Invariant submodels and exact solutions of Khokhlov–Zabolotskaya–Kuznetsov model of nonlinear hydroacoustics with dissipation

Abstract

We study three-dimensional Khokhlov–Zabolotskaya–Kuznetsov (KZK) model of the nonlinear hydroacoustics with dissipation. This model is described by third order quasilinear partial differential equation of the (KZK). We obtained that the (KZK) equation admits an infinite Lie group of the transformations, depending on the three arbitrary functions. This is due to the fact that in the (KZK) model the main direction of the wave’s propagation is singled out. The submodels of the (KZK) model.are described by the invariant solutions of the (KZK) equation. We studied essentially distinct, not linked by means of the point transformations, invariant solutions of rank 0 and 1 of this equation. Also we considered the invariant solutions of rank 2 and 3. The invariant solutions of rank 0 and 1 are found either explicitly, or their search is reduced to the solution of the nonlinear integro-differential equations. For example, we obtained the invariant solutions that we called by “Ultrasonic knife” and “Ultrasonic destroyer”. The submodel “Ultrasonic knife” have the following property: at each fixed moment of the time in the field of the existence of the solution near a some plane the pressure increases indefinitely and becomes infinite on this plane. The submodel “Ultrasonic destroyer” contains a countable number of “Ultrasonic knives”. The presence of the arbitrary constants in the integro-differential equations, that determine invariant solutions of rank 1 provides a new opportunities for analytical and numerical study of the boundary value problems for the received submodels, and, thus, for the original (KZK) model. With a help of these invariant solutions we researched a propagation of the intensive acoustic waves (one-dimensional, axisymmetric and planar) for which the acoustic pressure, speed and acceleration of its change, or the acoustic pressure , speed and acceleration of its change in the radial direction, or the acoustic pressure, speed and acceleration of its change in the direction of one of the axes are specified at the initial moment of the time at a fixed point. Under the certain additional conditions, we established the existence and the uniqueness of the solutions of boundary value problems, describing these wave processes. 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. Application of the obtained formula generating the new solutions for the found solutions gives families of the solutions containing three arbitrary functions.