Exact solutions for shock waves in dilute gases

In 1922 Becker found an exact solution for shock waves in gases using the Navier-Stokes-Fourier constitutive equations for a Prandtl number of value 3/4 with constant transport coefficients. His analysis has been extended to study some cases where an implicit solution can be found in an exact way. In this work we consider this problem for the so-called soft-spheres model in which the viscosity and thermal conductivity are proportional to a power of the temperature η,κ∝T^{σ}. In particular, we give implicit exact solutions for the Maxwell model (σ=1), hard spheres (σ=1/2), and when σ (the viscosity index) is a natural number.