Superposition formulas of multi-solution to a reduced (3+1)-dimensional nonlinear evolution equation

Abstract

In this paper, we gave the localized solutions, the interaction solutions and the mixed solutions to a reduced (3+1)-dimensional nonlinear evolution equation. These solutions were characterized by superposition formulas of positive quadratic functions, the exponential and hyperbolic functions. According to the known lump solution in the outset, we obtained the superposition formulas of positive quadratic functions by plausible reasoning. Next, we constructed the interaction solutions between the localized solutions and the exponential function solutions with the similar theory. These two kinds of solutions contained superposition formulas of positive quadratic functions, which were turned into general ternary quadratic functions, the coefficients of which were all rational operation of vector inner product. Then we obtained linear superposition formulas of exponential and hyperbolic function solutions. Finally, for aforementioned various solutions, their dynamic properties were showed by choosing specific values for parameters. From concrete plots, we observed wave characteristics of three kinds of solutions. Especially, we could observe distinct generation and separation situations when the localized wave and the stripe wave interacted at different time points.

Nonlinear Superposition and Composite Solution Formula for the Generalized (2+1)-Dimensional Variable-Coefficient Fifth-Order KdV Equation

“…Now, various systematic approaches to finding exact solutions have been established. For example, Bell polynomial method [5][6][7], Hirota bilinear method [8][9][10], Darboux transform method [11], nonlinear superposition method [12][13][14], variable separation method [15], rational function transformation method [16][17], etc.…”