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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.

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    $\begingroup$ If you are solving KP and haven't yet learned what an integrable system is, it's worth asking what your motivation is. I'm not being condescending and I'll provide you with references to help, but I can do that better if you explain your end goal. Which numerical method to use also depends partly on what kind of solutions you care about. $\endgroup$ – David Ketcheson Feb 10 '17 at 17:59
  • $\begingroup$ Maybe edit that comment? I don't think there is a way to write "If you are doing X and haven't learned Y, then..." that doesn't sound condescending. I agree with the general point, though. Answers are probably going to be different if you care about KP as integrable system, KP as numerical methods example, or KP as a model for a real-world problem. $\endgroup$ – AJK Feb 10 '17 at 23:27
  • $\begingroup$ I need a good numerical method with the aim of using KP as a model for real world problems. If you could also explain what a completely integrable system is that would be nice. Thanks a lot. $\endgroup$ – Millemila Feb 11 '17 at 3:41
  • $\begingroup$ David if you have some hints on some good numerical approach that would be great. thanks $\endgroup$ – Millemila Apr 8 '17 at 12:55

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