  We consider the asymptotic behavior of solutions to the Cauchy problem for a
dispersive-dissipative equation with a cubic nonlinearity. It is known that the
leading term of the asymptotic profile for the solution to this problem is the
Gaussian. Moreover, by analyzing the corresponding integral equation, the
higher-order asymptotic expansion for the solution to the linear part and the
first asymptotic profile for the Duhamel term have already been obtained. In
this paper, we construct the second asymptotic profile for the Duhamel term and
give the more detailed higher-order asymptotic expansion of the solutions,
which generalizes the previous works. Furthermore, we emphasize that the new
obtained higher-order asymptotic profiles have a good structure in the sense of
satisfying the parabolic self-similarity.
