Simple solutions of the Burgers and Hopf equations

We describe all solutions of the Burgers equation of analytic complexity not exceeding $1$. It turns out that all such solutions fall into four families of dimensions not exceeding $3$ that are represented by elementary functions. An example of a family of solutions of the Burgers equation of complexity $2$ is given. A similar problem is also solved for the Hopf equation. It turns out that all solutions to the Hopf equation of complexity $1$ form a two-parameter family of fractional-linear functions which coincides with one of the families of solutions of the Burgers equation.