I need to solve the Kadomtsev Petviashvili (KP) equations
$$\partial_x(\partial_t u+u \partial_x u+\epsilon^2\partial_{xxx}u)+\lambda\partial_{yy}u=0 $$
where $$\lambda=\pm 1 \;.$$ My questions are:
1) Which is the best numerical method to solve them? Or maybe a good method that is simple?
2) What does wikipidia mean with: "the KP equation is completely integrable"? I don't think it means KP has a general analytical solution, otherwise why are people still coming out with numerical methods for solving it? See for example this paper.