Purpose

The purpose of this study is to investigate the nonlinear Schrödinger equation (NLS) incorporating spatiotemporal dispersion and other dispersive effects. The goal is to derive various soliton solutions, including bright, dark, singular, periodic and exponential solitons, to enhance the understanding of soliton propagation dynamics in nonlinear metamaterials (MMs) and contribute new findings to the field of nonlinear optics.

Design/methodology/approach

The research uses a range of powerful mathematical approaches to solve the NLS. The proposed methodologies are applied systematically to derive a variety of optical soliton solutions, each demonstrating unique optical behaviors and characteristics. The approach ensures that both the theoretical framework and practical implications of the solutions are thoroughly explored.

Findings

The study successfully derives several types of soliton solutions using the aforementioned mathematical approaches. Key findings include bright optical envelope solitons, dark optical envelope solitons, periodic solutions, singular solutions and exponential solutions. These results offer new insights into the behavior of ultrashort solitons in nonlinear MMs, potentially aiding further research and applications in nonlinear wave studies.

Originality/value

This study makes an original contribution to nonlinear optics by deriving new soliton solutions for the NLS with spatiotemporal dispersion. The diversity of solutions, including bright, dark, periodic, singular and exponential solitons, adds substantial value to the existing body of knowledge. The use of distinct and reliable methodologies to obtain these solutions underscores the novelty and potential applications of the research in advancing optical technologies. The originality lies in the novel approaches used to obtain these diverse soliton solutions and their potential impact on the study and application of nonlinear waves in MMs.

中文翻译：

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光学超材料中的亮和暗光学孤子使用多种不同的方案来实现广义薛定谔方程

 目的

本研究的目的是研究结合时空色散和其他色散效应的非线性薛定谔方程（NLS）。目标是推导出各种孤子解，包括亮孤子、暗孤子、奇异孤子、周期孤子和指数孤子，以增强对非线性超材料（MM）中孤子传播动力学的理解，并为非线性光学领域贡献新发现。

设计/方法论/途径

该研究使用一系列强大的数学方法来解决 NLS。所提出的方法被系统地应用以获得各种光学孤子解决方案，每种解决方案都展示了独特的光学行为和特性。该方法确保解决方案的理论框架和实际意义都得到彻底探索。

 发现

该研究使用上述数学方法成功地导出了几种类型的孤子解。主要发现包括亮光学包络孤子、暗光学包络孤子、周期解、奇异解和指数解。这些结果为非线性MM中超短孤子的行为提供了新的见解，可能有助于非线性波研究的进一步研究和应用。

 原创性/价值

这项研究通过为具有时空色散的 NLS 推导新的孤子解，对非线性光学做出了原创性贡献。解决方案的多样性，包括明孤子、暗孤子、周期性孤子、奇异孤子和指数孤子，为现有知识体系增加了巨大的价值。使用独特且可靠的方法来获得这些解决方案强调了该研究在推进光学技术方面的新颖性和潜在应用。独创性在于用于获得这些不同孤子解的新方法及其对MM中非线性波的研究和应用的潜在影响。