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Modulational instability, interactions of two-component localized waves and dynamics in a semi-discrete nonlinear integrable system on a reduced two-chain lattice

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Abstract

In this paper, we investigate a semi-discrete two-component integrable system on zigzag-runged ladder lattice and find a variety of two-component localized waves. First of all, we investigate the modulational instability to reveal the formation mechanism of rogue waves from three distinct plane-wave solutions for this system. Secondly, the discrete generalized \((m,N-m)\)-fold Darboux transformation is first constructed to solve this system. As applications, three kinds of plane-wave seed solutions are chosen to exhibit discrete two-component localized wave solutions including rogue wave solutions, breather solutions, periodic wave solutions and interaction solutions. Moreover, the dynamical behaviors of such localized wave solutions are discussed via numerical simulation. Finally, we summarize a few mathematical features to obtain various two-component localized waves through the discrete generalized\((m,N-m)\)-fold Darboux transformation. It is found that there are simultaneously bright–dark, dark–bright and bright–bright two-component rogue wave solutions in this discrete two-component system, and moreover, the shapes of bright–dark rogue waves in two components appear alternately with the increase of order number of Darboux transformation when some special seed solutions are selected, which have not been reported before. It is hoped that the results obtained in this paper may be used to understand some related physical phenomena.

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References

  1. Y.F. Yue, L.L. Huang, Y. Chen, N-solitons, breathers, lumps and rogue wave solutions to a (3+1)-dimensional nonlinear evolution equation. Comput. Math. Appl. 75, 2538–2548 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  2. L.L. Huang, Y.F. Yue, Y. Chen, Localized waves and interaction solutions to a (3+1)-dimensional generalized KP equation. Comput. Math. Appl. 74, 831–844 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  3. L.L. Feng, S.F. Tian, X.B. Wang, T.T. Zhanga, Rogue waves, homoclinic breather waves and soliton waves for the (2+1)-dimensional B-type Kadomtsev–Petviashvili equation. Appl. Math. Lett. 65, 90–97 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. X.B. Wang, S.F. Tian, C.Y. Qin, T.T. Zhang, Dynamics of the breathers, rogue waves and solitary waves in the (2+1)-dimensional Ito equation. Appl. Math. Lett. 68, 40–47 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. W.X. Ma, X.L. Yong, H.Q. Zhang, Diversity of interaction solutions to the (2+1)-dimensional Ito equation. Comput. Math. Appl. 75, 289–295 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. W.X. Ma, Abundant lumps and their interaction solutions of (3+1)-dimensional linear PDEs. J. Geom. Phys. 133, 10–16 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. A.M. Wazwaz, Two new integrable fourth-order nonlinear equations: multiple soliton solutions and multiple complex soliton solutions. Nonlinear Dyn. 94, 2655–2663 (2018)

    Article  Google Scholar 

  8. A.M. Wazwaz, L. Kaur, New integrable Boussinesq equations of distinct dimensions with diverse variety of soliton solutions. Nonlinear Dyn. 97, 83–94 (2019)

    Article  MATH  Google Scholar 

  9. M. Onorato, S. Residori, U. Bortolozzo, A. Montina, F.T. Arecchi, Rogue waves and their generating mechanisms in different physical contexts. Phys. Rep. 528, 47–89 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  10. N. Akhmediev, A. Ankiewicz, M. Taki, Waves that appear from nowhere and disappear without a trace. Phys. Lett. A 373, 675–678 (2009)

    Article  ADS  MATH  Google Scholar 

  11. D.R. Solli, C. Ropers, P. Koonath et al., Optical rogue wave. Nature 450, 1054–1057 (2007)

    Article  ADS  Google Scholar 

  12. B. Kibler, J. Fatome, C. Finot et al., The Peregrine soliton in nonlinear fibre optics. Nat. Phys. 6, 790–795 (2010)

    Article  Google Scholar 

  13. E. Pelinovsky, C. Kharif, Extreme Ocean Waves (Springer, Berlin, 2008)

    Book  MATH  Google Scholar 

  14. A. Chabchoub, Tracking breather dynamics in irregular sea state conditions. Phys. Rev. Lett. 117, 144103 (2016)

    Article  ADS  Google Scholar 

  15. A. Chabchoub, N.P. Hoffmann, N. Akhmediev, Rogue wave observation in a water wave tank. Phys. Rev. Lett. 106, 204502 (2011)

