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A new derivation of (2 + 1)-dimensional Schrödinger equation with separated real and imaginary parts of the dependent variable and its solitary wave solutions

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Abstract

In this work, we have derived a new \((2+1)\)-dimensional Schrödinger equation that contains separated real and imaginary parts of the dependent variable as follows

$$\begin{aligned}{} & {} iw_{tx}+\left( iw_{t}-w_{xx}+w|w|^2\right) _{y}\\{} & {} \qquad \,\, +2\left( w\left( \partial _{x}^{-1}\left( \Re (w)\Re (w)_{y}\right. \right. \right. \\{} & {} \qquad \left. \left. \left. +\Im (w)\Im (w)_{y}\right) \right) \right) _{x}=0. \end{aligned}$$

This nonintegrable equation has been extracted by adopting the recursion operator of the known Schrödinger equation. However, the resulting model has proven too complex to solve or explore the physical meaning of its solutions. Therefore, we have transferred it into a system of three equations, and then, we have applied the Lie symmetry technique to reduce the latter to a simpler equivalent system. As a result, many solitary wave solutions have been attained, including bright solitary wave, dipole, kink, breather, periodic, and their interactions.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 11371326215 and No. 11975145). We thank the reviewers and editor for their recommendations to improve this manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China

    Nardjess Benoudina & Yi Zhang

  2. School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Nassim Bessaad

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  1. Nardjess Benoudina
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  2. Yi Zhang
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Correspondence to Nardjess Benoudina.

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Benoudina, N., Zhang, Y. & Bessaad, N. A new derivation of (2 + 1)-dimensional Schrödinger equation with separated real and imaginary parts of the dependent variable and its solitary wave solutions. Nonlinear Dyn 111, 6711–6726 (2023). https://doi.org/10.1007/s11071-022-08193-w

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