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International Journal of Theoretical Physics

, Volume 58, Issue 1, pp 92–102 | Cite as

Characteristics of the Lumps and Stripe Solitons with Interaction Phenomena in the (2 + 1)-Dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada Equation

  • Zhi-Hao Deng
  • Xia Chang
  • Jia-Ning Tan
  • Bing TangEmail author
  • Ke Deng
Article

Abstract

So far, the interaction between the lump waves and solitons has received much attention from many fields because of its significance to represent new physical phenomena occurring in various branches of physics. In this work, we study the interaction phenomenon between the lump waves and stripe solitons in the (2 + 1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation by making use of the Hirota bilinear method. Adopting the positive quadratic function solutions of the corresponding bilinear equation, a class of lump wave solutions are analytically constructed. What is more, we obtain the lump-single stripe soliton interaction solutions, and show that the one stripe soliton can split into a lump and a stripe soliton. In addition, we provide the interaction solutions between one lump and twin resonance stripe solitons, and present the law of the interaction between a lump and twin resonance stripe solitons by the related three-dimensional plots.

Keywords

The Caudrey–Dodd–gibbon–Sawada-Kotera equation Hirota’s bilinear form Lump waves Interaction phenomena Resonance solitons 

Notes

Acknowledgments

This study was supported by the National Natural Science Foundation of China (Grant Nos. 11604121, 11875126 and 11464012), the Natural Science Fund Project of Hunan Province (Grant No. 2017JJ3255), and the Natural Science Fund Project of Jishou University (Grant No. Jdy17032). We would like to thanks Professor Wen-Xiu Ma and Professor Sen-Yue Lou for useful suggestions on this work.

