Interactions of soliton and mean field in KdV equation with well type initial data

Abstract

For the KdV equation with well-type initial value, the interaction between the trial soliton and the mean field is studied. The well initial value will lead to the appearance of rarefaction wave and dispersion shock wave, and there will be a linear wave region after a long time. The interaction between trial soliton and mean field is described within the framework of Whitham modulation theory, and the trajectory of soliton is given. The predicted soliton amplitude and phase changes are numerically confirmed, verifying the correctness of the theoretical analysis.

Figures (11)

• Figure 1: Long-term evolution of the mean field of the well initial value.
• Figure 2: Dynamical evolution of the well type initial value without trial soliton in the $(x,t)$ plane.
• Figure 3: Construction of Riemann invariants: (a) Soliton tunnels before the plateau disappears; (b) Soliton tunnels after disappearing in the plateau; (c) Soliton embeds in the RW region; (d) Solito embeds in the LW region; (e) Soliton embeds in the DSW region.
• Figure 4: When $x_0<0$, the soliton on the left side passes through the DSW region and the RW region successively.
• Figure 5: (a) Initial value condition; (b) Behavior of soliton interacting with the mean field at $t=30$; (c) The process of soliton interaction with mean field. The black dashed line is the boundary of the region, and the red dashed line is the theoretically predicted soliton trajectory. The initial amplitude of the trial soliton is $a_L=8$, and the initial position is $x_0=-100$.