Advertisement

Nonlinear Dynamics

, Volume 98, Issue 1, pp 691–702 | Cite as

Evolution of initial discontinuity for the defocusing complex modified KdV equation

  • Liang-Qian Kong
  • Lei Wang
  • Deng-Shan WangEmail author
  • Chao-Qing Dai
  • Xiao-Yong Wen
  • Ling Xu
Original paper

Abstract

The complete classification of solutions to the defocusing complex modified Korteweg-de Vries (cmKdV) equation with the step-like initial condition is given by Whitham theory. The process of studying the solution of cmKdV equation can be reduced to explore four quasi-linear equations, which predicts the evolution of dispersive shock wave. The results obtained here are quite different from the defocusing nonlinear Schrödinger equation: the bidirectionality of defocusing nonlinear Schrödinger equation determines that there are two basic rarefaction and shock structures while in the cmKdV case three basic rarefaction structures and four basic dispersive shock structures are constructed which lead to more complicated classification of step-like initial condition, and wave patterns even consisted of six different regions while each of wave patterns is consisted of five regions in the defocusing nonlinear Schrödinger equation. Direct numerical simulations of cmKdV equation are agreed well with the solutions corresponding to Whitham theory.

Keywords

Whitham equations Dispersive shock wave Complex mKdV equation 

Notes

Acknowledgements

This work is supported by National Natural Science Foundation of China under Grant Nos. 11875126 and 11971067, Beijing Natural Science Foundation under Grant No. 1182009, the Beijing Great Wall Talents Cultivation Program under Grant No. CIT&TCD20180325 and Qin Xin Talents Cultivation Program (Nos. QXTCP A201702 and QXTCP B201704) of Beijing Information Science and Technology University.

Compliance with ethical standards

Conflict of Interest

We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the paper submitted.

