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

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Teoreticheskaya i Matematicheskaya Fizika, 2016, Volume 186, Number 2, Pages 191–220
DOI: https://doi.org/10.4213/tmf8958 (Mi tmf8958)

This article is cited in 36 scientific papers (total in 36 papers)

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

V. B. Matveev^ab, A. O. Smirnov^a

^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

Full-text PDF (3370 kB) Citations (36)

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

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.

Received: 28.04.2015
Revised: 31.08.2015

English version:
Theoretical and Mathematical Physics, 2016, Volume 186, Issue 2, Pages 156–182
DOI: https://doi.org/10.1134/S0040577916020033

Bibliographic databases:

MSC: 35Q55; 37C55

Language: Russian

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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https://www.mathnet.ru/eng/tmf/v186/i2/p191

This publication is cited in the following 36 articles:

A. B. Khasanov, Kh. N. Normurodov, T. G. Khasanov, “Integration of a Nonlinear Sine-Gordon–Liouville-Type Equation in the Class of Periodic Infinite-Gap Functions”, Ukr Math J, 76:8 (2025), 1381
A. B. Khasanov, R. Kh. Eshbekov, T. G. Hasanov, “Integration of a non-linear Hirota type equation with additional terms”, Izv. Math., 89:1 (2025), 196–219
A. B. Khasanov, T. G. Khasanov, “The Cauchy Problem for the Nonlinear Complex Modified Korteweg-de Vries Equation with Additional Terms in the Class of Periodic Infinite-Gap Functions”, Sib Math J, 65:4 (2024), 846
A. B. Khasanov, T. G. Khasanov, “Zadacha Koshi dlya nelineinogo kompleksnogo modifitsirovannogo uravneniya Kortevega — de Friza (kmKdF) s dopolnitelnymi chlenami v klasse periodicheskikh beskonechnozonnykh funktsii”, Sib. matem. zhurn., 65:4 (2024), 735–759
A. B. Khasanov, Kh. N. Normurodov, T. G. Khasanov, “Integration of a nonlinear sine-Gordon–Liouville-type equation in the class of periodic infinite-gap functions”, Ukr. Mat. Zhurn., 76:8 (2024), 1217
A. B. Khasanov, U. O. Xudayorov, “Integration of the Modified Korteweg–de Vries–Liouville Equation in the Class of Periodic Infinite-Gap Functions”, Math. Notes, 114:6 (2023), 1247–1259
Aknazar Khasanov, Khozhimurod Normurodov, Ulugbek Khudaerov, “Zadacha Koshi dlya nelineinogo uravneniya tipa sinus-Gordona v klasse periodicheskikh funktsii”, VOGUMFT, 2023, no. 1 (2), 210
A. B Khasanov, Kh. N Normurodov, U. O Khudaerov, “Cauchy Problem for the Nonlinear Liouville Equation in the Class of Periodic Infinite-Gap Functions”, Differentsialnye uravneniya, 59:10 (2023), 1412
A. B. Khasanov, Kh. N. Normurodov, U. O. Khudayorov, “Cauchy Problem for the Nonlinear Liouville Equation in the Class of Periodic Infinite-Gap Functions”, Diff Equat, 59:10 (2023), 1413
A. Khasanov, R. Eshbekov, Kh. Normurodov, “Integration of a Nonlinear Hirota Type Equation with Finite Density in the Class of Periodic Functions”, Lobachevskii J Math, 44:10 (2023), 4329
Aleksandr O. Smirnov, Eugeni A. Frolov, “On the Propagation Model of Two-Component Nonlinear Optical Waves”, Axioms, 12:10 (2023), 983
Rizvi S.T.R. Seadawy A.R. Akram U. Younis M. Althobaiti A., “Solitary Wave Solutions Along With Painleve Analysis For the Ablowitz-Kaup-Newell-Segur Water Waves Equation”, Mod. Phys. Lett. B, 36:02 (2022), 2150548
Zulfiqar A., Ahmad J., “Computational Solutions of Fractional (2+1)-Dimensional Ablowitz-Kaup-Newell-Segur Equation Using An Analytic Method and Application”, Arab. J. Sci. Eng., 47:1 (2022), 1003–1017
Ciancio A., Yel G., Kumar A., Baskonus H.M., Ilhan E., “On the Complex Mixed Dark-Bright Wave Distributions to Some Conformable Nonlinear Integrable Models”, Fractals-Complex Geom. Patterns Scaling Nat. Soc., 30:01 (2022), 2240018
V. S. Gerdjikov, Nianhua Li, V. B. Matveev, A. O. Smirnov, “On soliton solutions and soliton interactions of Kulish–Sklyanin and Hirota–Ohta systems”, Theoret. and Math. Phys., 213:1 (2022), 1331–1347
G. A. Mannonov, A. B. Khasanov, “The Cauchy problem for a nonlinear Hirota equation in the class of periodic infinite-zone functions”, St. Petersburg Math. J., 34:5 (2023), 821–845
Khalil Ahmad, Khudija Bibi, Baowei Feng, “New Function Solutions of Ablowitz-Kaup-Newell-Segur Water Wave Equation via Power Index Method”, Journal of Function Spaces, 2022 (2022), 1
Ufa Math. J., 13:2 (2021), 135–151
V. B. Matveev, A. O. Smirnov, “Elliptic solitons and «freak waves»”, St. Petersburg Math. J., 33:3 (2022), 523–551
V. B. Matveev, A. O. Smirnov, “Multiphase solutions of nonlocal symmetric reductions of equations of the AKNS hierarchy: General analysis and simplest examples”, Theoret. and Math. Phys., 204:3 (2020), 1154–1165

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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