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TMF, 2016, Volume 188, Number 3, Pages 397–415 (Mi tmf9095)  

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

The $N$-wave equations with $\mathcal{PT}$ symmetry

V. S. Gerdjikova, G. G. Grahovskiab, R. I. Ivanovc

a Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria
b Department of Mathematical Sciences, University of Essex, Colchester, UK
c School of Mathematical Sciences, Dublin Institute of Technology, Dublin, Ireland

Abstract: We study extensions of $N$-wave systems with $\mathcal{PT}$ symmetry and describe the types of (nonlocal) reductions leading to integrable equations invariant under the $\mathcal P$ (spatial reflection) and $\mathcal T$ (time reversal) symmetries. We derive the corresponding constraints on the fundamental analytic solutions and the scattering data. Based on examples of three-wave and four-wave systems (related to the respective algebras $sl(3,\mathbb C)$) and $so(5,\mathbb C)$), we discuss the properties of different types of one- and two-soliton solutions. We show that the $\mathcal{PT}$-symmetric three-wave equations can have regular multisoliton solutions for some specific choices of their parameters.

Keywords: integrable system, $\mathcal{PT}$ symmetry, inverse scattering transform, soliton solution

DOI: https://doi.org/10.4213/tmf9095

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English version:
Theoretical and Mathematical Physics, 2016, 188:3, 1305–1321

Bibliographic databases:

Document Type: Article

Citation: V. S. Gerdjikov, G. G. Grahovski, R. I. Ivanov, “The $N$-wave equations with $\mathcal{PT}$ symmetry”, TMF, 188:3 (2016), 397–415; Theoret. and Math. Phys., 188:3 (2016), 1305–1321

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. T. Xu, M. Li, Y. Huang, Ya. Chen, Ch. Yu, “Nonsingular localized wave solutions for the nonlocal Davey-Stewartson I equation with zero background”, Mod. Phys. Lett. B, 31:35 (2017), 1750338  crossref  mathscinet  isi  scopus
    2. M. Gurses, “Nonlocal Fordy-Kulish equations on symmetric spaces”, Phys. Lett. A, 381:21 (2017), 1791–1794  crossref  mathscinet  zmath  isi  scopus
    3. V. S. Gerdjikov, G. G. Grahovski, R. I. Ivanov, “On integrable wave interactions and Lax pairs on symmetric spaces”, Wave Motion, 71:SI (2017), 53–70  crossref  mathscinet  isi  scopus
    4. H. Sarfraz, U. Saleem, “Darboux transformation and multi-soliton solutions of local/nonlocal $N$-wave interactions”, Mod. Phys. Lett. A, 32:36 (2017), 1750196  crossref  mathscinet  zmath  isi  scopus
    5. M. Gurses, A. Pekcan, “Nonlocal nonlinear Schrödinger equations and their soliton solutions”, J. Math. Phys., 59:5 (2018), 051501  crossref  mathscinet  zmath  isi  scopus
    6. G. G. Grahovski, A. Mohammed, H. Susanto, “Nonlocal reductions of the Ablowitz–Ladik equation”, Theoret. and Math. Phys., 197:1 (2018), 1412–1429  mathnet  crossref  crossref  adsnasa  isi  elib
    7. G. G. Grahovski, A. J. Mustafa, H. Susanto, “Nonlocal reductions of the multicomponent nonlinear Schrödinger equation on symmetric spaces”, Theoret. and Math. Phys., 197:1 (2018), 1430–1450  mathnet  crossref  crossref  adsnasa  isi  elib
    8. J. Yang, “Physically significant nonlocal nonlinear Schrödinger equation and its soliton solutions”, Phys. Rev. E, 98:4 (2018), 042202  crossref  isi  scopus
    9. B.-F. Feng, X.-D. Luo, M. J. Ablowitz, Z. H. Musslimani, “General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions”, Nonlinearity, 31:12 (2018), 5385–5409  crossref  mathscinet  zmath  isi  scopus
    10. K. Manikandan, S. Stalin, M. Senthilvelan, “Dynamical behaviour of solitons in a $\mathcal{PT}$-invariant nonlocal nonlinear Schrödinger equation with distributed coefficients”, Eur. Phys. J. B, 91:11 (2018), 291  crossref  mathscinet  isi
    11. Gurses M., Pekcan A., “Nonlocal Modified Kdv Equations and Their Soliton Solutions By Hirota Method”, Commun. Nonlinear Sci. Numer. Simul., 67 (2019), 427–448  crossref  mathscinet  isi  scopus
    12. Gurses M., Pekcan A., “(2+1)-Dimensional Local and Nonlocal Reductions of the Negative Akns System: Soliton Solutions”, Commun. Nonlinear Sci. Numer. Simul., 71 (2019), 161–173  crossref  mathscinet  isi  scopus
    13. Yang J., “General N-Solitons and Their Dynamics in Several Nonlocal Nonlinear Schrodinger Equations”, Phys. Lett. A, 383:4 (2019), 328–337  crossref  mathscinet  isi  scopus
    14. Pekcan A., “Nonlocal Coupled Hi-Mkdv Systems”, Commun. Nonlinear Sci. Numer. Simul., 72 (2019), 493–515  crossref  mathscinet  isi  scopus
    15. Xu T. Lan Sh. Li M. Li L.-L. Zhang G.-W., “Mixed Soliton Solutions of the Defocusing Nonlocal Nonlinear Schrodinger Equation”, Physica D, 390 (2019), 47–61  crossref  mathscinet  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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