KP Example Pictures

Figure 1

A cnoidal wave solution of the KP equation. Every cnoidal wave solution is one-dimensional, and it is time-independent in a uniformly translating coordinate system.

Figure 2a
Figure 2b

A two-phase KP solution. This solution is stationary, as are all two-phase solutions that are genuinely two-dimensional. (a) Perspective view of the solution. (b) Overhead view, with contour lines shown.

Figure 3

A symmetric two-phase solution. Every symmetric two-phase solution is periodic in x and in y, and it translates purely in the x-direction.

Figure 4a	Figure 4b	Figure 4c	Figure 4d	Figure 4e

A three-phase KP solution. This solution is periodic in a moving frame, with a period T = 3.1908. It is shown at four times: (a) t = 0.0, (b) t = 0.5, (c) t = 1.5, (d) t = 3.1908, (e) t = 3.1908, but translated according to (1.14).

Figure 5

A three-phase KP solution that is nearly stationary. The period matrix of this solution is completely symmetric: z₁₁ = z₂₂ = z₃₃ = 3, z_ij = 6/5 for i != j. Consequently, there is another, identical solution with the same period matrix and with the phases renumbered: k₁ = 0.574979, k₂ = k₃ = 1.0, l₁ = 0, l₂ = -l₃ = 0.6412115.

Figure 6

Another three-phase KP solution that is nearly stationary.