Invariant submodels of the Westervelt model with dissipation

• We obtained all invariant submodel of the Westervelt model with dissipation.
• We researched a propagation of the intensive acoustic waves.
• We researched spherically symmetric ultrasonic field
• We established the existence and the uniqueness of the boundary value problems, describing some wave processes.
• We pointed the mechanical relevance of the solutions proposed.

We study three-dimensional Westervelt model of nonlinear hydroacoustics with dissipation. We received all its invariant submodels. With the help of invariant solutions, we explored some wave processes, specifying their physical meaning. The boundary value problems describing these processes are reduced to the nonlinear integro-differential equations. We established the existence and uniqueness of the solutions of these boundary value problems under some additional conditions. Also we considered the invariant solutions of rank 2 and 3. Mechanical relevance of the obtained solutions is as follows: (1) these solutions describe nonlinear and diffraction effects in ultrasonic fields of a special kind, (2) these solutions can be used as a test solutions in the numerical calculations performed in studies of ultrasonic fields generated by powerful emitters.