Two-dimensional stationary soliton gas

We study two-dimensional stationary soliton gas in the framework of the time-independent reduction of the Kadomtsev-Petviashvili (KPII) equation, which coincides with the integrable two-way ``good'' Boussinesq equation in the xy-plane. This (2+0)D reduction enables the construction of the kinetic equation for the stationary gas of KP solitons by invoking recent results on (1+1)D bidirectional soliton gases and generalised hydrodynamics of the Boussinesq equation. We then use the kinetic theory to analytically describe two basic types of 2D soliton gas interactions: (i) refraction of a line soliton by a stationary soliton gas, and (ii) oblique interference of two soliton gases. We verify the analytical predictions by numerically implementing the corresponding KPII soliton gases via exact N-soliton solutions with N-large and appropriately chosen random distributions for the soliton parameters. We also explicitly evaluate the long-distance correlations for the two-component interference configurations. The results can be applied to a variety of physical systems, from shallow water waves to Bose-Einstein condensates.