Dispersion conjecture for localized data in the two-dimensional Amick-Schonbek system

Consider localized smooth initial data for the two-dimensional Amick-Schonbek system satisfying the non-cavitation condition, namely positivity of the relevant total height. A structure localized in both spatial dimensions is localized in both coordinate directions.

Dispersion conjecture. Localised smooth initial data satisfying the non-cavitation condition do not lead to stable structures localised in both spatial dimension. The solutions exist for all times and are simply dispersed.

The conjecture is motivated by numerical simulations showing annular evolution and no lump solutions analogous to those of the Kadomtsev–Petviashvili I equation. The paper compares the observed behavior with the dispersive behavior of the Kadomtsev–Petviashvili II equation, but does not establish the asserted global existence or dispersion analytically.