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General soliton solutions to a nonlocal long-wave–short-wave resonance interaction equation with nonzero boundary condition

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Abstract

Under investigation in this work is a newly proposed nonlocal long-wave–short-wave resonance interaction (LSRI) equation with the self-induced parity-time (PT) symmetric potential. This equation offers PT symmetry analogues of the classical integrable LSRI equation and may be important for the occurence of such equations in nonlinear optics as the nonlocal NLS equation. General soliton solutions to the nonlocal LSRI equation with nonzero boundary condition are derived by using the Hirota’s bilinear method combined with the Kadomtsev–Petviashvili (KP) hierarchy reduction method. These solutions are expressed in terms of Gramian determinants and include dark–dark solitons, dark–antidark solitons and antidark–antidark solitons. Three typical cases of the two solitons, namely two dark–dark solitons, two dark–antidark solitons and two antidark–antidark solitons, are demonstrated.

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References

  1. Wang, L., Zhang, J.H., Liu, C., Li, M., Qi, F.H.: Breather transition dynamics, Peregrine combs and walls, and modulation instability in a variable-coefficient nonlinear Schördinger equation with higher-order effects. Phys. Rev. E 93, 062217 (2016)

    Article  MathSciNet  Google Scholar 

  2. Wang, L., Li, M., Qi, F.H., Xu, T.: Modulational instability, nonautonomous breathers and rogue waves for a variable-coefficient derivative nonlinear Schrödinger equation in the inhomogeneous plasmas. Phys Plasmas 22, 032308 (2015)

    Article  Google Scholar 

  3. Mihalache, D.: Multidimensional localized structures in optics and Bose-Einstein condensates: a selection of recent studies. Rom. J. Phys. 59, 295–312 (2014)

    Google Scholar 

  4. Bagnato, V.S., Frantzeskakis, D.J., Kevrekidis, P.G.: Bose-Einstein condensation: twenty years after[J]. Rom. Rep. Phys. 67, 251–253 (2015)

    Google Scholar 

  5. Malomed, B., Torner, L., Wise, F., Mihalache, D.: On multidimensional solitons and their legacy in contemporary atomic, molecular and optical physics. J. Phys. B At. Mol. Opt. Phys. 49, 17 (2016)

    Article  Google Scholar 

  6. Kevrekidis, P.G., Frantzeskakis, D.J.: Solitons in coupled nonlinear Schr?dinger models: a survey of recent developments. Rev. Phys 1, 140–153 (2016)

    Article  Google Scholar 

  7. Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

  8. Dickey, L.A.: Soliton Equations and Hamiltonian Systems. World Scientific, Singapore (2003)

    Book  MATH  Google Scholar 

  9. Matsuno, Y.: Bilinear Transformation Method. Academic, New York (1984)

    MATH  Google Scholar 

  10. Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  11. Bluman, G. W., Kumei, S.: Symmetries and differential equations. In: Grad. Texts in Math, 81, Springer, New York(1989)

  12. Matveev, V.B., Salle, M.A.: Darboux Transformation and Solitons. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  13. Gu, C.H., Hu, H.S., Zhou, Z.X.: Darboux Transformation in Soliton Theory and Geometric Applications. Shanghai Science and Technology Press, Shanghai (1999)

    Google Scholar 

  14. Date, E., Kashiwara, M., Jimbo, M., Miwa, T.: Transformation groups for soliton equations. In: Jimbo, M., Miwa, T. (eds.) Nonlinear Integrable Systems–Classical Theory and Quantum Theory, p. 39C119. World Scientific, Singapore (1983)

    Google Scholar 

  15. Jimbo, M., Miwa, T.: Solitons and infinite dimensional Lie algebras. Publ. Res. Inst. Math. Sci. 19(3), 943–1001 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ohta, Y., Wang, D.S., Yang, J.: General N-dark-dark solitons in the coupled nonlinear Schrödinger equations. Stud. Appl. Math. 127(4), 345–371 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Rao, J.G., Porsezian, K., He, J.S.: Semi–rational solutions of the third–type Davey–Stewartson equation. Chaos 27(8), 083115 (2017)

