S. Yu. Vernov, “Proof of the Absence of Elliptic Solutions of the Cubic Complex Ginzburg–Landau Equation”, TMF, 146:1 (2006), 161–171; Theoret. and Math. Phys., 146:1 (2006), 131

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Teoreticheskaya i Matematicheskaya Fizika, 2006, Volume 146, Number 1, Pages 161–171
DOI: https://doi.org/10.4213/tmf2016 (Mi tmf2016)

This article is cited in 14 scientific papers (total in 14 papers)

Proof of the Absence of Elliptic Solutions of the Cubic Complex Ginzburg–Landau Equation

S. Yu. Vernov

Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University

Full-text PDF (164 kB) Citations (14)

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DOI: https://doi.org/10.4213/tmf2016

Abstract: We consider the cubic complex Ginzburg–Landau equation. Using Hone's method, based on formal Laurent-series solutions and the residue theorem, we prove the absence of elliptic standing-wave solutions of this equation. This result complements a result by Hone, who proved the nonexistence of elliptic traveling-wave solutions. We show that it is more efficient to apply Hone's method to a system of polynomial differential equations rather than to an equivalent differential equation.

Keywords: standing wave, elliptic function, Laurent series, residue theorem, cubic complex Ginzburg–Landau equation.

English version:
Theoretical and Mathematical Physics, 2006, Volume 146, Issue 1, Pages 131–139
DOI: https://doi.org/10.1007/s11232-006-0013-9

Bibliographic databases:

Language: Russian

Citation: S. Yu. Vernov, “Proof of the Absence of Elliptic Solutions of the Cubic Complex Ginzburg–Landau Equation”, TMF, 146:1 (2006), 161–171; Theoret. and Math. Phys., 146:1 (2006), 131–139

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This publication is cited in the following 14 articles:

H. W. Schürmann, V. S. Serov, “On the existence of certain elliptic solutions of the cubically nonlinear Schrödinger equation”, Theoret. and Math. Phys., 219:1 (2024), 557–566
Robert Conte, Micheline Musette, Tuen Wai Ng, Chengfa Wu, “All meromorphic traveling waves of cubic and quintic complex Ginzburg-Landau equations”, Physics Letters A, 481 (2023), 129024
M. V. Demina, N. A. Kudryashov, “Dvoyako-periodicheskie meromorfnye resheniya avtonomnykh nelineinykh differentsialnykh uravnenii”, Model. i analiz inform. sistem, 21:5 (2014), 49–60
M. V. Demina, N. A. Kudryashov, “Doubly periodic meromorphic solutions of autonomous nonlinear differential equations”, Aut. Control Comp. Sci., 48:7 (2014), 633
Kudryashov N.A., Sinelshchikov D.I., “Elliptic solutions for a family of fifth order nonlinear evolution equations”, Applied Mathematics and Computation, 218:12 (2012), 6991–6997
Kudryashov N.A., Sinelshchikov D.I., “Exact solutions of the Swift-Hohenberg equation with dispersion”, Commun Nonlinear Sci Numer Simul, 17:1 (2012), 26–34
Kudryashov N.A., Sinelshchikov D.I., Demina M.V., “Exact solutions of the generalized Bretherton equation”, Phys. Lett. A, 375:7 (2011), 1074–1079
Demina M.V., Kudryashov N.A., “Explicit expressions for meromorphic solutions of autonomous nonlinear ordinary differential equations”, Commun. Nonlinear Sci. Numer. Simul., 16:3 (2011), 1127–1134
Demina M.V., Kudryashov N.A., “On elliptic solutions of nonlinear ordinary differential equations”, Applied Mathematics and Computation, 217:23 (2011), 9849–9853
Kudryashov N.A., Ryabov P.N., Sinelshchikov D.I., “Nonlinear waves in media with fifth order dispersion”, Phys Lett A, 375:20 (2011), 2051–2055
Demina M.V., Kudryashov N.A., “From Laurent series to exact meromorphic solutions: The Kawahara equation”, Phys. Lett. A, 374:39 (2010), 4023–4029
Vernov S.Yu., “Elliptic solutions of the quintic complex one-dimensional Ginzburg-Landau equation”, J. Phys. A, 40:32 (2007), 9833–9844
Vernov S.Yu., “Construction of Special Solutions for Nonintegrable Systems”, J. Nonlinear Math. Phys., 13:1 (2006), 50–63
Vernov S., “Interdependence Between the Laurent-Series and Elliptic Solutions of Nonintegrable Systems”, Computer Algebra in Scienfific Computing, Proceedings, Lecture Notes in Computer Science, 3718, eds. Ganzha V., Mayr E., Vorozhtsov E., Springer-Verlag Berlin, 2005, 457–468

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

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