The Russo–Smereka kinetic equation: Conservation laws, reductions and numerical solutions

• The Russo–Smereka equation is associated with an integrable hydrodynamic chain.
• A stability of bubbly flow in terms of hyperbolicity of the model is studied.
• Infinitely many particular solutions for the kinetic equation are found.
• A finite-dimensional approximation for the model is proposed.

The one-dimension Russo–Smereka kinetic equation describing the propagation of nonlinear concentration waves in a rarefied bubbly fluid is considered. Stability of the bubbly flow in terms of hyperbolicity of the kinetic equation is studied. It is proved that a hydrodynamical chain associated with the Russo–Smereka kinetic equation possesses infinitely many conservation laws. Reductions of the model to finite component systems are derived. Conservative form of the kinetic model is proposed and numerical solution of the Cauchy problem with discontinuous initial data is obtained.