comparing with the existing ones our solutions are much more general. Most of all, their formula forms reveal the influences of the arbitrary coefficients of the bubble-liquid-dispersion, the y-transverse-perturbation and the z-transverse-perturbation to the traveling soliton waves. Finally, the propagations of inelastic interactions of resonant two-soliton waves are shown in Figures 1-6. Particularly, by specifying special values to the resonant multi-soliton solution gives three-bound-soliton waves moving with the same speed and direction which is in the manner similar to the property of soliton molecule, as shown in Figure 6. The extracted results not only show the efficiency of the simplified LSP but also have potential applications to the physical phenomena of soliton molecules and the resonance of multi-soliton waves .