Two new hierarchies containing the sine-Gordon and sinh-Gordon equation, and their Lax representations

An approach for obtaining a new Lax pair from a given old Lax pair is proposed, and two hierarchies of nonlinear evolution equations, which include the well-known sine-Gordon and sinh-Gordon equation, are presented in this article. By the use of the general structure of commutator representations, the Lax representations of the two new hierarchies are obtained. Particularly, the new Lax pairs of the sine-Gordon and sinh-Gordon equation are given. All of these Lax representations are of operator form.