Аннотация: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).