u_t+u^pu_x+ u_{xxx}=\lambda \partial_x^{-1}u_{yy}, \quad p = 1,2, \dots

For 𝝀 = 1, this is the KP I equation, for 𝝀 = -1, this is the KP II equation. A blow-up of the solutions is possible for 𝑝 ≥ 4/3.

The video above shows the time evolution of the initial data

u(x,y,0)=12\partial_{xx}\exp(-x^2-y^2)

for 𝑝 = 4/3 and 𝝀 = 1.

The second video shows the time evolution of the initial data

u(x,y,0)=6\partial_{xx}\exp(-x^2-y^2)

for 𝑝 = 2 and 𝝀 = 1.

### References

See 1310.5215 for references and further details.