Geodesic
On solution of the Cauchy problem for the Korteweg--de~Vries equation with initial data the sum of a~periodic and a~rapidly decreasing function
Sbornik. Mathematics, Tome 63 (1989) no. 1, pp. 257-265

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A scheme is presented for solving the Cauchy problem for the KdV equation with initial data a sum of a periodic function $p(x)$ and a rapidly decreasing function $q(x)$. The scattering theory constructed earlier by the author for the pair of operators $H_0=-d^2/dx^2+p(x)$ and $H=H_0+q(x)$ is used to solve this problem. Evolution formulas for the scattering data are found. The solution $p(x,t)$ of the KdV equation with a periodic initial condition obtained by V. A. Marchenko and S. P. Novikov is assumed known. Bibliography: 11 titles. 
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     author = {N. E. Firsova},
     title = {On solution of the {Cauchy} problem for the {Korteweg--de~Vries} equation with initial data the sum of a~periodic and a~rapidly decreasing function},
     journal = {Sbornik. Mathematics},
     pages = {257--265},
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N. E. Firsova. On solution of the Cauchy problem for the Korteweg--de~Vries equation with initial data the sum of a~periodic and a~rapidly decreasing function. Sbornik. Mathematics, Tome 63 (1989) no. 1, pp. 257-265. http://geodesic.mathdoc.fr/item/SM_1989_63_1_a17/