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On the Darboux transformation of a generalized inhomogeneous higher-order nonlinear Schrödinger equation

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Abstract

Recently, a paper about the Nth-order rogue waves for an inhomogeneous higher-order nonlinear Schrödinger equation using the generalized Darboux transformation is published. Song et al. (Nonlinear Dyn 82(1):489–500. doi:10.1007/s11071-015-2170-6, 2015). However, the inhomogeneous equation which admits a nonisospectral linear eigenvalue problem is mistaken for having a constant spectral parameter by the authors. This basic error causes the results to be wrong, especially regarding the Darboux transformation (DT) in Sect. 2 when the inhomogeneous terms are dependent of spatial variable x. In fact, the DT for inhomogeneous equation has an essential difference from the isospectral case, and their results are correct only in the absence of inhomogeneity which was already discussed in detail before. Consequently, we firstly modify the DT based on corresponding nonisospectral linear eigenvalue problem. Then, the nonautonomous solitons are obtained from zero seed solutions. Properties of these solutions in the inhomogeneous media are discussed graphically to illustrate the influences of the variable coefficients. Finally, the failure of finding breather and rogue wave solutions from this modified DT is also discussed.

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Acknowledgments

This work is supported by the 13th Five-Year National Key Research and Development Program of China with Grant No. 2016YFC0401401, the NSF of China with Grant Nos.71271083, 11301179, the SSF of Bei**g with Grant No.15ZDA19, the Co-construction Project and Young Talents Plan of Bei**g Municipal Commission of Education. The authors also acknowledge the support by the Fundamental Research Funds of the Central Universities with the Grant Nos.2014ZZD08, 2014ZZD10, 2015MS56, 2016MS63. Xuelin Yong is partially supported by the State Scholarship Fund of China.

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Authors and Affiliations

  1. School of Mathematical Sciences and Physics, North China Electric Power University, Bei**g, 102206, China

    Xuelin Yong, Guo Wang, Wei Li & Yehui Huang

  2. School of Economics and Managements, North China Electric Power University, Bei**g, 102206, China

    Jianwei Gao

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Correspondence to Jianwei Gao.

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Yong, X., Wang, G., Li, W. et al. On the Darboux transformation of a generalized inhomogeneous higher-order nonlinear Schrödinger equation. Nonlinear Dyn 87, 75–82 (2017). https://doi.org/10.1007/s11071-016-3026-4

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