  The derivation of different models of non linear acoustic in thermo-ellastic
media as the Kuznetsov equation, the Khokhlov-Zabolotskaya-Kuznetsov (KZK)
equation and the Nonlinear Progressive wave Equation (NPE) from an isentropic
Navier-Stokes/Euler system is systematized using the Hilbert type expansion in
the corresponding perturbative and (for the KZK and NPE equations) paraxial
ansatz . The use of small, to compare to the constant state perturbations,
correctors allows to obtain the approximation results for the solutions of
these models and to estimate the time during which they keep closed in the L2
norm. The KZK and NPE equations are also considered as paraxial approximations
of the Kuznetsov equation, which is a model obtained only by perturbations from
the Navier-Stokes/Euler system. The Westervelt equation is obtained as a
nonlinear approximation of the Kuznetsov equation. In the aim to compare the
solutions of the exact and approximated systems in found approximation domains
the well-posedness results (for the Navier-Stokes system and the Kuznetsov
equation in a half-space with periodic in time initial and boundary data) were
obtained.
