Superposition behavior of the lump solutions and multiple mixed function solutions for the (3+1)-dimensional Sharma–Tasso–Olver-like equation

Abstract

In this paper, several analytical solutions of the (3+1)-dimensional Sharma–Tasso–Olver-like (STOL) equation describing the dynamical propagation of nonlinear dispersive waves in inhomogeneous media are obtained by means of the homoclinic test method. In order to study the interaction between the lump, breather and kink waves, we first construct the hybrid solutions between lump solutions and different functions of the (3+1)-dimensional STOL equation starting from the hybrid test functions, and then, the existence theorem and remarks about the superposition behavior of multiple quadratic functions, hyperbolic functions and trigonometric functions of the (3+1)-dimensional STOL equation are proved. With the help of the symbolic computing system Mathematica, the parameter relationship between the solutions is found, and the mixed solutions containing different functions are constructed. Besides, the superposition behaviors of multiple hyperbolic functions, trigonometric functions and function product forms are also studied. By selecting appropriate parameter values, the dynamical behaviors of these analytical solutions are illustrated and analyzed.

Appendix

$$\begin{aligned} \begin{aligned} \mu _{0}&=h_{1}^{2}+u_{0}^{2},\mu _{1}=\lambda _{1}h_{1}(h_{1}^{2}+3u_{0}^{2})+h_{4}, \quad \mu _{8}=h_{1}(3\lambda _{1}h_{1}+2\lambda _{2}h_{2}+2\lambda _{3}h_{3}), \\ \mu _{2}&=\lambda _{1}h_{1}^{2}(h_{1}^{2}-3u_{0}^{2})-2h_{1}h_{4}, \quad \mu _{4}=\lambda _{1}u_{0}^{2}(m_{i}^{2}-h_{1}^{2})(h_{1}^{2}-3u_{0}^{2}) +2h_{1}h_{4}(m_{i}^{2}+u_{0}^{2}), \\ \mu _{3}&=\lambda _{1}h_{1}^{2}(3u_{0}^{2}-h_{1}^{2})+2h_{1}h_{4}, \quad \mu _{5}=\lambda _{1}u_{0}^{2}(\alpha _{j}^{2}+h_{1}^{2})(h_{1}^{2}-3u_{0}^{2}) +2h_{1}h_{4}(\alpha _{j}^{2}-u_{0}^{2}), \\ \mu _{6}&=3\lambda _{1}(h_{1}^{2}-m_{i}^{2})+2\lambda _{3}(h_{1}h_{3}-m_{i}p_{i}), \quad \mu _{7}=3\lambda _{1}(h_{1}^{2}+\alpha _{j}^{2})+2\lambda _{2}(h_{1}h_{2}+\alpha _{j}\beta _{j}), \\ \mu _{9}&=\lambda _{1}m_{i}^{2}(m_{i}^{2}-3u_{0}^{2})-\mu _{8}(u_{0}^{2}+m_{i}^{2}), \quad \mu _{11}=3\lambda _{1}(h_{1}^{2}-m_{i}^{2})+2\lambda _{2}(h_{1}h_{2}-m_{i}n_{i}), \\ \mu _{10}&=\lambda _{1}\alpha _{j}^{2}(\alpha _{j}^{2}+3u_{0}^{2})-\mu _{8}(u_{0}^{2}-\alpha _{j}^{2}), \quad \mu _{12}=3\lambda _{1}(h_{1}^{2}+\alpha _{j}^{2})+2\lambda _{3}(h_{1}h_{3}+\alpha _{j}\gamma _{j}),\\ \mu _{13}&=\lambda _{1}u_{0}^{2}(h_{1}^{2}-3u_{0}^{2})(a_{i}^{2}-m_{i}^{2}+h_{1}^{2}) +2h_{1}h_{4}(a_{i}^{2}-m_{i}^{2}-u_{0}^{2}), \\ \mu _{14}&=\lambda _{1}u_{0}^{2}(h_{1}^{2}-3u_{0}^{2})(a_{i}^{2}-m_{i}^{2}-h_{1}^{2}) +2h_{1}h_{4}(a_{i}^{2}-m_{i}^{2}+u_{0}^{2}), \\ \mu _{15}&=\lambda _{1}u_{0}^{2}(h_{1}^{2}-3u_{0}^{2})(s_{j}^{2}-\alpha _{j}^{2}+h_{1}^{2}) +2h_{1}h_{4}(s_{j}^{2}-\alpha _{j}^{2}-u_{0}^{2}), \\ \mu _{16}&=\lambda _{1}u_{0}^{2}(h_{1}^{2}-3u_{0}^{2})(s_{j}^{2}-\alpha _{j}^{2}-h_{1}^{2}) +2h_{1}h_{4}(s_{j}^{2}-\alpha _{j}^{2}+u_{0}^{2}). \end{aligned} \end{aligned}$$

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