Self-similar solutions of the Korteweg–de Vries equation and potentials with a trivial $S$-matrix

It is shown that the Korteweg–de Vries equation with the initial condition $N(N+1)x^{-2}$ has self-similar solution of the form $u(x,t)=-2[\ln f(x,t)]_{xx}$, where $f$ is a polynomial in $x$ and $t$ whose coefficients are determined by recursion formulas.