Asymptotic stability conjecture for line solitary waves of the Amick-Schonbek system

The two-dimensional Amick-Schonbek system for $(\boldsymbol{ u},\boldsymbol{ u})$ admits one-dimensional solitary-wave solutions extended infinitely in the transverse direction, called line solitary waves. They are considered under two-dimensional perturbations in the Cauchy problem.

Asymptotic stability conjecture. The line solitary waves of the Amick-Schonbek system are asymptotically stable.

The paper presents numerical evidence for this qualitative stability property, while the global behavior of solutions to the two-dimensional system is otherwise unknown.