Figure 1.

Profile of solutions u and w. Figures (a) and (c) are three-dimensional plots with time t = 0, (b) and (d) are density plots with time t = 0."

Figure 2.

Figure (a) is the contour plot of the solutions u with δ1 = 1. Figure (b) is the contour plot of the solutions u with δ1 = 2. The blue lines of figures (a) and (b) are the relevant moving progress (17), i.e., $y=\tfrac{5}{4}x-1$ and $y=-\tfrac{3}{4}x-1$, respectively."

Figure 3.

Profile of equation (25). Figures (a) (b) are three-dimensional and density plots with time t = − 10, respectively. Figures (c) (d) are three-dimensional and density plots with time t = 0, respectively. Figures (e) (f) are three-dimensional and density plots with time t = 18, respectively."

Figure 4.

Profile of equation (36). Figure (a) is three-dimensional plot with time t = 0, (b) is the corresponding density plot."

Figure 5.

Profile of solutions u. Figure (a) is three-dimensional plot with time t = 0, (b) is density plot with time t = 0, (c) the contour plot shows the moving path with time t = − 55, 0, 55, (d) the wave propagation pattern along the x-axis by selecting different times t = − 20, 0, 20."

Figure 6.

Profile of solutions w. Figure (a) is three-dimensional plot with time t = 0, (b) is the corresponding density plot."