Projectors in nonlinear evolution problem: Acoustic solitons of bubbly liquid

The method of one-dimensional disturbances splitting into components of rightward propagating, leftward propagating, and stationary components by projection technique is applied to compressible liquid with bubbles. By such projecting, the fundamental system of equations is transformed to three nonlinear equations of the interacting components. A small parameter is introduced which determines input of nonlinear and dispersive terms. The system is reduced to one of a Korteweg-de Vries type. It is shown that these three-mode evolution equations are approximately reduced to intergrable KdV-MKdV equation on a class of initial conditions specified by projecting, soliton solution is presented.