    Article  ADS  Google Scholar 

  16. A. Chabchoub, N.P. Hoffmann, N. Akhmediev et al., Observation of a hierarchy of up to fifth-order rogue waves in a water tank. Phys. Rev. E 86, 056601 (2012)

    Article  ADS  Google Scholar 

  17. H. Bailung, S.K. Sharma, Y. Nakamura, Observation of Peregrine solitons in a multicomponent plasma with negative ions. Phys. Rev. Lett. 107, 255005 (2011)

    Article  ADS  Google Scholar 

  18. Z. Yan, Financial rogue waves. Commun. Theor. Phys. 54, 947 (2010)

    Article  ADS  MATH  Google Scholar 

  19. Z. Yan, Vector financial rogue waves. Phys. Lett. A 375, 4274–4279 (2011)

    Article  ADS  MATH  Google Scholar 

  20. B. Guo, L. Ling, Q.P. Liu, Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions. Phys. Rev. E 85, 026607 (2012)

    Article  ADS  Google Scholar 

  21. L. Xu, D.S. Wang, X.Y. Wen, Y.L. Jiang, Exotic localized vector waves in a two-component nonlinear wave system. J. Nonlinear Sci. 30, 537–564 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. L. Ling, L.C. Zhao, B. Guo, Darboux transformation and classification of solution for mixed coupled nonlinear Schrödinger equations. Commun. Nonlinear Sci. Numer. Simul. 32, 285–304 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. G. Zhang, Z. Yan, X.Y. Wen, Three-wave resonant interactions: multi-dark-dark-dark solitons, breathers, rogue waves, and their interactions and dynamics. Physica D 366, 27 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. G. Zhang, Z. Yan, Three-component nonlinear Schrödinger equations: modulational instability, \(N\)th-order vector rational and semi-rational rogue waves, and dynamics. Commun. Nonlinear Sci. Numer. Simul. 62, 117–133 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. H. Li, S.Y. Lou, Multiple soliton solutions of Alice–Bob Boussinesq equations. Chin. Phys. Lett. 36, 050501 (2019)

    Article  ADS  Google Scholar 

  26. L. Guo, J. He, L. Wang, Y. Cheng, D.J. Frantzeskakis, P.G. Kevrekidis, Two-dimensional rogue waves on zero background of the Davey–Stewartson II equation (2019). arXiv:1905.11541v1

  27. H.T. Wang, X.Y. Wen, D.S. Wang, Modulational instability, interactions of localized wave structures and dynamics in the modified self-steepening nonlinear Schrödinger equation. Wave Motion 91, 102396 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  28. X.S. Liu, L.C. Zhao, L. Duan, P. Gao, Z.Y. Yang, W.L. Yang, Interaction between breathers and rogue waves in a nonlinear optical fiber. Chin. Phys. Lett. 35, 020501 (2018)

    Article  ADS  Google Scholar 

  29. T. Xu, Y. Chen, Semirational solutions to the coupled Fokas—Lenells equations. Nonlinear Dyn. 95, 87–99 (2019)

    Article  MATH  Google Scholar 

  30. T. Xu, S. Lan, M. Li, L.L. Li, G.W. Zhang, Mixed soliton solutions of the defocusing nonlocal nonlinear Schrödinger equation. Physica D 390, 47–61 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. P. Liu, M. Jia, S.Y. Lou, Lax pair and exact solutions of a discrete coupled system related to coupled KdV and coupled mKdV equations. Phys. Scr. 76, 674 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. M. Toda, Theory of Nonlinear Lattices (Springer, Berlin, 1989)

    Book  MATH  Google Scholar 

  33. M. Wadati, Transformation theories for nonlinear discrete systems. Prog. Theor. Phys. Suppl. bf 59, 36–63 (1976)

    Article  ADS  Google Scholar 

  34. X.Y. Wu, B. Tian, H.M. Yin, Z. Du, Rogue-wave solutions for a discrete Ablowitz–Ladik equation with variable coefficients for an electrical lattice. Nonlinear Dyn. 93, 1635–1645 (2018)

    Article  MATH  Google Scholar 

  35. F. Yu, J. Yu, L. Li, Some discrete soliton solutions and interactions for the coupled Ablowitz–Ladik equations with branched dispersion. Wave Motion 94, 102500 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  36. L. Li, F. Yu, Optical discrete rogue wave solutions and numerical simulation for a coupled Ablowitz–Ladik equation with variable coefficients. Nonlinear Dyn. 91, 1993–2005 (2018)

