Higher-order asymptotic profiles for solutions to the Cauchy problem for a dispersive-dissipative equation with a cubic nonlinearity

TL;DR

This paper develops higher-order asymptotic profiles for solutions to a dispersive-dissipative PDE with cubic nonlinearity, extending previous results by constructing a second profile and detailing the asymptotic expansion.

Contribution

It introduces the second asymptotic profile for the Duhamel term and refines the higher-order expansion, highlighting the self-similar structure of the profiles.

Findings

Constructed the second asymptotic profile for the Duhamel term.

Provided a more detailed higher-order asymptotic expansion.

Profiles satisfy the parabolic self-similarity structure.

Abstract

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 newly obtained higher-order asymptotic profiles have a good structure in the sense of satisfying the parabolic self-similarity.