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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.

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.

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.

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.

This paper aims to explore new variable separation solutions for a new generalized (3 + 1)-dimensional breaking soliton equation, construct novel nonlinear excitations and discuss their dynamical behaviors that may exist in many realms such as fluid dynamics, optics and telecommunication.

By means of the multilinear variable separation approach, variable separation solutions for the new generalized (3 + 1)-dimensional breaking soliton equation are derived with arbitrary low dimensional functions with respect to {y, z, t}. The asymptotic analysis is presented to represent generally the evolutions of rogue waves.

Fixing several types of explicit expressions of the arbitrary function in the potential field U, various novel nonlinear wave excitations are fabricated, such as hybrid waves of kinks and line solitons with different structures and other interesting characteristics, as well as interacting waves between rogue waves, kinks, line solitons with translation and rotation.

The paper presents that a variable separation solution with an arbitrary function of three independent variables has great potential to describe localized waves.

The roles of parameters in the chosen functions are ascertained in this study, according to which, one can understand the amplitude, shape, background and other characteristics of the localized waves.

The work provides novel localized waves that might be used to explain some nonlinear phenomena in fluids, plasma, optics and so on.

The study proposes a new generalized (3 + 1)-dimensional breaking soliton equation and derives its nonlinear variable separation solutions. It is demonstrated that a variable separation solution with an arbitrary function of three independent variables provides a treasure-house of nonlinear waves.