  We consider the large time behavior of the solutions to the Cauchy problem
for the BBM-Burgers equation. We prove that the solution to this problem goes
to the self-similar solution to the Burgers equation called the nonlinear
diffusion wave. Moreover, we construct the appropriate second asymptotic
profiles of the solutions depending on the initial data. Based on that
discussion, we investigate the effect of the initial data on the large time
behavior of the solution, and derive the optimal asymptotic rate to the
nonlinear diffusion wave. Especially, the important point of this study is that
the second asymptotic profiles of the solutions with slowly decaying data,
whose case has not been studied, are obtained.
