QUALITATIVE ANALYSIS OF THE KHOKHLOV–ZABOLOTSKAYA–KUZNETSOV (KZK) EQUATION

Abstract

The Khokhlov–Zabolotskaya–Kuznetzov (KZK) equation is considered as a model of nonlinear acoustic which describes the nonlinear propagation of a finite-amplitude focused sound beam which is essentially one-directional, in the thermo-viscous medium.1,7,8 The aim of this paper is the study of the existence, uniqueness, stability, regularity, continuous dependence on the initial value and blow-up of solution of the KZK equation in Sobolev spaces Hs of periodic on x functions and with mean value zero. Existence of shock waves for the model with zero viscosity is proved using S. Alinhac's method.2 Global existence in time of the beam's propagation in viscous media is established for small enough initial data. Existence result is proved by two methods: first by the fractional step method in the particular case ℝ3 and s = 3 to justify the numerical results of Thierry Le Pollès25 and second for the general case ℝn and s > [n/2] + 1 by the approach used in Refs. 12 and 13 for the Kadomtsev–Petviashvili (KP) equation.