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

Buslaev, V. S.; Sukhanov, V. V.

Probl. Mat. Fiz. 11, 78-113 (1986).

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

MSC:

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

Keywords:

transition; reflection; spectral Schrödinger equation; inverse problem; scattering; Korteweg-de Vries equation; asymptotic description; Cauchy problem