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On the asymptotic behaviour of solutions of the equation \(\psi _{xx}+u(x,t)\psi +(\lambda /4)\psi =0\) with the potential u satisfying the Korteweg-de Vries equations, as t\(\to \infty\). II. (Russian. English summary) Zbl 0559.35074

The paper is one of a series of papers by the authors dedicated to the strict investigation of the asymptotic behaviour of the Korteweg-de Vries equation solutions as \(t\to \infty\) [see ibid. 120, 32-50 (1982; Zbl 0514.35077); for part I of this work see Probl. Mat. Fiz. 10, 70-102 (1982; Zbl 0513.35011)]. The aim of this work is an investigation of the solution \(\psi\) of the Schrödinger equation in the vicinity of the singular point \(x=3t\lambda\) for some special class of potentials introduced in the previous parts. As it will be shown in the end this class can be used for an asymptotic description of decreasing (at \(x\to \infty)\) of the solutions of the KdV equation as \(t\to \infty\). Out of the vicinity of the singular point the solution \(\psi\) has been investigated earlier. In the paper a series for the solution of the Schrödinger equation is studied and asymptotical properties of this series are considered.

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35J10 Schrödinger operator, Schrödinger equation
35B40 Asymptotic behavior of solutions to PDEs
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