KP-KdV Hierarchy and Pseudo-Differential Operators

Lesfari Ahmed

doi:10.33434/cams.478999

EN

KP-KdV Hierarchy and Pseudo-Differential Operators

Abstract

The study of KP-KdV equations are the archetype of integrable systems and are one of the most fundamental equations of soliton phenomena and a topic of active mathematical research. Our purpose here is to give a motivated and a sketchy overview of this interesting subject. One of the objectives of this paper is to study the KdV equation and the inverse scattering method (based on Schrödinger and Gelfand-Levitan equations) used to solve it exactly. We study some generalities on the algebra of infinite order differential operators. The algebras of Virasoro and Heisenberg, nonlinear evolution equations such as the KdV, Boussinesq and KP play a crucial role in this study. We make a careful study of some connection between pseudo-differential operators, symplectic structures, KP hierarchy and tau functions based on the Sato-Date-Jimbo-Miwa-Kashiwara theory. A few other connections and ideas concerning the KdV and Boussinesq equations, the Gelfand-Dickey flows, the Heisenberg and Virasoro algebras are given.

Keywords

Integrable systems,KdV equation,Gelfand-Levitan integral equation,KP hierarchy,Schr\"{o}dinger equation,Symplectic structures

References

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Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Lesfari Ahmed ^*
0000-0001-6213-4301
Türkiye

Publication Date

June 27, 2019

Submission Date

November 5, 2018

Acceptance Date

March 8, 2019

Published in Issue

Year 2019 Volume: 2 Number: 2

DOI

https://doi.org/10.33434/cams.478999

IZ

https://izlik.org/JA86EH66NX

APA

Ahmed, L. (2019). KP-KdV Hierarchy and Pseudo-Differential Operators. Communications in Advanced Mathematical Sciences, 2(2), 75-104. https://doi.org/10.33434/cams.478999

AMA

1.Ahmed L. KP-KdV Hierarchy and Pseudo-Differential Operators. Communications in Advanced Mathematical Sciences. 2019;2(2):75-104. doi:10.33434/cams.478999

Chicago

Ahmed, Lesfari. 2019. “KP-KdV Hierarchy and Pseudo-Differential Operators”. Communications in Advanced Mathematical Sciences 2 (2): 75-104. https://doi.org/10.33434/cams.478999.

EndNote

Ahmed L (June 1, 2019) KP-KdV Hierarchy and Pseudo-Differential Operators. Communications in Advanced Mathematical Sciences 2 2 75–104.

IEEE

[1]L. Ahmed, “KP-KdV Hierarchy and Pseudo-Differential Operators”, Communications in Advanced Mathematical Sciences, vol. 2, no. 2, pp. 75–104, June 2019, doi: 10.33434/cams.478999.

ISNAD

Ahmed, Lesfari. “KP-KdV Hierarchy and Pseudo-Differential Operators”. Communications in Advanced Mathematical Sciences 2/2 (June 1, 2019): 75-104. https://doi.org/10.33434/cams.478999.

JAMA

1.Ahmed L. KP-KdV Hierarchy and Pseudo-Differential Operators. Communications in Advanced Mathematical Sciences. 2019;2:75–104.

MLA

Ahmed, Lesfari. “KP-KdV Hierarchy and Pseudo-Differential Operators”. Communications in Advanced Mathematical Sciences, vol. 2, no. 2, June 2019, pp. 75-104, doi:10.33434/cams.478999.

Vancouver

1.Lesfari Ahmed. KP-KdV Hierarchy and Pseudo-Differential Operators. Communications in Advanced Mathematical Sciences. 2019 Jun. 1;2(2):75-104. doi:10.33434/cams.478999

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