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On the quintic time-dependent coefficient derivative nonlinear Schrödinger equation in hydrodynamics or fiber optics

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Abstract

Under investigation in this paper is a quintic time-dependent coefficient derivative nonlinear Schrödinger equation for certain hydrodynamic wave packets or a medium with the negative refractive index. A gauge transformation is found to obtain the equivalent form of the equation. With respect to the wave envelope for the free water surface displacement or envelope of the electric field, Painlevé integrable condition, different from that in the existing literature, is derived, with which the bilinear forms and N-soliton solutions are constructed. Asymptotic analysis illustrates that the interactions between the bright and bound solitons as well as between the bright solitons and Kuznetsov–Ma breathers are elastic with certain conditions, while some other interactions are inelastic under other conditions. Propagation paths and velocities for the solitons are both affected by the dispersion coefficient function when the relations among the coefficients are linear, or affected by the dispersion coefficient, self-steepening coefficient and cubic nonlinearity functions when the relations among the coefficients are nonlinear. Under different conditions, bell-shaped solitons can evolve into the bound solitons or Kuznetsov–Ma breathers, respectively. Interactions between the bright and parabolic (or hyperbolic) solitons are related to the dispersion coefficient, self-steepening coefficient and cubic nonlinearity functions. Compression effect on the propagation paths of the solitons, caused by the dispersion coefficient, is observed.

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Notes

  1. By the way, Eq. (3) has been demonstrated to belong to the Kaup–Newell hierarchy [59], and when \(\lambda (t)=\alpha (t)=1\) and \(\mu ( t )=\nu ( t ) =0\), Eq. (3) has been reduced to the Kaup–Newell equation describing the circular polarized nonlinear Alfvén waves in plasmas [60].

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Acknowledgements

We express our sincere thanks to the each member of our discussion group for their valuable suggestions. This work has been supported by the National Natural Science Foundation of China under Grant No. 11772017 and by the Fundamental Research Funds for the Central Universities under Grant No. 50100002016105010.

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  1. Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China

    Ting-Ting Jia, Yi-Tian Gao, Yu-Jie Feng, Lei Hu, Jing-Jing Su, Liu-Qing Li & Cui-Cui Ding

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  1. Ting-Ting Jia
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Jia, TT., Gao, YT., Feng, YJ. et al. On the quintic time-dependent coefficient derivative nonlinear Schrödinger equation in hydrodynamics or fiber optics. Nonlinear Dyn 96, 229–241 (2019). https://doi.org/10.1007/s11071-019-04786-0

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