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Multiphase solutions of nonlocal symmetric reductions of equations of the AKNS hierarchy: General analysis and simplest examples

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Abstract

We consider nonlocal symmetries that all or all even (all odd) equations of the AKNS hierarchy have. We construct examples of solutions simultaneously satisfying several nonlocal equations of the AKNS hierarchy. We present a detailed study of single-phase solutions.

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References

  1. V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in \( \mathcal{PT} \)-symmetric systems,” Rev. Modern Phys., 88, 035002 (2016); arXiv:1603.06826v1 [nlin.PS] (2016).

    ADS  Google Scholar 

  2. D. Christodoulides and J. Yang, eds., Parity–Time Symmetry and Its Applications (Springer Tracts Mod. Phys., Vol. 280), Springer, Singapore (2018).

    Google Scholar 

  3. M. J. Ablowitz and Z. H. Musslimani, “Integrable nonlocal nonlinear Schrödinger equation,” Phys. Rev. Lett., 110, 064105 (2013).

    ADS  Google Scholar 

  4. M. J. Ablowitz and Z. H. Musslimani, “Integrable discrete \(PT\) symmetric model,” Phys. Rev. E, 90, 032912 (2014).

    ADS  Google Scholar 

  5. M. J. Ablowitz and Z. H. Musslimani, “Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation,” Nonlinearity, 29, 915–946 (2016).

    ADS  MathSciNet  MATH  Google Scholar 

  6. M. J. Ablowitz and Z. H. Musslimani, “Integrable nonlocal nonlinear equations,” Stud. Appl. Math., 139, 7–59 (2017).

    MathSciNet  MATH  Google Scholar 

  7. T. P. Horikis and M. J. Ablowitz, “Rogue waves in nonlocal media,” Phys. Rev. E, 95, 042211 (2017); arXiv:1608.00927v1 [nlin.PS] (2016).

    ADS  MathSciNet  Google Scholar 

  8. M. J. Ablowitz, B.-F. Feng, X.-D. Luo, and Z. H. Musslimani, “Reverse space–time nonlocal sine-Gordon/sinh-Gordon equations with nonzero boundary conditions,” Stud. Appl. Math., 141, 267–307 (2018).

    MathSciNet  MATH  Google Scholar 

  9. V. S. Gerdjikov and A. Saxena, “Complete integrability of nonlocal nonlinear Schrödinger equation,” J. Math. Phys., 58, 013502 (2017).

    ADS  MathSciNet  MATH  Google Scholar 

  10. X.-Y. Tang, Z.-F. Liang, and X.-Z. Hao, “Nonlinear waves of a nonlocal modified KdV equation in the atmospheric and oceanic dynamical system,” Commun. Nonlinear Sci. Numer. Simulat., 60, 62–71 (2018).

    ADS  MathSciNet  Google Scholar 

  11. Z.-J. Yang, S.-M. Zhang, X.-L. Li, and Z.-G. Pang, “Variable sinh-Gaussian solitons in nonlocal nonlinear Schrödinger equation,” Appl. Math. Lett., 82, 64–70 (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  12. K. Manikandan, N. Vishnu Priya, M. Senthilvelan, and R. Sankaranarayanan, “Deformation of dark solitons in a \( \mathcal{PT} \)-invariant variable coefficients nonlocal nonlinear Schrodinger equation,” Chaos, 28, 083103 (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  13. D.-Y. Liu and W.-R. Sun, “Rational solutions for the nonlocal sixth-order nonlinear Schrödinger equation,” Appl. Math. Lett., 84, 63–69 (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  14. H.-Q. Zhang and M. Gao, “Rational soliton solutions in the parity–time-symmetric nonlocal coupled nonlinear Schrödinger equations,” Commun. Nonlinear Sci. Numer. Simul., 63, 253–260 (2018).

    ADS  MathSciNet  Google Scholar 

  15. P. S. Vinayagam, R. Radha, U. Al Khawaja, and L. Ling, “New classes of solutions in the coupled \( \mathcal{PT} \) symmetric nonlocal nonlinear Schrödinger equations with four wave mixing,” Commun. Nonlinear Sci. Numer. Simul., 59, 387–395 (2018); arXiv:1804.04914v1 [nlin.PS] (2018).

    ADS  MathSciNet  Google Scholar 

  16. K. Chen, X. Deng, S. Lou, and D.-J. Zhang, “Solutions of nonlocal equations reduced from the AKNS hierarchy,” Stud. Appl. Math., 141, 113–141 (2018).

