Article
Elliptic Solutions and Solitary Waves of a Higher Order KdV-BBM Long Wave Equation
Journal of Mathematical Analysis and Applications
(2017)
Abstract
We provide conditions for existence of hyperbolic, unbounded periodic and elliptic solutions in terms of Weierstrass ℘ functions of both third and fifth-order KdV–BBM (Korteweg-de Vries–Benjamin, Bona & Mahony) regularized long wave equation. An analysis for the initial value problem is developed together with a local and global well-posedness theory for the third-order KdV–BBM equation. Traveling wave reduction is used together with zero boundary conditions to yield solitons and periodic unbounded solutions, while for nonzero boundary conditions we find solutions in terms of Weierstrass elliptic ℘ functions. For the fifth-order KdV–BBM equation we show that a parameter γ = 1/12 , for which the equation has a Hamiltonian, represents a restriction for which there are constraint curves that never intersect a region of unbounded solitary waves, which in turn shows that only dark or bright solitons and no unbounded solutions exist. Motivated by the lack of a Hamiltonian structure for γ 6 /= 1/12 we develop Hk bounds, and we show for the non Hamiltonian system that dark and bright solitons coexist together with unbounded periodic solutions. For nonzero boundary conditions, due to the complexity of the nonlinear algebraic system of coefficients of the elliptic equation we construct Weierstrass solutions for a particular set of parameters only.
Keywords
- KdV–BBM equation,
- Solitary waves,
- Unidirectional waves,
- Long-crested waves,
- Elliptic solutions
Disciplines
Publication Date
August 15, 2017
DOI
https://doi.org/10.1016/j.jmaa.2017.03.057
Citation Information
S.C. Mancas and Ronald Adams. "Elliptic Solutions and Solitary Waves of a Higher Order KdV-BBM Long Wave Equation" Journal of Mathematical Analysis and Applications Vol. 452 Iss. 2 (2017) p. 1168 - 1181 Available at: http://works.bepress.com/stefani_mancas/97/