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I have the following nonlinear differential equation: $$\ddot z(t)-\sin(z(t))=0$$ where $z(t)$ is a complex variable. The solution of the same equation with $z(t)$ real, is a function of Jacobi amplitude integral. What happens when $z(t)=z_0(t)+iz_1(t)$?

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I don't think that anything happens; the same formula should give the solution in this case too, except that the modulus of the Jacobi elliptic function might be any complex number (instead of just a real number between 0 and 1). –  Hans Lundmark Nov 13 '12 at 15:42

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