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Teoreticheskaya i Matematicheskaya Fizika, 2025, Volume 222, Number 1, Pages 25–40
DOI: https://doi.org/10.4213/tmf10793
(Mi tmf10793)
 

Inverse scattering method via the Gel'fand–Levitan–Marchenko equation for some negative-order nonlinear wave equations

M. I. Ismailovab, C. Sabaza

a Department of Mathematics, Gebze Technical University, Gebze-Kocaeli, Turkey
b Center for Mathematics and its Applications, Khazar University, Baku, Azerbaijan
References:
Abstract: A class of negative-order Ablowitz–Kaup–Newell–Segur nonlinear evolution equations are obtained by applying the Lax hierarchy of a first-order linear system of three equations. The inverse scattering problem on the line is examined in the cases where the linear system becomes the classical Zakharov–Shabat system with real antisymmetric and real symmetric potentials. Referring to these results, the N-soliton solutions for the integro-differential version of the nonlinear Klein–Gordon equation coupled to a scalar field and the negative-order modified Korteweg-de Vries equation are obtained by using the inverse scattering method via the Gel'fand–Levitan–Marchenko equation.
Keywords: first-order linear system, negative-order AKNS hierarchy, Gel'fand–Levitan–Marchenko equation, inverse scattering method, Klein–Gordon equation coupled to a scalar field, negative-order mKdV.
Received: 12.07.2024
Revised: 14.08.2024
English version:
Theoretical and Mathematical Physics, 2025, Volume 222, Issue 1, Pages 20–33
DOI: https://doi.org/10.1134/S0040577925010039
Bibliographic databases:
Document Type: Article
MSC: 37K15, 35C08, 35L70
Language: Russian
Citation: M. I. Ismailov, C. Sabaz, “Inverse scattering method via the Gel'fand–Levitan–Marchenko equation for some negative-order nonlinear wave equations”, TMF, 222:1 (2025), 25–40; Theoret. and Math. Phys., 222:1 (2025), 20–33
Citation in format AMSBIB
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\by M.~I.~Ismailov, C.~Sabaz
\paper Inverse scattering method via the~Gel'fand--Levitan--Marchenko equation for some negative-order nonlinear wave equations
\jour TMF
\yr 2025
\vol 222
\issue 1
\pages 25--40
\mathnet{http://mi.mathnet.ru/tmf10793}
\crossref{https://doi.org/10.4213/tmf10793}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4855370}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2025TMP...222...20I}
\transl
\jour Theoret. and Math. Phys.
\yr 2025
\vol 222
\issue 1
\pages 20--33
\crossref{https://doi.org/10.1134/S0040577925010039}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-86000227806}
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    Теоретическая и математическая физика Theoretical and Mathematical Physics
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