Starting with the Miura transformation $v_x = - u -v^2$, how can I find the other half to the backlund transformation which will take solution of the KdV $u_t + 6 u u_x +u _{xxx}$ to a solution of the mKdV $v_t - 6 v^2 v_x +v_{xxx}$?
The unknown half should take the form of $v_t = f(u,v,u_x.v_x,u_{xxx})$.
I know the solution is $v_t =u_{xx} -2 u_x v -2u v_x$, but I don't know how to find this. I thought it should follow from the integrability conditions $v_{xt}=v_{tx}$, $u_{xt}=u_{tx}$ however I cannot get that to work.