Asymptotic behavior of solutions for t\(\to \infty\) of the equation \(\psi _{xx}+u(x,t)\psi +(\lambda /4)\psi =0\) with a potential u satisfying the Korteweg-de Vries equation. (Russian) Zbl 0606.35078

The formal transition \(\hat s\) and reflection \(\hat r\) coefficients are studied in detail for the spectral Schrödinger equation with the potential u asymptotic for \(t\to \infty\). In the distinguished classes of potentials and reflection coefficients the inverse problem of the scattering theory is considered. If the reflection coefficient \(\hat r\) can be reduced to the highest order term then the corresponding potential u satisfies the formal Korteweg-de Vries equation. These results are used for a rigorous proof of the complete asymptotic description of the solution of the Cauchy problem for the Korteweg-de Vries equation for \(t\to \infty\). The proof is based on the integral equation of the inverse problem of the scattering theory.
Reviewer: O.Dumbrajs


35Q99 Partial differential equations of mathematical physics and other areas of application
35R30 Inverse problems for PDEs
35P25 Scattering theory for PDEs
35B40 Asymptotic behavior of solutions to PDEs
35G25 Initial value problems for nonlinear higher-order PDEs