Geodesic
On the asymptotic behaviour as $t\to\infty$ of the solutions of the equation $\Psi_{xx}+u(x,t)\Psi+(\lambda/4)\Psi$ with a~potential~$u$ satisfying the Korteweg--De Vries Equation.~II.
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 16, Tome 138 (1984), pp. 8-32

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The paper is the third part of a series of authors papers dedicated to the strict investigation of the asymptotic behaviour of KdV equation solutions as $t\to\infty$. The aim of this work is an investigation of the solution $\Psi$ of the Sehroedinger 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 the decreasing (at $x\to\infty$) solutions of the KdV equation as $t\to\infty$. Out of the vicinity of the singular point the solution $\Psi$ nas been investigated earlier. In the paper a series for the solution of the Sehroedinger equation is studied and asymptotical properties of this serious are considered. 
@article{ZNSL_1984_138_a1,
     author = {V. S. Buslaev and V. V. Sukhanov},
     title = {On the asymptotic behaviour as $t\to\infty$ of the solutions of the equation $\Psi_{xx}+u(x,t)\Psi+(\lambda/4)\Psi$ with a~potential~$u$ satisfying the {Korteweg--De} {Vries} {Equation.~II.}},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {8--32},
     publisher = {mathdoc},
     volume = {138},
     year = {1984},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1984_138_a1/}
}
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V. S. Buslaev; V. V. Sukhanov. On the asymptotic behaviour as $t\to\infty$ of the solutions of the equation $\Psi_{xx}+u(x,t)\Psi+(\lambda/4)\Psi$ with a~potential~$u$ satisfying the Korteweg--De Vries Equation.~II.. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 16, Tome 138 (1984), pp. 8-32. http://geodesic.mathdoc.fr/item/ZNSL_1984_138_a1/