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SIGMA, 2010, том 6, 055, 27 страниц (Mi sigma512)  

Эта публикация цитируется в 28 научных статьях (всего в 28 статьях)

Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions

Aristophanes Dimakisa, Folkert Müller-Hoissenb

a Department of Financial and Management Engineering, University of the Aegean, 41, Kountourioti Str., GR-82100 Chios, Greece
b Max-Planck-Institute for Dynamics and Self-Organization, Bunsenstrasse 10, D-37073 Göttingen, Germany

Аннотация: We express AKNS hierarchies, admitting reductions to matrix NLS and matrix mKdV hierarchies, in terms of a bidifferential graded algebra. Application of a universal result in this framework quickly generates an infinite family of exact solutions, including e.g. the matrix solitons in the focusing NLS case. Exploiting a general Miura transformation, we recover the generalized Heisenberg magnet hierarchy and establish a corresponding solution formula for it. Simply by exchanging the roles of the two derivations of the bidifferential graded algebra, we recover “negative flows”, leading to an extension of the respective hierarchy. In this way we also meet a matrix and vector version of the short pulse equation and also the sine-Gordon equation. For these equations corresponding solution formulas are also derived. In all these cases the solutions are parametrized in terms of matrix data that have to satisfy a certain Sylvester equation.

Ключевые слова: AKNS hierarchy; negative flows; Miura transformation; bidifferential graded algebra; Heisenberg magnet; mKdV; NLS; sine-Gordon; vector short pulse equation; matrix solitons

DOI: https://doi.org/10.3842/SIGMA.2010.055

Полный текст: PDF файл (466 kB)
Полный текст: http://emis.mi.ras.ru/journals/SIGMA/2010/055/
Список литературы: PDF файл   HTML файл

Реферативные базы данных:

ArXiv: 1004.1627
Тип публикации: Статья
MSC: 37J35; 37K10; 16E45
Поступила: 12 апреля 2010 г.; в окончательном варианте 21 июня 2010 г.; опубликована 16 июля 2010 г.
Язык публикации: английский

Образец цитирования: Aristophanes Dimakis, Folkert Müller-Hoissen, “Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions”, SIGMA, 6 (2010), 055, 27 pp.

Цитирование в формате AMSBIB
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\paper Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions
\jour SIGMA
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    Эта публикация цитируется в следующих статьяx:
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    2. Matsuno Y., “A novel multi-component generalization of the short pulse equation and its multisoliton solutions”, J Math Phys, 52:12 (2011), 123702  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. Brunelli J.C., Sakovich S., “On Integrability of the Yao-Zeng Two-Component Short-Pulse Equation”, Phys. Lett. A, 377:1-2 (2012), 80–82  crossref  mathscinet  adsnasa  isi  elib  scopus
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    6. Tchokouansi H.T., Kuetche V.K., Kofane T.C., “Exact Soliton Solutions To a New Coupled Integrable Short Light-Pulse System”, Chaos Solitons Fractals, 68 (2014), 10–19  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
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    9. Oleksandr Chvartatskyi, Yuriy Sydorenko, “Darboux Transformations for $(2+1)$-Dimensional Extensions of the KP Hierarchy”, SIGMA, 11 (2015), 028, 20 pp.  mathnet  crossref  mathscinet
    10. Feng B.-F., Maruno K.-i., Ohta Ya., “Integrable Semi-Discretization of a Multi-Component Short Pulse Equation”, J. Math. Phys., 56:4 (2015), 043502  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    11. Feng B.-F., “Complex Short Pulse and Coupled Complex Short Pulse Equations”, Physica D, 297 (2015), 62–75  crossref  mathscinet  adsnasa  isi  elib  scopus
    12. Vekslerchik V.E., “Soliton Fay Identities: II. Bright Soliton Case”, J. Phys. A-Math. Theor., 48:44 (2015), 445204  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    13. Chvartatskyi O. Mueller-Hoissen F. Stoilov N., ““Riemann Equations” in Bidifferential Calculus”, J. Math. Phys., 56:10 (2015), 103512  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
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    16. Matsuno Y., “Integrable multi-component generalization of a modified short pulse equation”, J. Math. Phys., 57:11 (2016), 111507  crossref  mathscinet  zmath  isi  elib  scopus
    17. Fritzsche B., Kaashoek M.A., Kirstein B., Sakhnovich A.L., “Skew-selfadjoint Dirac systems with rational rectangular Weyl functions: explicit solutions of direct and inverse problems and integrable wave equations”, Math. Nachr., 289:14-15 (2016), 1792–1819  crossref  mathscinet  zmath  isi  elib  scopus
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    19. Feng B.-F., Maruno K.-I., Ohta Ya., “Geometric Formulation and Multi-dark Soliton Solution to the Defocusing Complex Short Pulse Equation”, Stud. Appl. Math., 138:3 (2017), 343–367  crossref  mathscinet  zmath  isi  scopus
    20. Li B.-Q., Ma Yu.-L., “Periodic Solutions and Solitons to Two Complex Short Pulse (Csp) Equations in Optical Fiber”, Optik, 144 (2017), 149–155  crossref  isi  scopus
    21. Popowicz Z., “Lax Representations For Matrix Short Pulse Equations”, J. Math. Phys., 58:10 (2017), 103506  crossref  mathscinet  zmath  isi  scopus
    22. Shen Sh., Feng B.-F., Ohta Ya., “A Modified Complex Short Pulse Equation of Defocusing Type”, J. Nonlinear Math. Phys., 24:2 (2017), 195–209  crossref  mathscinet  isi  scopus
    23. Brunelli J.C., “Super Extensions of the Short Pulse Equation”, Commun. Nonlinear Sci. Numer. Simul., 63 (2018), 356–364  crossref  mathscinet  isi  scopus
    24. Brunelli J.C., “Nonlocal Short Pulse Equations”, Braz. J. Phys., 48:4 (2018), 421–425  crossref  isi
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    27. Yu Sh., Yin X., “The Cauchy Problem For a Generalized Two-Component Short Pulse System With High-Order Nonlinearities”, J. Math. Anal. Appl., 475:2 (2019), 1427–1447  crossref  mathscinet  zmath  isi  scopus
    28. Tchidjo R.T., Tchokouansi H.T., Felenou E.T., Kuetche V.K., Bouetou T.B., “On the Dynamics of Magnetic Wave in Ferrites: Influence of Damping and Inhomogeneous Exchange Effects”, J. Magn. Magn. Mater., 484 (2019), 382–390  crossref  isi
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