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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>x</a:mi><a:mi>y</a:mi></a:mrow></a:math> plane. This <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mo>(</b:mo><b:mn>2</b:mn><b:mo>+</b:mo><b:mn>0</b:mn><b:mo>)</b:mo><b:mi mathvariant="normal">D</b:mi></b:mrow></b:math> reduction enables the construction of the spatial analog of the kinetic equation for the stationary gas of KP solitons by invoking recent results on <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mo>(</d:mo><d:mn>1</d:mn><d:mo>+</d:mo><d:mn>1</d:mn><d:mo>)</d:mo><d:mi mathvariant="normal">D</d:mi></d:mrow></d:math> bidirectional soliton gases and generalized hydrodynamics of the Boussinesq equation. We then use the spectral 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 <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>N</f:mi></f:math>-soliton solutions with <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>N</g:mi></g:math>-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.