Calculations from On the Existence of Periodic Traveling-Wave Solutions to Certain Systems of Nonlinear, Dispersive Wave Equations
Document Type
Other
Publisher
Utah State University
Publication Date
5-3-2024
Abstract
In the field of nonlinear waves, particular interest is given to periodic traveling-wave solutions of nonlinear, dispersive wave equations. This thesis aims to determine the existence of periodic traveling-wave solutions for several systems of water wave equations. These systems are the Schr¨odinger KdV-KdV, Schr¨odinger BBM-BBM, Schr¨odinger KdV-BBM, and Schr¨odinger BBM-KdV systems, and the abcd-system. In particular, it is shown that periodic traveling-wave solutions exist and are explicitly given in terms of cnoidal, the Jacobi elliptic function. Certain solitary-wave solutions are also established as a limiting case of the periodic traveling-wave solutions, that is, as the elliptic modulus approaches one.
Referenced by
Daniels, Jacob, "On the Existence of Periodic Traveling-Wave Solutions to Certain Systems of Nonlinear, Dispersive Wave Equations" (2024). All Graduate Theses and Dissertations, Fall 2023 to Present. 161. https://digitalcommons.usu.edu/etd2023/161
B. Brewer, J. Daniels, and N.V. Nguyen. “Exact Jacobi elliptic solutions of some models for the interaction of long and short waves”. In: AIMS Mathematics 9.2 (2024), pp. 2854–2873. https://doi.org/10.3934/math.2024141
J. Daniels and N.V. Nguyen. "Exact Jacobi elliptic solutions of the abcd-system". 2024. arXiv: 2402.16756 [math.AP]. https://doi.org/10.48550/arXiv.2402.16756
Recommended Citation
Daniels, J. (2024). Calculations from On the Existence of Periodic Traveling-Wave Solutions to Certain Systems of Nonlinear, Dispersive Wave Equations. Utah State University. https://doi.org/10.26078/ERRN-PJ37
Comments
There are five Maple worksheets, and five corresponding PDFs for those who do not have Maple to view the worksheets. Following the method presented in my thesis (#1 below), Maple was used to explicitly calculate solutions to the systems of partial differential equations mentioned in the abstract. These files outline how exactly this was done, and how a mathematical computer software like Maple can be used to perform these calculations.