Geodesic
Self-similar solutions of the Korteweg--de~Vries equation and potentials with a~trivial $S$-matrix
Teoretičeskaâ i matematičeskaâ fizika, Tome 34 (1978) no. 3, pp. 426-430

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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. 
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     author = {L. A. Bordag and V. B. Matveev},
     title = {Self-similar solutions of the {Korteweg--de~Vries} equation and potentials with a~trivial $S$-matrix},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {426--430},
     publisher = {mathdoc},
     volume = {34},
     number = {3},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1978_34_3_a12/}
}
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L. A. Bordag; V. B. Matveev. Self-similar solutions of the Korteweg--de~Vries equation and potentials with a~trivial $S$-matrix. Teoretičeskaâ i matematičeskaâ fizika, Tome 34 (1978) no. 3, pp. 426-430. http://geodesic.mathdoc.fr/item/TMF_1978_34_3_a12/