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

E. Ya. Khruslov

Sbornik. Mathematics, Tome 28 (1976) no. 2, pp. 229-248

Résumé

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.