    Article  MathSciNet  Google Scholar 

  18. Fokas, A.S.: Integrable multidimensional versions of the nonlocal nonlinear Schrödinger equation. Nonlinearity 29, 319–324 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ma, W.X.: Lump solutions to the Kadomtsev-Petviashvili equation. Phys. Lett. A 379(36), 1975–1978 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ma, W.X., Qin, Z.Y., Lü, X.: Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nolinear Dyn. 84(2), 923–931 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Yang, J.Y., Ma, W.X.: Lump solutions to the BKP equation by symbolic computation. Int. J. Mod. Phys. B 30, 1640028 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kheybari, S., Darvishi, M.T., Wazwaz, A.M.: A semi-analytical algorithm to solve systems of integro-differential equations under mixed boundary conditions. Appl. Math. Comput. 317, 72–89 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Darvishi, M.T., Najafi, M., Wazwaz, A.M.: Soliton solutions for Boussinesq-like equations with spatio-temporal dispersion. Ocean Eng. 130, 228–240 (2017)

    Article  Google Scholar 

  24. Wazwaz, A.M.: Abundant solutions of various physical features for the (2+1)-dimensional modified KdV-Calogero-Bogoyavlenskii-Schiff equation. Nolinear Dyn. 89, 1–6 (2017)

    Article  MathSciNet  Google Scholar 

  25. Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having \(PT\) symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. Bender, C.M., Boettcher, S., Melisinger, P.N.: PT-symmetric quantum mechanics. J. Math. Phys. 40, 2201–2229 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear Schrödinger equation. Phys. Rev. Lett. 110, 064105 (2013)

    Article  Google Scholar 

  28. Rao, J.G., Wang, L.H., Zhang, Y., He, J.S.: Rational solutions for the Fokas systems. Commun. Theor. Phys. 64(12), 605–618 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ablowitz, M.J., Luo, X. D., Musslimani, Z.H.: Inverse scattering transform for the nonlocal nonlinear Schrödinger equation with nonzero boundary conditions. arXiv:1612.02726 (2016)

  30. Li, M., Xu, T.: Dark and antidark soliton interactions in the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential. Phys. Rev. E 91, 033202 (2015)

    Article  MathSciNet  Google Scholar 

  31. Xu, Z., Chow, K.: Breathers and rogue waves for a third order nonlocal partial differential equation by a bilinear transformation. Appl. Math. Lett. 56, 72–77 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Rao. J.G., Zhang. Y.S., Fokas, A.S., He. J.S.: Rogue waves of the nonlocal Davey-Stewartson I equation (preprint 2016)

  33. Wen, X., Yan, Z.Y., Yang, Y.: Dynamics of higher-order rational solitons for the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential. Chaos 26, 063123 (2016)

    Article  MathSciNet  Google Scholar 

  34. Rao, J.G., Zhang, Y.S., Fokas, A.S., He, J.S.: Rogue waves of the nonlocal Davey-Stewartson I equation (preprint 2016)

  35. Yang, B., Yang, J.K.: Transformations between nonlocal and local integrable equations. arXiv:1705.00332 (2017)

  36. Yang, B., Yang, J.K.: General rogue waves in the nonlocal PT-symmetric nonlinear Schrödinger equation. arXiv:1711.05930 (2017)

  37. Yang, J.K.: General N–solitons and their dynamics in several nonlocal nonlinear Schrödinger equations. arXiv:1712.01181 (2017)

  38. Liu, Y.K., Li, B.: Rogue Waves in the \((2+1)\)-Dimensional Nonlinear Schrödinger equation with a parity-time-symmetric potential. Chin. Phy. Lett. 34, 010202 (2017)

    Article  Google Scholar 

  39. Lou, S.Y.: Alice-bob systems, \( Ps \)–\( Td \)-\( C\) principles and multi–soliton solutions. arXiv:1603.03975 (2016)

  40. Zhou, Z.X.: Darboux transformations and global solutions for a nonlocal derivative nonlinear Schrödinger equation. arXiv:1612.04892 (2016)

