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Evolution of initial discontinuity for the defocusing complex modified KdV equation

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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.

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References

  1. Whitham, G.B.: Nonlinear dispersive waves. Proc. R. Soc. Lond. Ser. A 283, 238 (1965)

    Article  MathSciNet  Google Scholar 

  2. Gurevich, A.V., Pitaevskii, L.P.: Nonstationary structure of a collisionless shock wave. Sov. Phys. JETP 2, 291 (1974)

    Google Scholar 

  3. Gurevich, A.V., Krylov, A.L., EL, G.A.: Evolution of a Riemann wave in dispersive hydrodynamics. Sov. Phys. JETP 74, 957 (1992)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  6. Tian, F.R.: Oscillations of the zero dispersion limit of the Korteweg-de Vries equation. Commmun. Pure Appl. Math. 46, 1093 (1993)

    Article  MathSciNet  Google Scholar 

  7. Tian, F.R.: The Whitham-type equations and linear overdetermined systems of Euler–Poisson–Darboux type. Duke Math. J. 74, 203 (1994)

    Article  MathSciNet  Google Scholar 

  8. Ma, W.X., Zhou, Y.: Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J. Differ. Equ. 264(4), 2633 (2018)

    Article  MathSciNet  Google Scholar 

  9. Zhang, J.B., Ma, W.X.: Mixed lump-kink solutions to the BKP equation. Comput. Math. Appl. 74, 591 (2017)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  12. Manukure, S., Zhou, Y., Ma, W.X.: Lump solutions to a (\(2+1\))-dimensional extended KP equation. Comput. Math. Appl. 75, 2414 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Xu, X.X., Sun, Y.P.: Two symmetry constraints for a generalized Dirac integrable hierarchy. J. Math. Analy. Appl. 458, 1073 (2018)

    Article  MathSciNet  Google Scholar 

  15. Zhao, H.Q., Ma, W.X.: Mixed lump–kink solutions to the KP equation. Comput. Math. Appl. 74, 1399 (2017)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  17. Wang, D.S., Liu, J.: Integrability aspects of some two-component KdV systems. Appl. Math. Lett. 79, 211 (2018)

    Article  MathSciNet  Google Scholar 

  18. Lax, P., Levermorem, C.: The small dispersion limit of the Korteweg-de Vries equation. Commun. Pure Appl. Math. 36, 253 (1983)

    Article  MathSciNet  Google Scholar 

  19. Buckingham, R., Venakides, S.: Long-time asymptotics of the nonlinear Schrödinger equation shock problem. Commun. Pure Appl. Math. 60, 1349 (2007)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  22. Zhang, X.E., Chen, Y.: Inverse scattering transformation for generalized nonlinear Schrödinger equation. Appl. Math. Lett. 98, 306 (2019)

    Article  MathSciNet  Google Scholar 

  23. Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. 137, 295 (1993)

    Article  MathSciNet  Google Scholar 

  24. Jenkins, R.: Regularization of a sharp shock by the defocusing nonlinear Schrödinger equation. Nonlinearity 28, 2131 (2015)

    Article  MathSciNet  Google Scholar 

  25. Ivanov, S.K., Kamchatnov, A.M.: Riemann problem for the photon fluid: self-steepening effects. Phys. Rev. A 96, 053844 (2017)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  30. Ablowitz, M.J., Biondini, G., Wang, Q.: Whitham modulation theory for the Kadomtsev–Petviashvili equation. Proc. R. Soc. A 473, 20160695 (2017)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  32. Ablowitz, M.J., Biondini, G., Wang, Q.: Whitham modulation theory for the two-dimensional Benjamin–Ono equation. Phy. Rev. E 96, 032225 (2017)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  36. Kamchatnov, A.M.: New approach to periodic solutions of integrable equations and nonlinear theory of modulational instability. Phys. Rep. 286, 199 (1997)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  38. Engquist, B., Lötstedt, P., Sjögreen, B.: Nonlinear filters for efficient shock computation. Math. Comput. 52, 509 (1989)

    Article  MathSciNet  Google Scholar 

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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.

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Authors and Affiliations

  1. Department of Mathematics and Physics, North China Electric Power University, Beijing, 102206, China

    Liang-Qian Kong & Lei Wang

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

    Deng-Shan Wang, Xiao-Yong Wen & Ling Xu

  3. School of Sciences, Zhejiang Agriculture and Forestry University, Hangzhou, 311300, Zhejiang, China

    Chao-Qing Dai

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  1. Liang-Qian Kong
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Correspondence to Deng-Shan Wang.

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Kong, LQ., Wang, L., Wang, DS. et al. Evolution of initial discontinuity for the defocusing complex modified KdV equation. Nonlinear Dyn 98, 691–702 (2019). https://doi.org/10.1007/s11071-019-05222-z

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