Simple solutions of three equations of mathematical physics

In this paper, we consider three equations of mathematical physics for functions of two variables: the heat equation, the Liouville equation, and the Korteweg-de Vries (KdV) equation. We obtain complete lists of simple solutions for all three equations, that is, solutions of analytic complexity not exceeding one. All solutions of this type for the heat equation can be expressed in terms of the error function (Theorem 1) and form a 4-parameter family; for the Liouville equation, the answer is the union of a 6-parameter family and a 3-parameter family of elementary functions (Theorem 2); for the Korteweg-de Vries equation, the list consists of four 3-parameter families containing elementary and elliptic functions (Theorem 3).