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Exotic rogue waves in an extended nonlocal nonlinear Schrödinger equation with self-induced PT-symmetric potentials

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Abstract

We consider the spectral problem and adjoint problem for an extended nonlocal nonlinear Schrödinger (NLS) equation. Utilizing the Schur polynomial theory and the Darboux transformation, we derive the high-order rogue wave solutions by providing three types of wave functions and adjoint ones. Our results show that the nonsingular and singular solutions, respectively, correspond to single peak rogue wave and collapsing rogue wave, which are two different rogue wave phenomena. The solutions obtained in this work exhibit rich rogue wave patterns, most of which have no counterparts in the extended local NLS equation.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant (No.11971133, No.12101159). We are grateful to the reviewers for their encouraging suggestions that were helpful in improving this paper further.

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  1. School of Mathematics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Xue-Wei Yan & Yong Chen

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Appendix

Appendix

$$\begin{aligned} D_{11}^{[1]}&=-16z^4t^2+8z^3g_{0}t-8z^3l_{0}t+4z^2g_{0}l_{0}-4{i}xz^2g_{0}-4{i}xz^2l_{0}+16{i}z^2t\\ &\quad -4{i}zg_{0}-4x^2z^2+4{i}zl_{0}+3,\\ D_{12}^{[1]}&=16t^2z^4+4{i}xz^2g_{0}+4{i}xz^2l_{0}-8tz^3g_{0}+8tz^3l_{0}+4x^2z^2-4z^2g_{0}l_{0}+1,\\ D_{21}^{[2]}&=48{i}t^2z^5l_{0}+4{i}z^3l_{0}^3+24{i}tz^4l_{0}^2-24{i}tx^2z^4-6{i}tz^2+48t^2xz^5-12{i}x^2z^3l_{0}\\&\quad -48{i}tz^3x+48tz^4l_{0}x+32{i}t^3z^6+48t^2z^4-4x^3z^3+12xz^3l_{0}^2+48tz^3l_{0}\\&\quad +12z^2l_{0}^2-12x^2z^2-24{i}z^2l_{0}x+6{i}zl_{1}-9{i}zl_{0}-9xz-3,\\ D_{22}^{[2]}&=32{i}t^3z^5+48{i}t^2z^4l_{0}-24{i}tx^2z^3+24{i}tz^3l_{0}^2+3{i}l_{0}+48t^2xz^4-12{i}x^2z^2l_{0}\\&\quad +4{i}z^2l_{0}^3+48tz^3xl_{0}+12xz^2l_{0}^2-4x^3z^2+18{i}zt+6{i}l_{1}+3x,\\ D_{31}^{[2]}&=-48{i}t^2xz^5g_{0}+48{i}t^2xz^5l_{0}-12{i}xz^3g_{0}l_{0}^2+48t^2z^3-64t^4z^7-48t^2z^5l_{0}^2\\&\quad -48{i}txz^4g_{0}l_{0}+32t^3z^6g_{0}-96t^3z^6l_{0}-8tz^4l_{0}^3-12x^2z^3l_{0}^2-12tz^2l_{1}\\&\quad +4z^3g_{0}l_{0}^3+6zg_{0}l_{1}-48{i}txz^2-12{i}z^2g_{0}l_{0}^2+6{i}l_{1}+64{i}t^3xz^6-4{i}xz^3l_{0}^3\\&\quad -24tx^2z^4g_{0}-24tx^2z^4l_{0}+24tz^4g_{0}l_{0}^2-3{i}l_{0}+4{i}x^3z^3g_{0}+12{i}x^3z^3l_{0}\\&\quad +144{i}t^2z^4l_{0}+12{i}x^2z^2g_{0}+12{i}x^2z^2l_{0}+9zl_{0}^2-48{i}t^2z^4g_{0}-24xz^2g_{0}l_{0}\\&\quad -9zg_{0}l_{0}+4{i}z^2l_{0}^3+128{i}t^3z^5+96t^2xz^4+48t^2z^5g_{0}l_{0}-9{i}xzl_{0}+3{i}g_{0}\\&\quad -6tz^2g_{0}-12x^2z^3g_{0}l_{0}+48{i}tz^3l_{0}^2-48{i}tz^3g_{0}l_{0}+48tz^3xl_{0}-48txz^3g_{0}\\&\quad +4x^4z^3-6{i}xzl_{1}+9{i}xzg_{0}+54tz^2l_{0}+8x^3z^2+16{i}tx^3z^4,\\ D_{32}^{[2]}&=48{i}t^2xz^4l_{0}-12{i}xz^2g_{0}l_{0}^2-48{i}txz^3g_{0}l_{0}+4z^2g_{0}l_{0}^3-3{i}xg_{0}-12x^2z^2g_{0}l_{0}\\&\quad -24tx^2z^3l_{0}+3{i}xl_{0}-12x^2z^2l_{0}^2-48t^2z^2+3g_{0}l_{0}+4x^4z^2-64t^4z^6-3l_{0}^2\\&\quad +4{i}x^3z^2g_{0}-4{i}xz^2l_{0}^3+64{i}t^3xz^5+16{i}tx^3z^3+18tzg_{0}-18tzl_{0}-8tz^3l_{0}^3\\&\quad +48t^2z^4g_{0}l_{0}-24tx^2z^3g_{0}+12{i}x^3z^2l_{0}-48t^2z^4l_{0}^2+32t^3z^5g_{0}-96t^3z^5l_{0}\\&\quad -6{i}xl_{1}-12tzl_{1}-48{i}t^2xz^4g_{0}+24tz^3g_{0}l_{0}^2+6g_{0}l_{1}. \end{aligned}$$

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Yan, XW., Chen, Y. Exotic rogue waves in an extended nonlocal nonlinear Schrödinger equation with self-induced PT-symmetric potentials. Eur. Phys. J. Plus 137, 1346 (2022). https://doi.org/10.1140/epjp/s13360-022-03536-3

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