    MathSciNet  MATH  Google Scholar 

  17. M. Gürses and A. Pekcan, “Nonlocal modified KdV equations and their soliton solutions by Hirota method,” Commun. Nonlinear Sci. Numer. Simul., 67, 427–448 (2019).

    ADS  MathSciNet  Google Scholar 

  18. B. Yang and J. Yang, “Rogue waves in the nonlocal \( \mathcal{PT} \)-symmetric nonlinear Schrödinger equation,” Lett. Math. Phys., 109, 945–973 (2019); arXiv:1711.05930v1 [nlin.SI] (2017).

    ADS  MathSciNet  MATH  Google Scholar 

  19. J. Yang, “General \(N\)-solitons and their dynamics in several nonlocal nonlinear Schrödinger equations,” Phys. Lett. A, 383, 328–337 (2019).

    ADS  MathSciNet  Google Scholar 

  20. A. O. Smirnov and E. E. Aman, “The simplest oscillating solutions of nonlocal nonlinear models,” J. Phys.: Conf. Ser., 1399, 022020 (2019).

    Google Scholar 

  21. Y. Yang, T. Suzuki, and X. Cheng, “Darboux transformations and exact solutions for the integrable nonlocal Lakshmanan–Porsezian–Daniel equation,” Appl. Math. Lett., 99, 105998 (2020).

    MathSciNet  MATH  Google Scholar 

  22. B. Yang and Y. Chen, “Reductions of Darboux transformations for the \( \mathcal{PT} \)-symmetric nonlocal Davey–Stewartson equations,” Appl. Math. Lett., 82, 43–49 (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  23. J. Rao, Y. Zhang, A. S. Fokas, and J. He, “Rogue waves of the nonlocal Davey–Stewartson I equation,” Nonlinearity, 31, 4090–4107 (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  24. C. Qian, J. Rao, D. Mihalache, and J. He, “Rational and semi-rational solutions of the \(y\)-nonlocal Davey–Stewartson I equation,” Comput. Math. Appl., 75, 3317–3330 (2018).

    MathSciNet  MATH  Google Scholar 

  25. B. Yang and Y. Chen, “Dynamics of rogue waves in the partially \( \mathcal{PT} \)-symmetric nonlocal Davey–Stewartson systems,” Commun. Nonlinear Sci. Numer. Simul., 69, 287–303 (2019).

    ADS  MathSciNet  Google Scholar 

  26. V. S. Gerdjikov, G. G. Grahovski, and R. I. Ivanov, “The \(N\)-wave equations with \( \mathcal{PT} \) symmetry,” Theor. Math. Phys., 188, 1305–1321 (2016).

    MathSciNet  MATH  Google Scholar 

  27. Z.-X. Zhou, “Darboux transformations and global solutions for a nonlocal derivative nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul., 62, 480–488 (2018).

    ADS  MathSciNet  Google Scholar 

  28. Y. Cao, B. A. Malomed, and J. He, “Two (2+1)-dimensional integrable nonlocal nonlinear Schrödinger equations: Breather, rational, and semi-rational solution,” Chaos Solitons Fractals, 114, 99–107 (2018); arXiv:1806.11107v2 [nlin.PS] (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  29. W. Liu and X. Li, “General soliton solutions to a (2+1)-dimensional nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions,” Nonlinear Dynamics, 93, 721–731 (2018).

    MATH  Google Scholar 

  30. Y. Cao, J. Rao, D. Mihalache, and J. He, “Semi-rational solutions for the (2+1)-dimensional nonlocal Fokas system,” Appl. Math. Lett., 80, 27–34 (2018).

    MathSciNet  MATH  Google Scholar 

  31. Q. Zhang, Y. Zhang, and R. Ye, “Exact solutions of nonlocal Fokas–Lenells equation,” Appl. Math. Lett., 98, 336–343 (2019).

    MathSciNet  MATH  Google Scholar 

  32. V. S. Gerdjikov, “On the integrability of Ablowitz–Ladik models with local and nonlocal reductions,” J. Phys.: Conf. Ser., 1205, 012015 (2019).

    Google Scholar 

  33. V. B. Matveev and A. O. Smirnov, “Solutions of the Ablowitz–Kaup–Newell–Segur hierarchy equations of the ‘rogue wave’ type: A unified approach,” Theor. Math. Phys., 186, 156–182 (2016).

