Soliton Regions Determined by Initial Values in the K-dV Equation

Shunji KAWAMOTO; Masao KIDO; Katsuya NAKAI

Soliton Regions Determined by Initial Values in the K-dV Equation

Shunji KAWAMOTO, Masao KIDO, Katsuya NAKAI

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We consider an initial value problem of the K-dV equation. The initial function is, in our case, assumed to be periodic with respect to the x-coordinate. Such a periodic function is inserted as the potential function in the associated Schrödlinger equation. The equation is, therefore, of Hill's type. A stable region in the stability chart in the λ-A plane for Hill's equation corresponds to a band which consists of continuous eigenvalues, were A is the amplitude of the periodic function and λ the eigenvalue. As the potential function, we consider a function in which dips of cosine functions of one-period stand side by with constant interval. When the interval between two consecutive cosine pulses is made infinite, the eigenvalues for one potential of cosine type is obtained. In this limiting case, clear correspondence between the numbers of eigenvalues and solitons is obtained, and soliton regions are precisely determined. This analysis can be compared with results of computer experiments on the K-dV equation. On basis of our method, it is seen that the determination of soliton region is also usable for the determination of the number of solitons.

Publication: IEICE TRANSACTIONS on transactions Vol.E62-E No.12 pp.869-873

Publication Date: 1979/12/25

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Shunji KAWAMOTO
Masao KIDO
Katsuya NAKAI

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Shunji KAWAMOTO, Masao KIDO, Katsuya NAKAI, "Soliton Regions Determined by Initial Values in the K-dV Equation" in IEICE TRANSACTIONS on transactions, vol. E62-E, no. 12, pp. 869-873, December 1979, doi: .
Abstract: We consider an initial value problem of the K-dV equation. The initial function is, in our case, assumed to be periodic with respect to the x-coordinate. Such a periodic function is inserted as the potential function in the associated Schrödlinger equation. The equation is, therefore, of Hill's type. A stable region in the stability chart in the λ-A plane for Hill's equation corresponds to a band which consists of continuous eigenvalues, were A is the amplitude of the periodic function and λ the eigenvalue. As the potential function, we consider a function in which dips of cosine functions of one-period stand side by with constant interval. When the interval between two consecutive cosine pulses is made infinite, the eigenvalues for one potential of cosine type is obtained. In this limiting case, clear correspondence between the numbers of eigenvalues and solitons is obtained, and soliton regions are precisely determined. This analysis can be compared with results of computer experiments on the K-dV equation. On basis of our method, it is seen that the determination of soliton region is also usable for the determination of the number of solitons.
URL: https://global.ieice.org/en_transactions/transactions/10.1587/e62-e_12_869/_p

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@ARTICLE{e62-e_12_869,
author={Shunji KAWAMOTO, Masao KIDO, Katsuya NAKAI, },
journal={IEICE TRANSACTIONS on transactions},
title={Soliton Regions Determined by Initial Values in the K-dV Equation},
year={1979},
volume={E62-E},
number={12},
pages={869-873},
abstract={We consider an initial value problem of the K-dV equation. The initial function is, in our case, assumed to be periodic with respect to the x-coordinate. Such a periodic function is inserted as the potential function in the associated Schrödlinger equation. The equation is, therefore, of Hill's type. A stable region in the stability chart in the λ-A plane for Hill's equation corresponds to a band which consists of continuous eigenvalues, were A is the amplitude of the periodic function and λ the eigenvalue. As the potential function, we consider a function in which dips of cosine functions of one-period stand side by with constant interval. When the interval between two consecutive cosine pulses is made infinite, the eigenvalues for one potential of cosine type is obtained. In this limiting case, clear correspondence between the numbers of eigenvalues and solitons is obtained, and soliton regions are precisely determined. This analysis can be compared with results of computer experiments on the K-dV equation. On basis of our method, it is seen that the determination of soliton region is also usable for the determination of the number of solitons.},
keywords={},
doi={},
ISSN={},
month={December},}

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TY - JOUR
TI - Soliton Regions Determined by Initial Values in the K-dV Equation
T2 - IEICE TRANSACTIONS on transactions
SP - 869
EP - 873
AU - Shunji KAWAMOTO
AU - Masao KIDO
AU - Katsuya NAKAI
PY - 1979
DO -
JO - IEICE TRANSACTIONS on transactions
SN -
VL - E62-E
IS - 12
JA - IEICE TRANSACTIONS on transactions
Y1 - December 1979
AB - We consider an initial value problem of the K-dV equation. The initial function is, in our case, assumed to be periodic with respect to the x-coordinate. Such a periodic function is inserted as the potential function in the associated Schrödlinger equation. The equation is, therefore, of Hill's type. A stable region in the stability chart in the λ-A plane for Hill's equation corresponds to a band which consists of continuous eigenvalues, were A is the amplitude of the periodic function and λ the eigenvalue. As the potential function, we consider a function in which dips of cosine functions of one-period stand side by with constant interval. When the interval between two consecutive cosine pulses is made infinite, the eigenvalues for one potential of cosine type is obtained. In this limiting case, clear correspondence between the numbers of eigenvalues and solitons is obtained, and soliton regions are precisely determined. This analysis can be compared with results of computer experiments on the K-dV equation. On basis of our method, it is seen that the determination of soliton region is also usable for the determination of the number of solitons.
ER -