Geodesic
The Korteweg-de Vries equation in classes of increasing functions with prescribed asymptotics as $|x|\to\infty$
Sbornik. Mathematics, Tome 50 (1985) no. 1, pp. 125-135 Cet article a éte moissonné depuis la source Math-Net.Ru

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The Cauchy problem is considered for the Korteweg–de Vries equation with an increasing initial function admitting an asymptotic expansion in decreasing powers of $x$ as $|x|\to\infty$. It is proved that asymptotic solutions having the form of series in decreasing powers of $x$ differ from the actual solutions by a function $w(x,t)$ smooth in $t$ with values in $S(\mathbf R_x)$. Bibliography: 3 titles. 
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     author = {I. N. Bondareva},
     title = {The {Korteweg-de} {Vries} equation in classes of increasing functions with prescribed asymptotics as $|x|\to\infty$},
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I. N. Bondareva. The Korteweg-de Vries equation in classes of increasing functions with prescribed asymptotics as $|x|\to\infty$. Sbornik. Mathematics, Tome 50 (1985) no. 1, pp. 125-135. http://geodesic.mathdoc.fr/item/SM_1985_50_1_a7/

[1] Menikoff A., “The existences of unbounded solutions of the Korteweg–de Vries equation”, Commun. Pure Appl. Math., 25 (1972), 407–432 | DOI | MR | Zbl

[2] Bondareva I. N., Shubin M. A., “Rastuschie asimptoticheskie resheniya uravneniya Kortevega–de Friza i ego vysshikh analogov”, DAN SSSR, 267:5 (1982), 1035–1038 | MR | Zbl

[3] Stummel F., “Ellipitishe Differenzenoperatoren unter Dirichletrandbedingungen”, Math. Zeitschrift, 97 (1967), 169–211 | DOI | MR | Zbl