References

  1. 1.
    Whitham, G.B.: Nonlinear dispersive waves. Proc. R. Soc. Lond. Ser. A 283, 238 (1965)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Gurevich, A.V., Pitaevskii, L.P.: Nonstationary structure of a collisionless shock wave. Sov. Phys. JETP 2, 291 (1974)Google Scholar
  3. 3.
    Gurevich, A.V., Krylov, A.L., EL, G.A.: Evolution of a Riemann wave in dispersive hydrodynamics. Sov. Phys. JETP 74, 957 (1992)MathSciNetGoogle Scholar
  4. 4.
    Wright, O.C.: Korteweg-de Vries zero dispersion limit: through first breaking for cubic-like analytic initial data. Commmun. Pure Appl. Math. 46, 423 (1993)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Tian, F.R., Ye, J.: On the Whitham equations for the semiclassical limit of the defocusing nonlinear Schrödinger equation. Commmun. Pure Appl. Math. 52, 655 (1999)CrossRefGoogle Scholar
  6. 6.
    Tian, F.R.: Oscillations of the zero dispersion limit of the Korteweg-de Vries equation. Commmun. Pure Appl. Math. 46, 1093 (1993)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Tian, F.R.: The Whitham-type equations and linear overdetermined systems of Euler–Poisson–Darboux type. Duke Math. J. 74, 203 (1994)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ma, W.X., Zhou, Y.: Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J. Differ. Equ. 264(4), 2633 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Zhang, J.B., Ma, W.X.: Mixed lump-kink solutions to the BKP equation. Comput. Math. Appl. 74, 591 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    McAnally, M., Ma, W.X.: An integrable generalization of the D-Kaup–Newell soliton hierarchy and its bi-Hamiltonian reduced hierarchy. Appl. Math. Comput. 323, 220 (2018)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Dong, H.H., Zhao, K., Yang, H.Q., Li, Y.Q.: Generalised (\(2+1\))-dimensional super MKdV hierarchy for integrable systems in soliton theory. East Asian J. Appl. Math. 5, 256 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Manukure, S., Zhou, Y., Ma, W.X.: Lump solutions to a (\(2+1\))-dimensional extended KP equation. Comput. Math. Appl. 75, 2414 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ma, W.X.: Abundant lumps and their interaction solutions of (\(3+1\))-dimensional linear PDEs. J. Geom. Phys. 133, 10 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Xu, X.X., Sun, Y.P.: Two symmetry constraints for a generalized Dirac integrable hierarchy. J. Math. Analy. Appl. 458, 1073 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Zhao, H.Q., Ma, W.X.: Mixed lump–kink solutions to the KP equation. Comput. Math. Appl. 74, 1399 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ma, W.X., Yong, X.L., Zhang, H.Q.: Diversity of interaction solutions to the (\(2+1\))-dimensional Ito equation. Comput. Math. Appl. 75, 289 (2018)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Wang, D.S., Liu, J.: Integrability aspects of some two-component KdV systems. Appl. Math. Lett. 79, 211 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lax, P., Levermorem, C.: The small dispersion limit of the Korteweg-de Vries equation. Commun. Pure Appl. Math. 36, 253 (1983)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Buckingham, R., Venakides, S.: Long-time asymptotics of the nonlinear Schrödinger equation shock problem. Commun. Pure Appl. Math. 60, 1349 (2007)CrossRefGoogle Scholar
  20. 20.
    Wang, D.S., Wang, X.L.: Long-time asymptotics and the bright N-soliton solutions of the Kundu–Eckhaus equation via the Riemann–Hilbert approach. Nonlinear Anal. Real World Appl. 41, 334 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wang, D.S., Guo, B.L., Wang, X.L.: Long-time asymptotics of the focusing Kundu–Eckhaus equation with nonzero boundary conditions. J. Differ. Equ. 266, 5209 (2019)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Zhang, X.E., Chen, Y.: Inverse scattering transformation for generalized nonlinear Schrödinger equation. Appl. Math. Lett. 98, 306 (2019)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. 137, 295 (1993)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Jenkins, R.: Regularization of a sharp shock by the defocusing nonlinear Schrödinger equation. Nonlinearity 28, 2131 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ivanov, S.K., Kamchatnov, A.M.: Riemann problem for the photon fluid: self-steepening effects. Phys. Rev. A 96, 053844 (2017)CrossRefGoogle Scholar
  26. 26.
    Ivanov, S.K., Kamchatnov, A.M., Congy, T., Pavloff, N.: Solution of the Riemann problem for polarization waves in a two-component Bose–Einstein condensate. Phys. Rev. E 96, 062202 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kamchatnov, A.M., Kuo, Y.H., Lin, T.C., Horng, T.L., Gou, S.C., Clift, R., El, G.A., Grimshaw, R.H.: Undular bore theory for the Gardner equation. Phys. Rev. E 86, 036605 (2012)CrossRefGoogle Scholar
  28. 28.
    Kodama, Y., Pierce, V.U., Tian, F.R.: On the Whitham equations for the defocusing complex modified KdV equation. SIAM J. Math. Anal. 41, 26 (2008)MathSciNetGoogle Scholar
  29. 29.
    El, G.A., Nguyen, L.T.K., Smyth, N.: Dispersive shock waves in systems with nonlocal dispersion of Benjamin–Ono type. Nonlinearity 31, 1392 (2018)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Ablowitz, M.J., Biondini, G., Wang, Q.: Whitham modulation theory for the Kadomtsev–Petviashvili equation. Proc. R. Soc. A 473, 20160695 (2017)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Ablowitz, M.J., Biondini, G., Rumanov, I.: Whitham modulation theory for (\(2+1\))-dimensional equations of Kadomtsev–Petviashvili type. J. Phys. A 51, 215501 (2018)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Ablowitz, M.J., Biondini, G., Wang, Q.: Whitham modulation theory for the two-dimensional Benjamin–Ono equation. Phy. Rev. E 96, 032225 (2017)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Grava, T., Klein, C.: Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations. Commun. Pure Appl. Math. 60, 1623 (2007)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Ablowitz, M.J., Demirci, A., Ma, Y.P.: Dispersive shock waves in the Kadomtsev–Petviashvili and two dimensional Benjamin–Ono equations. Physica D 333, 84 (2016)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Pierce, V.U., Tian, F.R.: Self-similar solutions of the non-strictly hyperbolic Whitham equations for the KdV hierarchy. Dyn. Partial Differ. Equ. 4, 263 (2007)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Kamchatnov, A.M.: New approach to periodic solutions of integrable equations and nonlinear theory of modulational instability. Phys. Rep. 286, 199 (1997)MathSciNetCrossRefGoogle Scholar
  37. 37.
    El, G.A., Geogjaev, V.V., Gurevich, A.V., Krylov, A.L.: Decay of an initial discontinuity in the defocusing NLS hydrodynamics. Physica D 86, 186 (1995)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Engquist, B., Lötstedt, P., Sjögreen, B.: Nonlinear filters for efficient shock computation. Math. Comput. 52, 509 (1989)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Liang-Qian Kong
    • 1
  • Lei Wang
    • 1
  • Deng-Shan Wang
    • 2
    Email author
  • Chao-Qing Dai
    • 3
  • Xiao-Yong Wen
    • 2
  • Ling Xu
    • 2
  1. 1.Department of Mathematics and PhysicsNorth China Electric Power UniversityBeijingChina
  2. 2.School of ScienceBeijing Information Science and Technology UniversityBeijingChina
  3. 3.School of SciencesZhejiang Agriculture and Forestry UniversityHangzhouChina

Personalised recommendations