    Article  Google Scholar 

  37. T. Xu, D.E. Pelinovsky, Darboux transformation and soliton solutions of the semi-discrete massive Thirring model. Phys. Lett. A 383, 125948 (2019)

    Article  MathSciNet  Google Scholar 

  38. Z. Du, B. Tian, H.P. Chai, X.H. Zhao, Lax pair, Darboux transformation, vector rational and semi-rational rogue waves for the three-component coupled Hirota equations in an optical fiber. Eur. Phys. J. Plus 134, 213 (2019)

    Article  Google Scholar 

  39. D.W. Zuo, Y.T. Gao, Y.J. Feng, L. Xue, Rogue-wave interaction for a higher-order nonlinear Schrödinger–Maxwell–Bloch system in the optical-fiber communication. Nonlinear Dyn. 78, 2309–2318 (2014)

    Article  MATH  Google Scholar 

  40. S. Chen, Y. Ye, J.M. Soto-Crespo, P. Grelu, F. Baronio, Peregrine solitons beyond the threefold limit and their two-soliton interactions. Phys. Rev. Lett. 121, 104101 (2018)

    Article  ADS  Google Scholar 

  41. S. Chen, J.M. Soto-Crespo, F. Baronio, P. Grelu, D. Mihalache, Rogue-wave bullets in a composite (2+1)D nonlinear medium. Opt. Express 24, 15251 (2016)

    Article  ADS  Google Scholar 

  42. G. Zhang, S. Chen, Z. Yan, Focusing and defocusing Hirota equations with non-zero boundary conditions: inverse scattering transforms and soliton solutions. Commun. Nonlinear Sci. Numer. Simul. 80, 104927 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  43. X. Zhang, Y. Chen, Inverse scattering transformation for generalized nonlinear Schrödinger equation. Appl. Math. Lett. 98, 306–313 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  44. A.M. Wazwaz, Negative-order integrable modified KdV equations of higher orders. Nonlinear Dyn. 93, 1371–1376 (2018)

    Article  MATH  Google Scholar 

  45. A.M. Wazwaz, L. Kaur, Complex simplified Hirota’s forms and Lie symmetry analysis for multiple real and complex soliton solutions of the modified KdV-Sine-Gordon equation. Nonlinear Dyn. 95, 2209–2215 (2019)

    Article  MATH  Google Scholar 

  46. D.Y. Liu, B. Tian, Y. Jiang, W.R. Sun, Soliton solutions and Bäcklund transformations of a (2+1)-dimensional nonlinear evolution equation via the Jaulent–Miodek hierarchy. Nonlinear Dyn. 78, 2341–2347 (2014)

    Article  MATH  Google Scholar 

  47. X.H. Zhao, B. Tian, X.Y. Xie, X.Y. Wu, Y. Sun, Y.J. Guo, Solitons, Bäcklund transformation and Lax pair for a (2+1)-dimensional Davey–Stewartson system on surface waves of finite depth. Wave Random Complex 28, 356–366 (2018)

    Article  ADS  Google Scholar 

  48. Q. Ding, S.F. Tian, On differential form method to find Lie symmetries of two types of Toda lattices. Rep. Math. Phys. 74, 323 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. S.F. Tian, Lie symmetry analysis, conservation laws and solitary wave solutions to a fourth-order nonlinear generalized Boussinesq water wave equation. Appl. Math. Lett. 100, 106056 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  50. B. Yang, Y. Chen, High-order soliton matrices for Sasa–Satsuma equation via local Riemann–Hilbert problem. Nonlinear Anal.-Real 45, 918–941 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  51. D.S. Wang, X. Wang, Long-time asymptotics and the bright \(N\)-soliton solutions of the Kundu–Eckhaus equation via the Riemann–Hilbert approach. Nonlinear Anal.-Real 41, 334–361 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  52. R. Guo, H.H. Zhao, Y. Wang, A higher-order coupled nonlinear Schrödinger system: solitons, breathers, and rogue wave solutions. Nonlinear Dyn. 83, 2475–2484 (2016)

