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Solitons and rogue wave solutions of focusing and defocusing space shifted nonlocal nonlinear Schrödinger equation

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Abstract

In this paper, we are concerned with the explicit analytic solutions for the focusing and defocusing space shifted nonlocal nonlinear Schrödinger (NLS) equation introduced by Ablowitz and Musskimani (Phys Lett A 409:127516, 2021). The nonsingular N-soliton solutions of the defocusing space shifted nonlocal NLS equation are obtained, while the multi-rogue wave solutions are constructed for focusing space shifted nonlocal NLS equation by Darboux transformation. The asymptotic analysis of the soliton solutions is investigated theoretically and numerically. The dynamic features of first-, second-order RW solutions are analysed explicitly. It shows that the space shift \(x_0\) reveals more general dynamic behaviors in the space shifted nonlocal NLS equation.

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Acknowledgements

The work of JY is supported by National Natural Science Foundation of China under Grant No.12001361, Young Teachers Training Assistance Program of Shanghai under Grant No. ZZEGDD20005, that of LYM by National Natural Science Foundation of China under Grant No.11701510.

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Authors and Affiliations

  1. College of Arts and Sciences, Shanghai Polytechnic University, Shanghai, 201209, People’s Republic of China

    Jun Yang

  2. Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, 310023, People’s Republic of China

    Hai-Fang Song, Miao-Shuang Fang & Li-Yuan Ma

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  1. Jun Yang
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  2. Hai-Fang Song
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  3. Miao-Shuang Fang
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  4. Li-Yuan Ma
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Correspondence to Li-Yuan Ma.

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Appendix

Appendix

The second-order RW solutions of (16) are expressed in details as follows

$$\begin{aligned} q^{[2]}(x,t)=e^{-2it}\left( 1+2i\frac{\tau _{1,2}^{(1)}}{\tau _{0,2}^{(1)}}\right) , \end{aligned}$$
(20)