    MathSciNet  MATH  Google Scholar 

  34. V. B. Matveev and A. O. Smirnov, “AKNS and NLS hierarchies, MRW solutions, \(P_n\) breathers, and beyond,” J. Math. Phys., 59, 091419 (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  35. V. B. Matveev and A. O. Smirnov, “Two-phase periodic solutions to the AKNS hierarchy equations,” J. Math. Sci., 242, 722–741 (2019).

    MathSciNet  MATH  Google Scholar 

  36. E. D. Belokolos, A. I. Bobenko, V. Z. Enol’skii, A. R. Its, and V. B. Matveev, Algebro-geometrical Approach to Nonlinear Evolution Equations, Springer, Berlin (1994).

    Google Scholar 

  37. B. Yang and J. Yang, “On general rogue waves in the parity–time-symmetric nonlinear Schrödinger equation,” J. Math. Anal. Appl., 487, 124023 (2020); arXiv:1903.06203v1 [nlin.SI] (2019).

    MathSciNet  MATH  Google Scholar 

  38. M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform–Fourier analysis for nonlinear problems,” Stud. Appl. Math., 53, 249–315 (1974).

    MathSciNet  MATH  Google Scholar 

  39. M. Lakshmanan, K. Porsezian, and M. Daniel, “Effect of discreteness on the continuum limit of the Heisenberg spin chain,” Phys. Lett. A, 133, 483–488 (1988).

    ADS  Google Scholar 

  40. K. Porsezian, M. Daniel, and M. Lakshmanan, “On the integrability aspects of the one-dimensional classical continuum isotropic biquadratic Heisenberg spin chain,” J. Math. Phys., 33, 1807–1816 (1992).

    ADS  MathSciNet  MATH  Google Scholar 

  41. M. Daniel, K. Porsezian, and M. Lakshmanan, “On the integrable models of the higher order water wave equation,” Phys. Lett. A, 174, 237–240 (1993).

    ADS  MathSciNet  Google Scholar 

  42. R. Hirota, “Exact envelope-soliton solutions of a nonlinear wave equation,” J. Math. Phys., 14, 805–809 (1973).

    ADS  MathSciNet  MATH  Google Scholar 

  43. C. Q. Dai and J. F. Zhang, “New solitons for the Hirota equation and generalized higher-order nonlinear Schrodinger equation with variable coefficients,” J. Phys. A: Math. Theor., 39, 723–737 (2006).

    ADS  MathSciNet  MATH  Google Scholar 

  44. A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Rogue waves and rational solutions of the Hirota equation,” Phys. Rev. E, 81, 046602 (2010).

    ADS  MathSciNet  Google Scholar 

  45. L. Li, Z. Wu, L. Wang, and J. He, “High-order rogue waves for the Hirota equation,” Ann. Phys., 334, 198–211 (2013).

    ADS  MathSciNet  MATH  Google Scholar 

  46. J. S. He, Ch. Zh. Li, and K. Porsezian, “Rogue waves of the Hirota and the Maxwell–Bloch equations,” Phys. Rev. E, 87, 012913 (2013); arXiv:1205.1191v3 [nlin.SI] (2012).

    ADS  Google Scholar 

  47. I. M. Krichever, “Methods of algebraic geometry in the theory of non-linear equations,” Russian Math. Surveys, 32, 185–213 (1977).

    ADS  MATH  Google Scholar 

  48. N. I. Akhiezer, Elements of the Theory of Elliptic Functions [in Russian], Nauka, Moscow (1970); English. transl.,, Amer. Math. Soc., Providence, R. I. (1990).

    Google Scholar 

  49. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U.S. Govt. Print. Off., Washington (1964).

    MATH  Google Scholar 

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Funding

This research was supported by the Russian Foundation for Basic Research (Grant No. 19-01-00734).

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Authors and Affiliations

  1. St. Petersburg Department of Steklov Institute of Mathematics of the Russian Academy of Sciences, St. Petersburg, Russia

    V. B. Matveev

  2. Institut de Mathématiques de Bourgogne (IMB), Université de Bourgogne – France Comté, Dijon, France

    V. B. Matveev

  3. St. Petersburg State University of Aerospace Instrumentation, St. Petersburg, Russia

    A. O. Smirnov

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  1. V. B. Matveev
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Correspondence to V. B. Matveev.

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Matveev, V.B., Smirnov, A.O. Multiphase solutions of nonlocal symmetric reductions of equations of the AKNS hierarchy: General analysis and simplest examples. Theor Math Phys 204, 1154–1165 (2020). https://doi.org/10.1134/S0040577920090056

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