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薛定谔型方程是一类十分重要的微分方程.高维及变系数薛定谔型方程的研究具有一定的价值和意义.本文利用相似变换推导了（n+1）维（2m+1）次变系数非线性薛定谔方程的一类新的孤子解，给出了系数之间满足的关系.并利用定态薛定谔方程的解，得到了（n+1）维（2m+1）次变系数非线性薛定谔方程的明暗孤子解.最后，对于特殊的情况，给出了明暗孤立子解的图像，并系统分析了孤子解的空间结构和传播特性.
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关键词:
- （n+1）维（2m+1）次非线性薛定谔方程 /
- 相似变换 /
- 精确解
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
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