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On the complex modified Korteweg–de Vries equation with a self-consistent source and nonzero boundary conditions

UDC 517.957 We study the method of inverse scattering transform for defocusing complex modified Korteweg–de-Vries (cmKdV) equation with self-consistent sources (SCS) and nonzero boundary conditions at infinity. We prove a theorem on the evolution of the scattering data for a self-adjoin...

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Date:2025
Main Authors: Khasanov, A. B., Reyimberganov, A. A.
Format: Article
Language:English
Published: Institute of Mathematics, NAS of Ukraine 2025
Online Access:https://umj.imath.kiev.ua/index.php/umj/article/view/7993
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Journal Title:Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal
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author Khasanov, A. B.
Reyimberganov, A. A.
Khasanov, A. B.
Reyimberganov, A. A.
author_facet Khasanov, A. B.
Reyimberganov, A. A.
Khasanov, A. B.
Reyimberganov, A. A.
author_sort Khasanov, A. B.
baseUrl_str https://umj.imath.kiev.ua/index.php/umj/oai
collection OJS
datestamp_date 2025-11-01T09:34:12Z
description UDC 517.957 We study the method of inverse scattering transform for defocusing complex modified Korteweg–de-Vries (cmKdV) equation with self-consistent sources (SCS) and nonzero boundary conditions at infinity. We prove a theorem on the evolution of the scattering data for a self-adjoint Zakharov–Shabat system in which the potential is a solution of the defocusing cmKdV equation with SCS. The obtained results enable us to solve the inverse scattering problem for the self-adjoint Zakharov–Shabat system with a potential from the class of functions nonvanishing at infinity.
doi_str_mv 10.3842/umzh.v77i1.7993
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spelling umjimathkievua-article-79932025-11-01T09:34:12Z On the complex modified Korteweg–de Vries equation with a self-consistent source and nonzero boundary conditions On the complex modified Korteweg–de Vries equation with a self-consistent source and nonzero boundary conditions Khasanov, A. B. Reyimberganov, A. A. Khasanov, A. B. Reyimberganov, A. A. Inverse scattering transform Complex modified Korteweg-de Vries equations Zakharov-Shabat system Self-consistent source Non-zero boundary conditions UDC 517.957 We study the method of inverse scattering transform for defocusing complex modified Korteweg–de-Vries (cmKdV) equation with self-consistent sources (SCS) and nonzero boundary conditions at infinity. We prove a theorem on the evolution of the scattering data for a self-adjoint Zakharov–Shabat system in which the potential is a solution of the defocusing cmKdV equation with SCS. The obtained results enable us to solve the inverse scattering problem for the self-adjoint Zakharov–Shabat system with a potential from the class of functions nonvanishing at infinity. УДК 517.957 Про складне модифіковане рівняння Кортевега–де Фріза з самоузгодженим джерелом і ненульовими крайовими умовами Досліджується метод оберненого перетворення розсіяння для дефокусування складного модифікованого рівняння Кортевега–де Фріза (смКдФ) з самоузгодженими джерелами (СУД) і ненульовими граничними умовами на нескінченності. Доведено теорему про еволюцію даних розсіяння самоспряженої системи Захарова–Шабата, де потенціал є розв'язком дефокусуючого рівняння смКдФ з СУД. Отримані результати дають змогу розв'язати обернену задачу розсіяння для самоспряженої системи Захарова–Шабата з потенціалом у класі функцій, що є ненульовими на нескінченності. Institute of Mathematics, NAS of Ukraine 2025-10-31 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7993 10.3842/umzh.v77i1.7993 Ukrains’kyi Matematychnyi Zhurnal; Vol. 77 No. 1 (2025); 80 Український математичний журнал; Том 77 № 1 (2025); 80 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7993/10301 Copyright (c) 2025 Anvar Reyimberganov
spellingShingle Khasanov, A. B.
Reyimberganov, A. A.
Khasanov, A. B.
Reyimberganov, A. A.
On the complex modified Korteweg–de Vries equation with a self-consistent source and nonzero boundary conditions
title On the complex modified Korteweg–de Vries equation with a self-consistent source and nonzero boundary conditions
title_alt On the complex modified Korteweg–de Vries equation with a self-consistent source and nonzero boundary conditions
title_full On the complex modified Korteweg–de Vries equation with a self-consistent source and nonzero boundary conditions
title_fullStr On the complex modified Korteweg–de Vries equation with a self-consistent source and nonzero boundary conditions
title_full_unstemmed On the complex modified Korteweg–de Vries equation with a self-consistent source and nonzero boundary conditions
title_short On the complex modified Korteweg–de Vries equation with a self-consistent source and nonzero boundary conditions
title_sort on the complex modified korteweg–de vries equation with a self-consistent source and nonzero boundary conditions
topic_facet Inverse scattering transform
Complex modified Korteweg-de Vries equations
Zakharov-Shabat system
Self-consistent source
Non-zero boundary conditions
url https://umj.imath.kiev.ua/index.php/umj/article/view/7993
work_keys_str_mv AT khasanovab onthecomplexmodifiedkortewegdevriesequationwithaselfconsistentsourceandnonzeroboundaryconditions
AT reyimberganovaa onthecomplexmodifiedkortewegdevriesequationwithaselfconsistentsourceandnonzeroboundaryconditions
AT khasanovab onthecomplexmodifiedkortewegdevriesequationwithaselfconsistentsourceandnonzeroboundaryconditions
AT reyimberganovaa onthecomplexmodifiedkortewegdevriesequationwithaselfconsistentsourceandnonzeroboundaryconditions

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