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TMF, 1975, Volume 23, Number 1, Pages 51–68 (Mi tmf3750)  

This article is cited in 72 scientific papers (total in 72 papers)

Schrödinger operators with finite-gap spectrum and $N$-soliton solutions of the Korteweg–de Vries equation

A. R. Its, V. B. Matveev

Abstract: Explicit description of periodic potentials for which the corresponding Schrodinger operator $N$ possesses only the finite number of energy gaps is obtained. Using this result the solution of the Korteveg–de Vries equation with the “finite-gap” initial condition is expressed, by means of the $N$-dimensional $\Theta$-function, $N$ being the number of the nondegenerate energy gaps. The following characteristic property of the $N$-gap periodic potentials and the $N$-soliton decreasing potentials is discovered: the existence of two solutions $\psi_1(x,\lambda), \psi_2(x,\lambda)$ of the Schrodinger equation, for which the product $\psi_1,\psi_2$ is the polynomial $P$ ($\operatorname{deg}P=N$. $N$ is the number of gaps or the number of bound states of $H$) from the spectral parameter $\lambda$.

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Theoretical and Mathematical Physics, 1975, 23:1, 343–355

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Received: 09.07.1974

Citation: A. R. Its, V. B. Matveev, “Schrödinger operators with finite-gap spectrum and $N$-soliton solutions of the Korteweg–de Vries equation”, TMF, 23:1 (1975), 51–68; Theoret. and Math. Phys., 23:1 (1975), 343–355

Citation in format AMSBIB
\by A.~R.~Its, V.~B.~Matveev
\paper Schr\"odinger operators with finite-gap spectrum and $N$-soliton solutions of the Korteweg--de~Vries equation
\jour TMF
\yr 1975
\vol 23
\issue 1
\pages 51--68
\jour Theoret. and Math. Phys.
\yr 1975
\vol 23
\issue 1
\pages 343--355

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