Dynamics of a differential-difference integrable $(2+1)$-dimensional system

A Kadomtsev-Petviashvili- (KP-) type equation appears in fluid mechanics, plasma physics, and gas dynamics. In this paper, we propose an integrable semidiscrete analog of a coupled $(2+1)$-dimensional system which is related to the KP equation and the Zakharov equation. $N$-soliton solutions of the discrete equation are presented. Some interesting examples of soliton resonance related to the two-soliton and three-soliton solutions are investigated. Numerical computations using the integrable semidiscrete equation are performed. It is shown that the integrable semidiscrete equation gives very accurate numerical results in the cases of one-soliton evolution and soliton interactions.

Figure 1

Numerical solution of the one-soliton solution at $t=4$.

Figure 2

Exact solution for the collision of the two-soliton solution at $t=4$.

Figure 3

Numerical solution for the collision of two solitons at $t=4$.

Figure 4

The elastic interaction of two solitons at $t=20$.

Figure 5

Resonant interactions between two solitons at $t=-10$ when $b(1,2)=0$.

Figure 6

Resonant interactions between two solitons at $t=20$ when $b(1,2)\to 0$.

Figure 7

Elastic interactions between three solitons at $t=20$.

Figure 8

Resonant interactions between three solitons at $t=20$.