Prolongation structure of the variable coefficient KdV equation

The prolongation structure methodologies of Wahlquist–Estabrook Wahlquist H D and Estabrook F B 1975 J. Math. Phys. 16 1 for nonlinear differential equations are applied to a variable-coefficient KdV equation. Based on the obtained prolongation structure, a Lie algebra with five parameters is constructed. Under certain conditions, a Lie algebra representation and three kinds of Lax pairs for the variable- coefficient KdV equation are derived.