  41. Ma, L.Y., Shou, S.F., Zhu, Z.N.: Soliton solution and gauge equivalence for an integrable nonlocal complex modified Korteweg-de Vries equation. J. Math. Phys. 58, 103501 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ma, L.Y., Zhu, Z.N.: Nonlocal nonlinear Schrödinger equation and its discrete version: soliton solutions and gauge equivalence. J. Math. Phys. 57, 083507 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  43. Huang, X., Ling, L.: Soliton solutions for the nonlocal nonlinear Schrödinger equation. Eur. Phys. J. Plus 131, 148 (2016)

    Article  Google Scholar 

  44. Wen, X.Y., Yan, Z., Yang, Y.: Dynamics of higher-order rational solitons for the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential. Chaos 26, 063123 (2016)

    Article  MathSciNet  Google Scholar 

  45. Yang, B., Chen, Y.: General rogue waves and their dynamics in several reverse time integrable nonlocal nonlinear equations. arXiv:1712.05974 (2017)

  46. Rao, J.G., Cheng, Y., He, J.S.: Rational and semi-rational solutions of the nonlocal Davey-Stewartson equations. Stud. Appl. Math. 139, 568–598 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  47. Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear equations. Stud. Appl. Math. 139, 7–59 (1026)

    Article  MathSciNet  MATH  Google Scholar 

  48. Zhou, Z.X.: Darboux transformations and global solutions for nonlocal Davey-Stewartson I equation. arXiv:1612.05689 (2016)

  49. Yang, B., Chen, Y.: Dynamics of Rogue Waves in the Partially \(PT\)-symmetric Nonlocal Davey-Stewartson systems. arXiv:1710.07061 (2017)

  50. Feng, B.F., Luo, X.D., Ablowitz, M.J., Musslimani, Z.H.: General soliton solutions to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions. arXiv:1712.09172 (2017)

  51. Benny, D.J.: A general theory for interactions between short and long waves. Stud. Appl. Math. 56, 15 (1977)

    Google Scholar 

  52. Yajima, N., Oikawa, M.: Formation and interaction of Sonic- Langmuir solitons inverse scattering method. Prog. Theor. Phys. 56, 1719–1739 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  53. Zakharov, V.E.: Collapse of Langmuir waves. Sov. Phys. JETP 35, 908–914 (1972)

    Google Scholar 

  54. Ma, Y.C.: Complete solution of the long wave-short wave resonance equations. Stud. Appl. Math. 59, 201–221 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  55. Kivshar, Y.S.: Stable vector solitons composed of bright and dark pulses. Opt. Lett. 17, 1322 (1992)

    Article  Google Scholar 

  56. Chowdhury, A., Tataronis, J.A.: Long wave-short wave resonance in nonlinear negative refractive index media. Phys. Rev. Lett. 100, 153905 (2008)

    Article  Google Scholar 

  57. Rao, J.G., Porsezian, K., He, J.S., Kanna, T.: Dynamics of lumps and dark-dark solitons in the multi-component long-wave-short-wave resonance interaction system. Proc. R. Soc. A 474, 20170627 (2018)

    Article  Google Scholar 

  58. Nistazakis, H.E., Frantzeskakis, D.J., Kevrekidis, P.G., Malomed, B.A., Carretero -Gonzàlez, R.: Phys. Rev. A 77, 03361 (2008)

    Article  Google Scholar 

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Acknowledgements

The author thanks Prof. Y. Zhang for his helpful discussions. This work was supported by the National Key Research and Development Program of China (Grant Nos. 2016YFC1402000, 2016YFC1402304), NSFC–Shandong Joint Fund for Marine Science Research Centers (Grant No. U1606405).

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  1. Laboratory of Marine Science and Numerical Modeling, The First Institute of Oceanography, Qingdao, 266061, China

    Baonan Sun

  2. Laboratory for Regional Oceanography and Numerical Modeling, Qingdao National Laboratory for Marine Science and Technology, Qingdao, 266237, China

    Baonan Sun

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Sun, B. General soliton solutions to a nonlocal long-wave–short-wave resonance interaction equation with nonzero boundary condition. Nonlinear Dyn 92, 1369–1377 (2018). https://doi.org/10.1007/s11071-018-4132-2

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  • DOI: https://doi.org/10.1007/s11071-018-4132-2

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