where

$$\begin{aligned} \tau _{1,2}^{(1)}=&\frac{1}{24}(-1280it^4-1024t^5\\&-128t^3(1+(-2x+x_{0})^2) \\&-96it^2(3+(-2x+x_{0})^2)\\ {}&-4t(-15+(-2x+x_{0})^2(-6\\&+(-2x+x_{0})^2))-i(-3+(-2x+x_{0})^2\\&(6+(-2x+x_{0})^2)) \\&-4r_{0}^3(4t+i(1+2x-x_{0})-2s_{0})^2-6r_{1}(4t\\&+i(1+2x-x_{0})-2s_{0})^2\\&-9s_{0}+96its_{0}+1280it^3s_{0} \\&+1280t^4s_{0}+24xs_{0}+96itxs_{0}\\&+384t^2xs_{0}-512it^3xs_{0}\\&+192itx^2s_{0}+384t^2x^2s_{0}+32x^3s_{0}\\&-128itx^3s_{0}+16x^4s_{0} \\&-12x_{0}s_{0}-48itx_{0}s_{0}-192t^2x_{0}s_{0}+256it^3x_{0}s_{0}\\&-192itxx_{0}s_{0}-384t^2xx_{0}s_{0}\\&-48x^2x_{0}s_{0}+192itx^2x_{0}s_{0}\\&-32x^3x_{0}s_{0}+48itx_{0}^2s_{0}+96t^2x_{0}^2s_{0}+24xx_{0}^2s_{0}\\&-96itxx_{0}^2s_{0}+24x^2x_{0}^2s_{0}-4x_{0}^3s_{0}\\&+16itx_{0}^3s_{0}-8xx_{0}^3s_{0}+x_{0}^4s_{0}\\&-384it^2s_{0}^2-512t^3s_{0}^2-192txs_{0}^2+384it^2xs_{0}^2\\&+32ix^3s_{0}^2+96tx_{0}s_{0}^2-192it^2x_{0}s_{0}^2 \\&-48ix^2x_{0}s_{0}^2+24ixx_{0}^2s_{0}^2-4ix_{0}^3s_{0}^2\\&-4s_{0}^3+32its_{0}^3+64t^2s_{0}^3+16xs_{0}^3\\&-64itxs_{0}^3-16x^2s_{0}^3-8x_{0}s_{0}^3+32itx_{0}s_{0}^3\\&+16xx_{0}s_{0}^3-4x_{0}^2s_{0}^3\\&-4r_{0}^2((4t+2ix-ix_{0})(32t^2+4it(6+2x-x_{0})\\&+(-2x+x_{0})^2)+s_{0}(-3(-1+48t^2\\&-4x+8it(3+2x-x_{0})\\&+2x_{0}+(-2x+x_{0})^2)+4(3i+12t-s_{0})s_{0})\\&-6s_{1})-6s_{1}+6(4t-2ix+ix_{0})\\&(4t+i(2-2x+x_{0}))s_{1}\\&+r_{0}(-1280t^4-96t^2(-2+2x-x_{0})(2x-x_{0})\\&-256it^3(5+2x-x_{0})\\&-(-3+4x^2-4x(2+x_{0})\\&+x_{0}(4+x_{0}))(3+(-2x+x_{0})^2)-16it(6+8x^3\\&-12x^2(-1+x_{0})+x_{0}(3-(-3+x_{0})x_{0})\\&+6x(-1+(-2+x_{0})x_{0}))\\&+24(i+4t)(1+8t(i+2t)\\&+(-2x+x_{0})^2)s_{0}-12\\&(-1+48t^2+4x-8it(-3+2x-x_{0})\\&-2x_{0}+(-2x+x_{0})^2)s_{0}^2\\&+16(4t+i(1-2x+x_{0}))s_{0}^3\\&+24(4t+i(1-2x+x_{0}))s_{1})),\\ \tau _{0,2}^{(1)}&=\frac{1}{144}(-9(1+12x^2)-16(256t^6+x^4(3+4x^2)\\&+48t^4(9+4x^2)+t^2(99-72x^2+48x^4))\\&+12x((3-16t^2)^2\\&+8(1+16t^2)x^2+16x^4)x_{0}\\&-3((3-16t^2)^2+24(1+16t^2)x^2+80x^4)\\&x_{0}^2+8x(3+48t^2+20x^2)x_{0}^3\\&-3(1+16t^2+20x^2)x_{0}^4+12xx_{0}^5-x_{0}^6\\&-8r_{0}^3(64t^3+48it^2(2x-x_{0})\\&-12t(-3+(-2x+x_{0})^2)\\&-i(2x-x_{0})(-3+(-2x+x_{0})^2)+2s_{0}\\&(-3(4t+i(-1+2x-x_{0}))(4t+i(1+2x-x_{0}))\\&+2(12t+6ix-3ix_{0}-2s_{0})s_{0})-12s_{1})\\&-12r_{1}(64t^3+48it^2(2x-x_{0})\\&-12t(-3+(-2x+x_{0})^2)\\&-i(2x-x_{0})(-3+(-2x+x_{0})^2)+2s_{0}\\&(-3(4t+i(-1+2x-x_{0}))\\&(4t+i(1+2x-x_{0}))+2(12t\\&+6ix-3ix_{0}-2s_{0})s_{0})-12s_{1})\\&-12r_{0}^2(192t^2+256t^4+128it^3(2x-x_{0})\\&+8it(2x-x_{0})^3-(-2x+x_{0})^4\\&-6(64t^3+16it^2(2x-x_{0})\\&+i(2x-x_{0})(1+(-2x+x_{0})^2)\\&+4t(3+(-2x+x_{0})^2))s_{0}+12(1+16t^2\\&+(-2x +x_{0})^2)s_{0}^2-8(4t-2ix+ix_{0})s_{0}^3\\&-12(4t-2ix+ix_{0})s_{1})\\&+2(3(1024t^5-9i(2x-x_{0})-256it^4(2x-x_{0})\\&-32it^2(2x-x_{0})(-3+(-2x+x_{0})^2)\\&-i(2x-x_{0})^3(2+(-2x+x_{0})^2)\\&+128t^3(8+(-2x+x_{0})^2)+4t(15\\&+(-2x+x_{0})^4))s_{0}-6(192t^2\\&+256t^4-128it^3(2x-x_{0})\\&-8it(2x-x_{0})^3-(-2x+x_{0})^4)s_{0}^2+4(64t^3\\&-48it^2(2x-x_{0})-12t(-3+(-2x+x_{0})^2)\\&+i(2x-x_{0})(-3+(-2x+x_{0})^2))s_{0}^3\\&+6(64t^3-48it^2(2x-x_{0})\\&-12t(-3+(-2x+x_{0})^2)+i(2x-x_{0})\\&(-3+(-2x+x_{0})^2))s_{1})\\&-6r_{0}(1024t^5+9i(2x-x_{0})+256it^4(2x-x_{0})\\&+32it^2(2x-x_{0})(-3+(-2x+x_{0})^2)\\&+i(2x-x_{0})^3(2+(-2x+x_{0})^2)\\&+128t^3(8+(-2x+x_{0})^2)+4t(15\\&+(-2x+x_{0})^4)-12s_{1}\\&+2(-3(3+256t^4+(-2x+x_{0})^2 \\&(2+(-2x+x_{0})^2)+32t^2(3+(-2x+x_{0})^2))s_{0}\\&+6(64t^3-16it^2(2x-x_{0})-i(2x-x_{0})\\&(1+(-2x+x_{0})^2)+4t(3+(-2x+x_{0})^2))s_{0}^2\\&-4(4t-i(1+2x-x_{0}))(4t+i(1-2x+x_{0}))\\&s_{0}^3-6(4t-2ix+ix_{0})^2s_{1}))). \end{aligned}$$

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Yang, J., Song, HF., Fang, MS. et al. Solitons and rogue wave solutions of focusing and defocusing space shifted nonlocal nonlinear Schrödinger equation. Nonlinear Dyn 107, 3767–3777 (2022). https://doi.org/10.1007/s11071-021-07147-y

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