Unifying the steady state resonant solutions of the periodically forced KdVB, mKdVB, and eKdVB equations

The periodically forced extended KdVB (eKdVB) equation, which contains both KdVB and modified KdVB (mKdVB) equations as special cases, is known to possess a rich array of resonant steady solutions. We present an analytic methodology based on singular perturbation and asymptotic matching in order to illustrate and approximate these solutions in the limit that the dispersive effects are small relative to the nonlinear and forcing terms. Weak Burgers damping is also included at the same order as dispersion. Solutions across the resonant band may be constructed and show good agreement with solutions of the full equation, showing clearly the role of the various physical effects. In this way, direct comparisons and connections are made between the various classes of KdVB equations, illustrating, in particular, the underlying mathematical connections between the KdVB and mKdVB equations.