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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.
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Keywords:
- (n+1)-dimensional (2m+1)-th nonlinear Schrödinger equation /
- Similarity transformations /
- Exact solutions
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