Hybrid soliton and breather waves, solution molecules and breather molecules of a \((3+1)\)-dimensional Geng equation in shallow water waves. (English) Zbl 1525.76022
The authors present a comprehensive study on a \((3+1)\)-dimensional nonlinear Geng equation, a subject of considerable interest in the field of mathematical physics. Utilizing the Hirota bilinear method, the authors delve into the calculation of first- to fourth-order solutions of the equation, thereby contributing significantly to the existing body of literature. The paper is particularly noteworthy for its three main contributions: the observation of hybrid solitons and breather waves, the elucidation of soliton molecules, and the corresponding constraint conditions under various coordinates.
The authors start by discussing the fourth-order solutions, providing a detailed mathematical framework for their derivation. They introduce an auxiliary function \(f\) and demonstrate how it can be substituted into the equation to compute the fourth-order solutions. This section serves as a cornerstone for the subsequent discussions on soliton and breather molecules.
One of the paper’s most compelling sections is its exploration of soliton molecules. The authors meticulously outline the conditions necessary for generating these molecules, offering mathematical expressions and constraints that govern their formation. They extend their discussion to include the conditions for two soliton molecules under different coordinates, thereby providing a comprehensive understanding of the phenomenon.
Furthermore, the paper touches upon the intriguing subject of breather molecules. While the authors successfully demonstrate that breather molecules can be observed by adjusting certain dispersion parameters, they acknowledge that the explicit conditions for generating these molecules remain unknown. This gap in the research invites further investigation and could be considered as a fertile ground for future studies.
In summary, the paper by Li and Ma offers a robust and mathematically sound exploration of a \((3+1)\)-dimensional nonlinear Geng equation. Through their use of the Hirota bilinear method, they not only calculate various orders of solutions but also make significant strides in understanding the complex phenomena of soliton and breather molecules. While the paper is comprehensive, it leaves room for future research, particularly in the area of breather molecules.
The authors start by discussing the fourth-order solutions, providing a detailed mathematical framework for their derivation. They introduce an auxiliary function \(f\) and demonstrate how it can be substituted into the equation to compute the fourth-order solutions. This section serves as a cornerstone for the subsequent discussions on soliton and breather molecules.
One of the paper’s most compelling sections is its exploration of soliton molecules. The authors meticulously outline the conditions necessary for generating these molecules, offering mathematical expressions and constraints that govern their formation. They extend their discussion to include the conditions for two soliton molecules under different coordinates, thereby providing a comprehensive understanding of the phenomenon.
Furthermore, the paper touches upon the intriguing subject of breather molecules. While the authors successfully demonstrate that breather molecules can be observed by adjusting certain dispersion parameters, they acknowledge that the explicit conditions for generating these molecules remain unknown. This gap in the research invites further investigation and could be considered as a fertile ground for future studies.
In summary, the paper by Li and Ma offers a robust and mathematically sound exploration of a \((3+1)\)-dimensional nonlinear Geng equation. Through their use of the Hirota bilinear method, they not only calculate various orders of solutions but also make significant strides in understanding the complex phenomena of soliton and breather molecules. While the paper is comprehensive, it leaves room for future research, particularly in the area of breather molecules.
Reviewer: Denys Dutykh (Le Bourget-du-Lac)
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