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Dynamic behavior and stability analysis of nonlinear modes in the fourth-order generalized Ginzburg–Landau model with near \(\mathcal{PT}\)-symmetric potentials

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Abstract

We investigate the fourth-order generalized Ginzburg–Landau equation and the nonlinear modes modulated by \(\mathcal{PT}\)-symmetric potentials. By means of Hirota method, we obtained the bilinear form of the equation and further derived the analytic soliton solution. Dynamic behaviors of the solitons under the modulation of near \(\mathcal{PT}\)-symmetric potentials were studied by numerical simulation: The nonlinear modes tend to be unstable when the potential is closer to conventional \(\mathcal{PT}\)-symmetric potential, and the amplitude of the nonlinear modes oscillates periodically when the imaginary part of the \(\mathcal{PT}\)-symmetric potentials is sufficiently large. Moreover, we obtained new nonlinear modes that are different from the above analytic soliton solutions by numerical excitation and tested their stability. These new findings of nonlinear modes in the generalized Ginzburg–Landau model can be potentially applied to hydrodynamics, optics and matter waves in Bose–Einstein condensates.

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Acknowledgements

We express our sincere thanks to the editor, referees and all the members of our discussion group for their valuable comments. This work was supported by the National Training Program of Innovation (Grant numbers 202210019045). The funding body plays an important role in the design of the study, in analysis, calculation, and in writing of the manuscript.

Funding

This work was supported by the National Training Program of Innovation (Grant numbers 202210019045). The funding body plays an important role in the design of the study, in analysis, calculation, and in writing of the manuscript.

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  1. College of Science, China Agricultural University, Beijing, 100083, China

    Jia-Rui Zhang, Zhao-Lin Zheng, Da Lin & Yu-Jia Shen

  2. College of Engineering, China Agricultural University, Beijing, 100083, China

    Jia-Qi Zhang

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  1. Jia-Rui Zhang
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Zhang, JR., Zhang, JQ., Zheng, ZL. et al. Dynamic behavior and stability analysis of nonlinear modes in the fourth-order generalized Ginzburg–Landau model with near \(\mathcal{PT}\)-symmetric potentials. Nonlinear Dyn 109, 1005–1017 (2022). https://doi.org/10.1007/s11071-022-07441-3

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