Similarity transformations and exact solutions of the generalized (<inline-formula><tex-math id="M3">\begin{document}$ n+1 $\end{document}</tex-math></inline-formula>)-dimensional Schrödinger equation with (<inline-formula><tex-math id="M4">\begin{document}$ 2 m+1 $\end{document}</tex-math></inline-formula>)-th order nonlinear terms and spatiotemporally varying coefficients

WANG Gangwei; TAN Zixuan

doi:10.7498/aps.74.20250225

Abstract
Schrödinger-type equations represent a fundamentally important class of differential equations. The study of high-dimensional and variable-coefficient Schrödinger-type equations is of significant theoretical and practical value, providing critical insights into the dynamics of complex wave phenomena. In this paper, we employ similarity transformations to derive a novel class of soliton solutions for the ($ n+1 $)-dimensional ($ 2m+1 $)th-order variable-coefficient nonlinear Schrödinger equation. By extending similarity transformations from lower-dimensional equations to higher dimensions, we establish the intrinsic relationships among the equation’s coefficients. Furthermore, utilizing the solutions of the stationary Schrödinger equation and applying the balancing-coefficient method, we construct both bright and dark soliton solutions for the ($ n+1 $)-dimensional ($ 2m+1 $)th-order variable-coefficient nonlinear Schrödinger equation. Finally, for specific cases, we present graphical representations of the bright and dark soliton solutions and conduct a systematic analysis of their spatial structures and propagation characteristics. Our results reveal that bright solitons exhibit a single-peak structure, while dark solitons form trough-like profiles, further confirming the stability of soliton wave propagation.

Keywords:
(n+1)-dimensional (2m+1)-th nonlinear Schrödinger equation /

Similarity transformations /

Exact solutions
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图 1 方程(29)的一种明孤子解, 展示的图像为$ |u|^2 $. 其中$ x_1\in[-10, 10] $, $ t\in[0, 10] $, $ x_2=-3, x_3 = 2 $.　(a) 三维图; (b) 相位图

Figure 1. Bright soliton solution of Eq.(29), shown as $ |u|^2 $, where $ x_1\in[-10, 10] $, $ t\in[0, 10] $, $ x_2=-3, x_3 = 2 $. (a) 3-dimensional image; (b) Phase image.

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图 2 方程(29)的一种明孤子解, 展示的图像为$ |u|^2 $. 其中$ x_1\in[-10, 10] $, $ t\in[3, 5] $, $ x_2=-3, x_3 = 2 $. 　(a) 三维图; (b) 相位图

Figure 2. Bright soliton solution of Eq.(29), shown as $ |u|^2 $, where $ x_1\in[-10, 10] $, $ t\in[3, 5] $, $ x_2=-3, x_3 = 2 $. (a) 3-dimensional image; (b) Phase image.

DownLoad: Full-Size Img PowerPoint

图 3 方程(30)的一种暗孤子解, 展示的图像为$ |u|^2 $. 其中 $ x\in[-10, 10] $, $ t\in[0, 10] $, $ x_2=-3, x_3 = 2 $. 　(a) 三维图; (b) 相位图

Figure 3. Dark soliton solution of Eq.(30), shown as $ |u|^2 $, where $ x_1\in[-10, 10] $, $ t\in[0, 10] $, $ x_2=-3, x_3 = 2 $. (a) 3-dimensional image; (b) Phase image.

DownLoad: Full-Size Img PowerPoint

图 4 方程(30)的一种暗孤子解, 展示的图像为$ |u|^2 $. 其中 $ x\in[-10, 10] $, $ t\in[0, 2] $, $ x_2=-3, x_3 = 2 $. 　(a) 三维图; (b) 相位图

Figure 4. Dark soliton solution of Eq.(30), shown as $ |u|^2 $, where $ x_1\in[-10, 10] $, $ t\in[0, 2] $, $ x_2=-3, x_3 = 2 $. (a) 3-dimensional image; (b) Phase image.

DownLoad: Full-Size Img PowerPoint

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Similarity transformations and exact solutions of the generalized ($ n+1 $)-dimensional Schrödinger equation with ($ 2 m+1 $)-th order nonlinear terms and spatiotemporally varying coefficients

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Similarity transformations and exact solutions of the generalized ($ n+1 $)-dimensional Schrödinger equation with ($ 2 m+1 $)-th order nonlinear terms and spatiotemporally varying coefficients

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