    Article  MATH  Google Scholar 

  53. X.J. Zhao, R. Guo, H.Q. Hao, \(N\)-fold Darboux transformation and discrete soliton solutions for the discrete Hirota equation. Appl. Math. Lett. 75, 114–120 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  54. H.T. Wang, X.Y. Wen, Soliton elastic interactions and dynamical analysis of a reduced integrable nonlinear Schrödinger system on a triangular-lattice ribbon. Nonlinear Dyn. 100, 1571–1587 (2020)

    Article  Google Scholar 

  55. X.Y. Wen, Elastic interaction and conservation laws for the Nonlinear self-dual network equation in electric circuit. J. Phys. Soc. Jpn. 81, 114006 (2012)

    Article  ADS  Google Scholar 

  56. H.T. Wang, X.Y. Wen, Dynamics of multi-soliton and breather solutions for a new semi-discrete coupled system related to coupled NLS and coupled complex mKdV equations. Mod. Phys. Lett. B 32, 1850340 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  57. H.T. Wang, X.Y. Wen, Dynamics of discrete soliton propagation and elastic interaction in a higher-order coupled Ablowitz–Ladik equation. Appl. Math. Lett. 100, 106013 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  58. X.Y. Wen, Y. Yang, Z. Yan, Generalized perturbation \((n, M)\)-fold Darboux transformations and multi-rogue-wave structures for the modified self-steepening nonlinear Schrödinger equation. Phys. Rev. E 92, 012917 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  59. X.Y. Wen, Z. Yan, Modulational instability and higher-order rogue waves with parameters modulation in a coupled integrable AB system via the generalized Darboux transformation. Chaos 25, 123115 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  60. X.Y. Wen, Z. Yan, Higher-order rational solitons and rogue-like wave solutions of the (2+1)-dimensional nonlinear fluid mechanics equations. Commun. Nonlinear. Sci. Numer. Simulat. 43, 311–329 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  61. X.Y. Wen, Z. Yan, B.A. Malomed, Higher-order vector discrete rogue-wave states in the coupled Ablowitz–Ladik equations: exact solutions and stability. Chaos 26, 123110 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  62. X.Y. Wen, Z. Yan, Modulational instability and dynamics of multi-rogue wave solutions for the discrete Ablowitz–Ladik equation. J. Math. Phys. 59, 073511 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  63. X.Y. Wen, D.S. Wang, Modulational instability and higher order-rogue wave solutions for the generalized discrete Hirota equation. Wave Motion 79, 84–97 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  64. O.O. Vakhnenko, Solitons on a zigzag-runged ladder lattice. Phys. Rev. E 64, 067601 (2001)

    Article  ADS  Google Scholar 

  65. O.O. Vakhnenko, Inverse scattering transform for the nonlinear Schrödinger system on a zigzag-runged ladder lattice. J. Math. Phys. 51, 103518 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  66. O.O. Vakhnenko, Integrable nonlinear ladder system with background-controlled intersite resonant coupling. J. Phys. A Math. Gen. 39, 11013–11027 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  67. S.A. Bronsard, D.E. Pelinovsky, New integrable semi-discretizations of the coupled nonlinear Schrödinger equations (2017). arXiv:1705.05974v1

  68. L.C. Zhao, L. Ling, Quantitative relations between modulational instability and several well-known nonlinear excitations. J. Opt. Soc. Am. B 33, 850–856 (2016)

    Article  ADS  Google Scholar 

  69. O.O. Vakhnenko, Semi-discrete integrable nonlinear Schrödinger system with background-controlled inter-site resonant coupling. J. Nonlinear Math. Phys. 24, 250–302 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank the anonymous reviewer’s valuable suggestions. We would like to express our sincere thanks to other members of our discussion group for their valuable comments. This work has been partially supported by National Natural Science Foundation of China under Grant Nos. 12071042 and 61471406, and Beijing Natural Science Foundation under Grant No. 1202006.

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Authors and Affiliations

  1. School of Applied Science, Beijing Information Science and Technology University, Beijing, 100192, China

    Hao-Tian Wang & Xiao-Yong Wen

  2. State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, P. O. Box 122, Beijing, 100876, China

    Hao-Tian Wang

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Wang, HT., Wen, XY. Modulational instability, interactions of two-component localized waves and dynamics in a semi-discrete nonlinear integrable system on a reduced two-chain lattice. Eur. Phys. J. Plus 136, 461 (2021). https://doi.org/10.1140/epjp/s13360-021-01454-4

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