Hierarchies of the Korteweg–de Vries Equation Related to Complex Expansion and Perturbation

We consider the possibility of constructing a hierarchy of the complex extension of the Korteweg–de Vries equation (cKdV), which under the assumption that the function is real passes into the KdV hierarchy. A hierarchy is understood here as a family of nonlinear partial differential equations with a...

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Главные авторы:	Redkina, T. V., Редькина, Т. В., Zakinyan, A. R., Закинян, А. Р., Zakinyan, R. G., Закинян, Р. Г., Surneva, O. B., Сурнева, О. Б.
Формат:	Статья
Язык:	English
Опубликовано:	2023
Темы:	Lax pairs Complexification of the Korteweg–de Vries equation Korteweg–de Vries hierarchies Integrable partial differential equations Perturbations of the Korteweg–de Vries equation
Online-ссылка:	https://dspace.ncfu.ru/handle/20.500.12258/23801
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spelling	ir-20.500.12258-238012025-02-11T07:56:35Z Hierarchies of the Korteweg–de Vries Equation Related to Complex Expansion and Perturbation Redkina, T. V. Редькина, Т. В. Zakinyan, A. R. Закинян, А. Р. Zakinyan, R. G. Закинян, Р. Г. Surneva, O. B. Сурнева, О. Б. Lax pairs Complexification of the Korteweg–de Vries equation Korteweg–de Vries hierarchies Integrable partial differential equations Perturbations of the Korteweg–de Vries equation We consider the possibility of constructing a hierarchy of the complex extension of the Korteweg–de Vries equation (cKdV), which under the assumption that the function is real passes into the KdV hierarchy. A hierarchy is understood here as a family of nonlinear partial differential equations with a Lax pair with a common scattering operator. The cKdV hierarchy is obtained by examining the equation on the eigenvalues of the fourth-order Hermitian self-conjugate operator on the invariant transformations of the eigenvector-functions. It is proved that for an operator ˆHn to transform a solution of the equation on eigenvalues ( ˆM − λE)V = 0 into a solution of the same equation, it is necessary and sufficient that the complex function u(x, t) of the operator ˆM satisfies special conditions that are the complexifications of the KdV hierarchy equations. The operators ˆHn are constructed as differential operators of order 2n + 1. We also construct a hierarchy of perturbed KdV equations (pKdV) with a special perturbation function, the dynamics of which is described by a linear equation. It is based on the system of operator equations obtained by Bogoyavlensky. Since the elements of the hierarchies are united by a common scattering operator, it remains unchanged in the derivation of the equations. The second differential operator of the Lax pair has increasing odd derivatives while retaining a skew-symmetric form. It is shown that when perturbation tends to zero, all hierarchy equations are converted to higher KdV equations. It is proved that the pKdV hierarchy equations are a necessary and sufficient condition for the solutions of the equation on eigenvalues to have invariant transformations. 2023-06-30T07:45:33Z 2023-06-30T07:45:33Z 2023 Статья Redkina, T.V., Zakinyan, A.R., Zakinyan, R.G., Surneva, O.B. Hierarchies of the Korteweg–de Vries Equation Related to Complex Expansion and Perturbation // Axioms. - 2023. - 12(4), art. no. 371. - DOI: 10.3390/axioms12040371 http://hdl.handle.net/20.500.12258/23801 en Axioms application/pdf application/pdf
institution	СКФУ
collection	Репозиторий
language	English
topic	Lax pairs Complexification of the Korteweg–de Vries equation Korteweg–de Vries hierarchies Integrable partial differential equations Perturbations of the Korteweg–de Vries equation
spellingShingle	Lax pairs Complexification of the Korteweg–de Vries equation Korteweg–de Vries hierarchies Integrable partial differential equations Perturbations of the Korteweg–de Vries equation Redkina, T. V. Редькина, Т. В. Zakinyan, A. R. Закинян, А. Р. Zakinyan, R. G. Закинян, Р. Г. Surneva, O. B. Сурнева, О. Б. Hierarchies of the Korteweg–de Vries Equation Related to Complex Expansion and Perturbation
description	We consider the possibility of constructing a hierarchy of the complex extension of the Korteweg–de Vries equation (cKdV), which under the assumption that the function is real passes into the KdV hierarchy. A hierarchy is understood here as a family of nonlinear partial differential equations with a Lax pair with a common scattering operator. The cKdV hierarchy is obtained by examining the equation on the eigenvalues of the fourth-order Hermitian self-conjugate operator on the invariant transformations of the eigenvector-functions. It is proved that for an operator ˆHn to transform a solution of the equation on eigenvalues ( ˆM − λE)V = 0 into a solution of the same equation, it is necessary and sufficient that the complex function u(x, t) of the operator ˆM satisfies special conditions that are the complexifications of the KdV hierarchy equations. The operators ˆHn are constructed as differential operators of order 2n + 1. We also construct a hierarchy of perturbed KdV equations (pKdV) with a special perturbation function, the dynamics of which is described by a linear equation. It is based on the system of operator equations obtained by Bogoyavlensky. Since the elements of the hierarchies are united by a common scattering operator, it remains unchanged in the derivation of the equations. The second differential operator of the Lax pair has increasing odd derivatives while retaining a skew-symmetric form. It is shown that when perturbation tends to zero, all hierarchy equations are converted to higher KdV equations. It is proved that the pKdV hierarchy equations are a necessary and sufficient condition for the solutions of the equation on eigenvalues to have invariant transformations.
format	Статья
author	Redkina, T. V. Редькина, Т. В. Zakinyan, A. R. Закинян, А. Р. Zakinyan, R. G. Закинян, Р. Г. Surneva, O. B. Сурнева, О. Б.
author_facet	Redkina, T. V. Редькина, Т. В. Zakinyan, A. R. Закинян, А. Р. Zakinyan, R. G. Закинян, Р. Г. Surneva, O. B. Сурнева, О. Б.
author_sort	Redkina, T. V.
title	Hierarchies of the Korteweg–de Vries Equation Related to Complex Expansion and Perturbation
title_short	Hierarchies of the Korteweg–de Vries Equation Related to Complex Expansion and Perturbation
title_full	Hierarchies of the Korteweg–de Vries Equation Related to Complex Expansion and Perturbation
title_fullStr	Hierarchies of the Korteweg–de Vries Equation Related to Complex Expansion and Perturbation
title_full_unstemmed	Hierarchies of the Korteweg–de Vries Equation Related to Complex Expansion and Perturbation
title_sort	hierarchies of the korteweg–de vries equation related to complex expansion and perturbation
publishDate	2023
url	https://dspace.ncfu.ru/handle/20.500.12258/23801
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Hierarchies of the Korteweg–de Vries Equation Related to Complex Expansion and Perturbation

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