Extended two dimensional equation for the description of nonlinear waves in gas–liquid mixture

• We derive two-dimensional equation describing waves in a gas–liquid mixture.
• We investigate integrability of this equation using the Painlevé approach.
• We construct some exact solutions of the equation derived.
• We perform numerical investigation of the waves, described by the equation.

We consider a system of equations for the description of nonlinear waves in a liquid with gas bubbles. Taking into account high order terms with respect to a small parameter, we derive a new nonlinear partial differential equation for the description of density perturbations of mixture in the two-dimensional case. We investigate integrability of this equation using the Painlevé approach. We show that traveling wave reduction of the equation is integrable under some conditions on parameters. Some exact solutions of the equation derived are constructed. We also perform numerical investigation of the nonlinear waves described by the derived equation.