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

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.