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TMF, 1987, Volume 72, Number 2, Pages 183–196 (Mi tmf5321)  

This article is cited in 15 scientific papers (total in 15 papers)

Exact first-order solutions of the nonlinear Schrödinger equation

N. N. Akhmediev, V. M. Eleonskii, N. E. Kulagin


Abstract: A method of obtaining exact solutions of the nonlinear Schrödinger equation (NSE) is suggested which is based on the substitution connecting real and imaginary parts of the solution by a linear relationship with coefficients depending on time only. The method is essentially the construction of a certain system of ordinary differential equations the solutions of which determine the solutions of NSE. The solutions obtained form a three-parameter family and are expressed in terms of the Jacobi elliptic functions and the third kind incomplete elliptic integral. In general case, the solutions are periodic in spatial variable and double-periodic in time. Particular cases for which the solutions can be expressed in terms of the Jacobi functions and elementary functions are studied in detail. Possibilities of practical applications of the solutions found are pointed out.

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English version:
Theoretical and Mathematical Physics, 1987, 72:2, 809–818

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Received: 12.03.1986

Citation: N. N. Akhmediev, V. M. Eleonskii, N. E. Kulagin, “Exact first-order solutions of the nonlinear Schrödinger equation”, TMF, 72:2 (1987), 183–196; Theoret. and Math. Phys., 72:2 (1987), 809–818

Citation in format AMSBIB
\Bibitem{AkhEleKul87}
\by N.~N.~Akhmediev, V.~M.~Eleonskii, N.~E.~Kulagin
\paper Exact first-order solutions of the nonlinear Schr\"odinger equation
\jour TMF
\yr 1987
\vol 72
\issue 2
\pages 183--196
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=915545}
\zmath{https://zbmath.org/?q=an:0656.35135}
\transl
\jour Theoret. and Math. Phys.
\yr 1987
\vol 72
\issue 2
\pages 809--818
\crossref{https://doi.org/10.1007/BF01017105}
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    Citing articles on Google Scholar: Russian citations, English citations
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    2. Yakhshimuratov A., “The Nonlinear Schrodinger Equation with a Self-consistent Source in the Class of Periodic Functions”, Math Phys Anal Geom, 14:2 (2011), 153–169  crossref  isi
    3. V. P. Ruban, “On the nonlinear Schrцdinger equation for waves on a nonuniform current”, JETP Letters, 95:9 (2012), 486–491  mathnet  crossref  isi  elib  elib
    4. Turitsyn S.K., Bale B.G., Fedoruk M.P., “Dispersion-Managed Solitons in Fibre Systems and Lasers”, Phys. Rep.-Rev. Sec. Phys. Lett., 521:4 (2012), 135–203  crossref  isi
    5. V. P. Ruban, “On the modulation instability of surface waves on a large-scale shear flow”, JETP Letters, 97:4 (2013), 188–193  mathnet  crossref  crossref  isi  elib  elib
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    10. Van Gorder R.A., “Breathers and Nonlinear Waves on Open Vortex Filaments in the Nonrelativistic Abelian Higgs Model”, Phys. Rev. D, 95:9 (2017), 096007  crossref  isi
    11. P. G. Grinevich, P. M. Santini, “Phase resonances of the NLS rogue wave recurrence in the quasisymmetric case”, Theoret. and Math. Phys., 196:3 (2018), 1294–1306  mathnet  crossref  crossref  adsnasa  isi  elib
    12. Li M., Shui J.-J., Huang Y.-H., Wang L., Li H.-J., “Localized-Wave Interactions For the Discrete Nonlinear Schrodinger Equation Under the Nonvanishing Background”, Phys. Scr., 93:11 (2018), 115203  crossref  isi  scopus
    13. Hoffmann C., Charalampidis E.G., Frantzeskakis D.J., Kevrekidis P.G., “Peregrine Solitons and Gradient Catastrophes in Discrete Nonlinear Schrodinger Systems”, Phys. Lett. A, 42-43 (2018), 3064–3070  crossref  isi  scopus
    14. P. G. Grinevich, P. M. Santini, “Konechnozonnyi podkhod v periodicheskoi zadache Koshi dlya anomalnykh voln v nelineinom uravnenii Shredingera pri nalichii neskolkikh neustoichivykh mod”, UMN, 74:2(446) (2019), 27–80  mathnet  crossref  elib
    15. Karachalios I N., Kyriazopoulos P., Vetas K., “Excitation of Peregrine-Type Waveforms From Vanishing Initial Conditions in the Presence of Periodic Forcing”, Z. Naturfors. Sect. A-J. Phys. Sci., 74:5 (2019), 371–382  crossref  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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