Nonlinear equations linearized using the generalized Cole-Hopf substitutions and the exactly integrable models of the one-dimensional compressible fluid flows

Abstract

A new method to derive the nonlinear equations linearized using the substitutions extending the Cole-Hopf substitution to the Burgers equation is considered. A method to analyze the general structure of the solutions and to calculate the exact solutions in the problems of one-dimensional compressible fluid flows is developed on the basis of this approach. The cases of the ideal and viscous fluids are considered. The problem of the dynamics of dust-like matter at a zero pressure and the gas-dust mixture is analyzed.