Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 2 / 31C / c / Solution

The given Riccati ansatz is $v^{\prime }=-i{v}^2+iz/6$. Differentiating it and substituting for $v^{\prime }$ showsIt therefore selects $C=i/6$ in the preceding scaled Painleve equation. Write $v=-i{\Psi }^{\prime }/\Psi$. The quadratic terms cancel in the ansatz, leavingThus every nonzero solution of the indicated Airy equation gives the local mKdV solutionon intervals where $\Psi \ne 0$. For example $\Psi (z)=A\mathrm{Ai}(-z/6^{1/3})+B\mathrm{Bi}(-z/6^{1/3})$, with constants not both zero, has the required equation. This branch is generally complex and meromorphic; the printed ansatz itself includes $i$. A real $v$ on a real interval would require simultaneously $v^{\prime }=0$ and $v^2=z/6$, which is impossible on an interval. Hence this particular ansatz does not furnish a nontrivial real-valued mKdV profile.