Elliptic solutions and solitary waves of a higher order KdV--BBM long wave equation

TL;DR

This paper investigates elliptic and solitary wave solutions of higher order KdV--BBM equations, providing existence conditions, analyzing well-posedness, and exploring solution types including solitons and elliptic functions.

Contribution

It introduces new conditions for solutions, analyzes well-posedness, and explores solution structures for third and fifth-order KdV--BBM equations, including Hamiltonian restrictions and elliptic solutions.

Findings

Existence conditions for elliptic, hyperbolic, and unbounded solutions.

Well-posedness theory for the third-order KdV--BBM equation.

Only dark or bright solitons exist when the fifth-order equation is Hamiltonian.

Abstract

We provide conditions for existence of hyperbolic, unbounded periodic and elliptic solutions in terms of Weierstrass $\wp$ 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 $\wp$ functions. For the fifth-order KdV--BBM equation we show that a parameter $\gamma=\frac {1}{12}$, 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 $\gamma\neq\frac{1}{12}$ we develop $H^k$ 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.