The Derivation of a Fifth-Order Equation via the Lax and the Alternate Lax Methods

We present the derivation of a fifth-order integrable nonlinear partial differential equation via the Lax method and the alternate Lax method in the continuous case. The Lax method uses a pair of differential operators, L and A, satisfying a compatibility condition. The pair (L,A) is known as the Lax pair. The alternate Lax method is a variation of the Lax method and use a consistency relation equivalent to the commutation of a certain derivative operator associated with the time evolution and the spacial evolution. In the paper we also show that the Lax method and the alternate Lax method are equivalent.