References

  1. 1.
    Leblond, H., Manna, M.: Single-oscillation two-dimensional solitons of magnetic polaritons. Phys. Rev. Lett. 99, 064102 (2007)ADSCrossRefGoogle Scholar
  2. 2.
    Leblond, H., Kremer, D., Mihalache, D.: Ultrashort spatiotemporal optical solitons in quadratic nonlinear media: generation of line and lump solitons from few-cycle input pulses. Phys. Rev. A. 80, 053812 (2009)ADSCrossRefGoogle Scholar
  3. 3.
    Yang, C., Li, W., Yu, W., Liu, M., Zhang, Y., Ma, G., Lei, M., Liu, W.: Amplification, reshaping, fission and annihilation of optical solitons in dispersion-decreasing fiber. Nonlinear Dynam. 92, 203–213 (2018)CrossRefzbMATHGoogle Scholar
  4. 4.
    Lin, F.H., Chen, S.T., Qu, Q.X., Wang, J.P., Zhou, X.W., Lü, X.: Resonant multiple wave solutions to a new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation: linear superposition principle. Appl. Math. Lett. 78, 112–117 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Liu, W., Yang, C., Liu, M., Yu, W., Zhang, Y., Lei, M.: Effect of high-order dispersion on three-soliton interactions for the variable-coefficients Hirota equation. Phys. Rev. E. 96, 042201 (2017)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Lü, X., Chen, S.T., Ma, W.X.: Constructing lump solutions to a generalized Kadomtsev-Petviashvili-Boussinesq equation. Nonlinear Dynam. 86, 523–534 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Sun, W.R.: Breather-to-soliton transitions and nonlinear wave interactions for the nonlinear Schrödinger equation with the sextic operators in optical fibers. Ann. Phys. 529, 1600227 (2017)CrossRefzbMATHGoogle Scholar
  8. 8.
    Li, W.Y., Ma, G.L., Yu, W.T., Zhang, Y.J., Liu, M.L., Yang, C.Y., Liu, W.J.: Soliton structures in the (1+1)-dimensional Ginzburg-Landau equation with a parity-time-symmetric potential in ultrafast optics. Chinese Phys. B. 27, 030504 (2018)ADSCrossRefGoogle Scholar
  9. 9.
    Lü, X., Lin, F.: Soliton excitations and shape-changing collisions in alpha helical proteins with interspine coupling at higher order. Commun. Nonlinear Sci. Numer. Simul. 32, 241–261 (2016)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Cai, L.Y., Wang, X., Wang, L., Li, M., Liu, Y., Shi, Y.Y.: Nonautonomous multi-peak solitons and modulation instability for a variable-coefficient nonlinear Schrödinger equation with higher-order effects. Nonlinear Dynam. 90, 2221–2230 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gao, L.N., Zi, Y.Y., Yin, Y.H., Ma, W.X., Lü, X.: Bäcklund transformation, multiple wave solutions and lump solutions to a (3+1)-dimensional nonlinear evolution equation. Nonlinear Dynam. 89, 2233–2240 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mertens, F.G., Quintero, N.R., Cooper, F., Khare, A., Saxena, A.: Nonlinear Dirac equation solitary waves in external fields. Phys. Rev. E. 86, 046602 (2012)ADSCrossRefGoogle Scholar
  13. 13.
    Liu, M.L., Liu, W.J., Pang, L.H., Teng, H., Fang, S.B., Wei, Z.Y.: Ultrashort pulse generation in model-locked erbium-doped fiber lasers with tungsten disulfide saturable absorber. Opt. Commun. 406, 72–75 (2018)ADSCrossRefGoogle Scholar
  14. 14.
    Tomizawa, S., Mishima, T.: New cylindrical gravitational soliton waves and gravitational Faraday rotation. Phys. Rev. D. 90, 044036 (2014)ADSCrossRefGoogle Scholar
  15. 15.
    Lü, X., Ma, W.X.: Soliton structures in the (1+1)-dimensional Ginzburg-Landau equation with a parity-time-symmetric potential in ultrafast optics. Nonlinear Dynam. 85, 1217–1222 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hoefer, M.A., Sommacal, M., Silva, T.J.: Propagation and control of nanoscale magnetic-droplet solitons. Phys. Rev. B. 85, 214433 (2012)ADSCrossRefGoogle Scholar
  17. 17.
    Solli, D.R., Ropers, C., Koonath, P., Jalali, B.: Optical rogue waves. Nature. 450, 1054–1057 (2007)ADSCrossRefGoogle Scholar
  18. 18.
    Zhang, J.H., Wang, L., Liu, C.: Superregular breathers, characteristics of nonlinear stage of modulation instability induced by higher-order effects. Proc. R. Soc. A. 473, 20160681 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Wang, L., Zhu, Y.J., Wang, Z.Q., Xu, T., Qi, F.H., Xue, Y.S.: Asymmetric rogue waves, breather-to-soliton conversion, and nonlinear wave interactions in the Hirota-Maxwell-Bloch system. J. Phys. Soc. Jpn. 85, 024001 (2016)ADSCrossRefGoogle Scholar
  20. 20.
    Liu, W.J., Liu, M.L., Han, H.N., Fang, S.B., Teng, H., Lei, M., Wei, Z.Y.: Nonlinear optical properties of WSe2 and MoSe2 films and their applications in passively Q-switched erbium doped fiber lasers [invited]. Photonics. Res. 6, C15–C21 (2018)CrossRefGoogle Scholar
  21. 21.
    Wang, L., Zhu, Y.J., Wang, Z.Z., Qi, F.H., Guo, R.: Higher-order semirational solutions and nonlinear wave interactions for a derivative nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul. 33, 218–228 (2016)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Wang, L., Zhu, Y.J., Qi, F.H., Li, M., Guo, R.: Modulational instability, higher-order localized wave structures, and nonlinear wave interactions for a nonautonomous Lenells-Fokas equation in inhomogeneous fibers. Chaos. 25, 063111 (2015)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Wang, L., Li, X., Qi, F.H., Zhang, L.L.: Breather interactions and higher-order nonautonomous rogue waves for the inhomogeneous nonlinear Schrödinger Maxwell-Bloch equations. Ann. Phys. 359, 97 (2015)CrossRefzbMATHGoogle Scholar
  24. 24.
