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Teoreticheskaya i Matematicheskaya Fizika, 2020, Volume 202, Number 3, Pages 382–392
DOI: https://doi.org/10.4213/tmf9814 (Mi tmf9814) 		 
This article is cited in 5 scientific papers (total in 5 papers)

Primitive solutions of the Korteweg–de Vries equation

S. A. Dyachenkoa, P. Nabelekb, D. V. Zakharovc, V. E. Zakharovde

a Department of Mathematics, University of Washington, Seattle, Washington, USA
b Department of Mathematics, Oregon State University, Corvallis, Oregon, USA
c Department of Mathematics, Central Michigan University, Mount Pleasant, Michigan, USA
d Department of Mathematics, University of Arizona, Tucson, Arizona, USA
e Skolkovo Institute of Science and Technology, Skolkovo, Moscow Oblast, Russia
Full-text PDF (839 kB) Citations (5)
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DOI: https://doi.org/10.4213/tmf9814
Abstract: We survey recent results connected with constructing a new family of solutions of the Korteweg–de Vries equation, which we call primitive solutions. These solutions are constructed as limits of rapidly vanishing solutions of the Korteweg–de Vries equation as the number of solitons tends to infinity. A primitive solution is determined nonuniquely by a pair of positive functions on an interval on the imaginary axis and a function on the real axis determining the reflection coefficient. We show that elliptic one-gap solutions and, more generally, periodic finite-gap solutions are special cases of reflectionless primitive solutions.
Keywords: integrable system, Korteweg–de Vries equation, primitive solution.
Funding agency Grant number
National Science Foundation DMS-1716822
DMS-1715323
Russian Science Foundation 19-72-30028
The research of S. A. Dyachenko and D. V. Zakharov was supported by the National Science Foundation (Grant No. DMS-1716822).
The research of V. E. Zakharov was supported by the National Science Foundation (Grant No. DMS-1715323).
The results in Secs. 3–5 were obtained with support of a grant from the Russian Science Foundation (Project No. 19-72-30028).
Received: 08.09.2019
Revised: 08.09.2019
English version:
Theoretical and Mathematical Physics, 2020, Volume 202, Issue 3, Pages 334–343
DOI: https://doi.org/10.1134/S0040577920030058
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: S. A. Dyachenko, P. Nabelek, D. V. Zakharov, V. E. Zakharov, “Primitive solutions of the Korteweg–de Vries equation”, TMF, 202:3 (2020), 382–392; Theoret. and Math. Phys., 202:3 (2020), 334–343
Citation in format AMSBIB
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\paper Primitive solutions of the~Korteweg--de~Vries equation
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  • https://www.mathnet.ru/eng/tmf9814
  • https://doi.org/10.4213/tmf9814
  • https://www.mathnet.ru/eng/tmf/v202/i3/p382
    • This publication is cited in the following 5 articles:
    1. H. G. Abdelwahed, A. F. Alsarhana, E. K. El-Shewy, Mahmoud A. E. Abdelrahman, “Higher-order dispersive and nonlinearity modulations on the propagating optical solitary breather and super huge waves”, Fractal Fract, 7:2 (2023), 127  crossref
    2. S. M. Mabrouk, E. Y. Abu El Seoud, A.-M. Wazwaz, “The nonlocal potential transformation method for solitary wave packets of a shock-breaking dynamics system”, Waves in Random and Complex Media, 2022, 1–14  crossref  mathscinet
    3. P. V. Nabelek, “On solutions to the nonlocal $\overline{\partial}$ -problem and $(2+1)$ dimensional completely integrable systems”, Lett. Math. Phys., 111:1 (2021), 16  crossref  mathscinet  isi
    4. Nabelek V P., “Algebro-Geometric Finite Gap Solutions to the Korteweg-de Vries Equation as Primitive Solutions”, Physica D, 414 (2020), 132709  crossref  mathscinet  isi  scopus
    5. V. E. Zakharov, D. V. Zakharov, “Generalized primitive potentials”, Dokl. Math., 101:2 (2020), 117–121  mathnet  crossref  crossref  zmath  elib
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    		Теоретическая и математическая физика Theoretical and Mathematical Physics
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    Full-text PDF :187
    References:70
    First page:29
     
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