  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.
