Similarity transformations and exact solutions of the generalized (<i>n</i> + 1)-dimensional Schr&#246;dinger equation with (2<i>m</i> + 1)-th order nonlinear terms and spatiotemporally varying coeffcients

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-coeffcient Schrödingertype 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-coeffcient nonlinear Schrödinger equation. By extending similarity transformations from lower-dimensional equations to higher dimensions, we establish the intrinsic relationships among the equation’ s coeffcients. Furthermore, utilizing the solutions of the stationary Schrödinger equation and applying the balancing-coeffcient method, we construct both bright and dark soliton solutions for the (n+1)-dimensional (2m+1) th-order variable-coeffcient 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
Cited By

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

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

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