    Wang, L., Wu, X., Zhang, H.Y.: Superregular breathers and state transitions in a resonant erbium-doped fiber system with higher-order effects. Phys. Lett. A. 382, 2650–2654 (2018)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Li, P., Wang, L., Kong, L.Q., Wang, X., Xie, Z.Y.: Nonlinear waves in the modulation instability regime for the ffth-order nonlinear Schrödinger equation. Appl. Math. Lett. 85, 110–117 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Liu, W.J., Liu, M.L., Ou Yang, Y.Y., Hou, H.R., Ma, G.L., Lei, M., Wei, Z.Y.: Tungsten diselenide for mode-locked erbium-doped fiber lasers with short pulse duration. Nanotechnology. 29, 174002 (2018)ADSCrossRefGoogle Scholar
  27. 27.
    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)ADSCrossRefGoogle Scholar
  28. 28.
    Liu, W.J., Zhu, Y.N., Liu, M.L., Wen, B., Fang, S.B., Teng, H., Lei, M., Liu, L.M., Wei, Z.Y.: Optical properties and applications for MoS2-Sb2Te3-MoS2 heterostructure materials. Photonics. Res. 6, 220–227 (2018)CrossRefGoogle Scholar
  29. 29.
    Sun, W.R., Wang, L.: Matter rogue waves for the three-component Gross-Pitaevskii equations in the spinor Bose-Einstein condensates. Proc. R. Soc. A. 474, 20170276 (2018)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wang, L., Liu, C., Wu, X., Wang, X., Sun, W.R.: Dynamic of superregular breathers in the quintic nonlinear Schrödinger equation. Nonlinear Dynam. 94, 977–989 (2018).  https://doi.org/10.1007/s11071-018-4404-x
  31. 31.
    Wang, L., Jiang, D.Y., Qi, F.H., Shi, Y.Y., Zhao, Y.C.: Dynamics of the higher-order rogue waves for a generalized mixed nonlinear Schrödinger model. Commun. Nonlinear Sci. Numer. Simul. 42, 502 (2017)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Wang, L., Wang, Z.Q., Sun, W.R., Shi, Y.Y., Li, M., Xu, M.: Dynamics of Peregrine combs and Peregrine walls in an inhomogeneous Hirota and Maxwell-Bloch system. Commun. Nonlinear Sci. Numer. Simul. 47, 190 (2017)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: Classifying the hierarchy of nonlinear-Schrödinger-equation rogue-wave solutions. Phys. Rev. E. 88, 013207 (2013)ADSCrossRefGoogle Scholar
  34. 34.
    He, J., Wang, L., Li, L., Porsezian, K., Erdélyi, R.: Few-cycle optical rogue waves: Complex modified Korteweg-de Vries equation. Phys. Rev. E. 89, 062917 (2014)ADSCrossRefGoogle Scholar
  35. 35.
    Wang, L.H., Porsezian, K., He, J.S.: Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation. Phys. Rev. E. 87, 053202 (2013)ADSCrossRefGoogle Scholar
  36. 36.
    Sun, W.R., Liu, D.Y., Xie, X.Y.: Vector semirational rogue waves and modulation instability for the coupled higher-order nonlinear Schrödinger equations in the birefringent optical fibers. Chaos. 27, 043114 (2017)ADSCrossRefzbMATHGoogle Scholar
  37. 37.
    Manakov, S.V., Zakharov, V.E., Bordag, L.A., Its, A.R., Matveev, V.B.: Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction. Phys. Lett. A. 63, 205–206 (1977)ADSCrossRefGoogle Scholar
  38. 38.
    Estévez, P.G., Díaz, E., Domínguez-Adame, F., Cerveró, J.M., Diez, E.: Lump solitons in a higher-order nonlinear equation in 2 + 1 dimensions. Phys. Rev. E. 93, 062219 (2016)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Lou, S.-Y.: On the coherent structures of the Nizhnik-Novikov-Veselov equation. Phys. Lett. A. 277, 94–100 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ma, W.-X.: Lump solutions to the Kadomtsev-Petviashvili equation. Phys. Lett. A. 379, 1975–1978 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Satsuma, J., Ablowitz, M.J.: Two-dimensional lumps in nonlinear dispersive systems. J. Math. Phys. 20, 1496–1503 (1979)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Ma, W.X., Qin, Z.Y., Xing, L.: Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nonlinear Dynam. 84, 923–931 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Huang, L.-L., Chen, Y.: Lump solutions and interaction phenomenon for (2+1) -dimensional Sawada-Kotera equation. Commun. Theor. Phys. 67, 473–478 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Wang, H.: Lump and interaction solutions to the (2 + 1)-dimensional burgers equation. Appl. Math. Lett. 85, 27–34 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Konopelchenko, B.G., Dubrovsky, V.G.: Some new Integrable nonlinear evolution equations in (2+1)-dimensions. Phys. Lett. A. 102, 15–17 (1984)ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    Cao, C.W., Wu, Y.T., Geng, X.G.: On quasi-periodic solutions of the 2+1 dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation. Phys. Lett. A. 256, 59–65 (1999)ADSCrossRefGoogle Scholar
  47. 47.
    Wang, L., Xian, D.: Homoclinic breather-wave solutions,periodic-wave solutions and kink solitary-wave solutions for CDGKS equations. Chin. J. Quantum Elect. 29, 417–420 (2012)Google Scholar
  48. 48.
    Meng, X.H.: The periodic solitary wave solutions for the (2+1)-dimensional fifth-order KdV equation. J. Appl. Math. Phys. 2, 639–643 (2014)CrossRefGoogle Scholar
  49. 49.
    Gao, L.-N., Zhao, X.-Y., Zi, Y.-Y., Yu, J., Lü, X.: Resonant behavior of multiple wave solutions to a Hirota bilinear equation. Computers and Mathematics with Applications. 72, 1225–1229 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Yang, J.-Y., Ma, W.-X.: Lump solutions to the BKP equation by symbolic computation. Int. J. Mod. Phys. B. 30, 1640028 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Hossen, M.B., Roshida, H.-O., Ali, M.Z.: Characteristics of the solitary waves and rogue waves with interaction phenomena in a (2 +1)-dimensional breaking soliton equation. Phys. Lett. A. 352, 1268–1274 (2018)ADSMathSciNetCrossRefGoogle Scholar
  52. 52.
    Yang, H.W., Chen, X., Guo, M., Chen, Y.D.: A new ZK–BO equation for three-dimensional algebraic Rossby solitary waves and its solution as well as fission property. Nonlinear Dynam. 91, 2019–2032 (2018)CrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Physics, Mechanical and Electrical EngineeringJishou UniversityJishouChina

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