Some methods for generating solutions to the Korteweg–de Vries equation

The constraint equation which must hold for the reciprocal of a known solution for the Korteweg–de Vries (KdV) equation to be a solution itself is derived. These reciprocal solutions are required to satisfy a differential equation which is in fact a Painlevé equation. A differential constraint is also derived which allows the product of two solutions of the KdV equation to be a new solution as well.