  In this paper the effect of a small dissipation on waves is included to find
exact solutions to the modified BBM equation. Using Lyapunov functions and
dynamical systems theory, we prove that when viscosity is added to the BBM
equation, in certain regions there still exist bounded traveling wave solutions
in the form of solitary waves, periodic, and elliptic functions. By using the
canonical form of Abel equation, the polynomial Appell invariant make the
equation integrable in terms of Weierstrass $\wp$ functions. We will use a
general formalism based on Ince's transformation to write the general solution
of dissipative BBM in terms of $\wp$ functions, from which all the other known
solutions can be obtained via simplifying assumptions. Using ODE analysis we
show that the traveling wave speed is a bifurcation parameter that makes
transition between different classes of waves.
