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

Abstract

In this paper we provide conditions for existence of hyperbolic, unbounded periodic and elliptic solutions in terms of Weierstrass \(\wp\) functions of both third and fifth-order (Korteweg-de Vries) KdV or (Benjamin, Bona \& Mahony) BBM-type 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 type equation. The traveling wave solutions are obtained for zero boundary conditions to yield solitons and periodic unbounded traveling waves, while for nonzero boundary conditions we show how the solutions are generalized in terms of \(\wp\) functions. For the fifth-order KdV-BBM type equation we show that a parameter \(\gamma=\frac {1}{12}\) represents a restriction for which there are four constrain 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.