Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 2 / 31C (Integrable Systems)

For a localized traveling wave put $q(x,t)=Q(\xi )$ with $\xi =x-ct-x_0$, and require $Q,Q^{\prime },Q^{\prime \prime }\to 0$ at infinity. The modified KdV equation integrates once to $Q^{\prime \prime }-cQ+2Q^3=0$. Multiplying by $Q^{\prime }$ and integrating again givesA nonzero decaying real solution requires $c>0$. Separating on the rising or falling half of the wave and using the indicated inverse-hyperbolic-secant integral yieldsThe signs and translations are arbitrary. Differentiating the sech profile verifies $Q^{\prime \prime }=cQ-2Q^3$ directly. The relevant antiderivative is the inverse hyperbolic secant, since $d(\mathrm{arcsech}x)/dx=-1/[x\sqrt{1-x^2}]$ for $0<x<1$.

The scaling of the equation suggests $q=t^{-1/3}v(z)$, $z=x{t}^{-1/3}$, for $t>0$. Substitution givesIntegrating in $z$ gives the scaled Painlevé II equationwhere $C$ is an arbitrary integration constant. If the standard normalization $w^{\prime \prime }=2w^3+sw+\alpha$ is desired, set $z=3^{1/3}s$ and $v(z)=i{3}^{-1/3}w(s)$. Then $\alpha =-3iC$ andThe direct scaled equation avoids introducing imaginary variables when a real $v$ is required.

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.