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TMF, 2016, Volume 186, Number 2, Pages 191–220 (Mi tmf8958)  

This article is cited in 8 scientific papers (total in 8 papers)

Solutions of the Ablowitz–Kaup–Newell–Segur hierarchy equations of the “rogue wave” type: A unified approach

V. B. Matveevab, A. O. Smirnova

a St. Petersburg State University for Aerospace Instrumentation (SUAI), St. Petersburg, Russia
b Institut de Mathématiques de Bourgogne, Université de Bourgogne-Franche Comté, Dijon, France

Abstract: We describe a unified structure of solutions for all equations of the Ablowitz–Kaup–Newell–Segur hierarchy and their combinations. We give examples of solutions that satisfy different equations for different parameter values. In particular, we consider a rank-$2$ quasirational solution that can be used to investigate many integrable models in nonlinear optics. An advantage of our approach is the possibility to investigate changes in the behavior of a solution resulting from changing the model.

Keywords: rogue wave, freak wave, nonlinear Schrödinger equation, Hirota equation, AKNS hierarchy
Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/tmf8958

Full text: PDF file (3370 kB)
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English version:
Theoretical and Mathematical Physics, 2016, 186:2, 156–182

Bibliographic databases:

MSC: 35Q55; 37C55
Received: 28.04.2015
Revised: 31.08.2015

Citation: V. B. Matveev, A. O. Smirnov, “Solutions of the Ablowitz–Kaup–Newell–Segur hierarchy equations of the “rogue wave” type: A unified approach”, TMF, 186:2 (2016), 191–220; Theoret. and Math. Phys., 186:2 (2016), 156–182

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Jiguang Rao, Lihong Wang, Wei Liu, Jingsong He, “Rogue-wave solutions of the Zakharov equation”, Theoret. and Math. Phys., 193:3 (2017), 1783–1800  mathnet  crossref  crossref  adsnasa  isi  elib
    2. V. Kotlyarov, D. Shepelsky, “Planar unimodular Baker–Akhiezer function for the nonlinear Schödinger equation”, Ann. Math. Sci. Appl., 2:2 (2017), 343–384  mathscinet  zmath  isi
    3. L. Wang, Ch. Yang, J. Wang, J. He, “The height of an $n$th-order fundamental rogue wave for the nonlinear Schrödinger equation”, Phys. Lett. A, 381:20 (2017), 1714–1718  crossref  mathscinet  zmath  isi  scopus
    4. L. L. Feng, T. T. Zhang, “Breather wave, rogue wave and solitary wave solutions of a coupled nonlinear Schrödinger equation”, Appl. Math. Lett., 78 (2018), 133–140  crossref  mathscinet  zmath  isi  scopus
    5. A. Ali, A. R. Seadawy, D. Lu, “Computational methods and traveling wave solutions for the fourth-order nonlinear Ablowitz–Kaup–Newell–Segur water wave dynamical equation via two methods and its applications”, Open Phys., 16:1 (2018), 219–226  crossref  isi  scopus
    6. V. P. Kotlyarov, “A Matrix Baker–Akhiezer Function Associated with the Maxwell–Bloch Equations and their Finite-Gap Solutions”, SIGMA, 14 (2018), 082, 27 pp.  mathnet  crossref
    7. V. B. Matveev, A. O. Smirnov, “AKNS and NLS hierarchies, MRW solutions, $P_n$ breathers, and beyond”, J. Math. Phys., 59:9, SI (2018), 091419  crossref  mathscinet  zmath  isi  scopus
    8. A. O. Smirnov, V. B. Matveev, “Dvukhfaznye periodicheskie resheniya uravnenii iz AKNS ierarkhii”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 25, K 70-letiyu M. A. Semenova-Tyan-Shanskogo, Zap. nauchn. sem. POMI, 473, POMI, SPb., 2018, 205–227  mathnet
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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