Asymptotics of the solution of the Cauchy problem for the Korteweg–de Vries equation with initial data of step type

The method of the inverse scattering problem is used to solve the Cauchy problem for the Korteweg–deVries equation with initial data of step type: $u(x,0)\to-c^2$ ($x\to-\infty$), $u(x,0)\to0$ ($x\to\infty$). Formulas are obtained for transforming the scattering data with respect to the time, making it possible to obtain a solution $u(x,t)$ of the problem for arbitrary $t$ with the aid of linear integral equations of scattering theory. The asymptotic behavior of the solution as $t\to+\infty$ is investigated in a neighborhood of the wave front $\bigl(x>4c^2t-\frac1{2c}\ln t^N\bigr)$. It is shown that in this region the solution splits up into solitons, the distance between which increases as $\ln t^{1/c}$, and an explicit form for these solitons is derived.
Bibliography: 12 titles.