A Note on Lax Pairs of the Sawada-Kotera Equation

We prove that the new Lax pair of the Sawada-Kotera equation, discovered recently by Hickman, Hereman, Larue, and G？kta？, and the well-known old Lax pair of this equation, considered in the form of zero-curvature representations, are gauge equivalent to each other if and only if the spectral parameter is nonzero, while for zero spectral parameter a nongauge transformation is required. 1. Introduction Recently, the following interesting result was obtained by Hickman et al. [1]. It turned out that the Sawada-Kotera equation [2, 3] possesses two different Lax representations in the operator form where subscripts of the scalar functions and denote respective derivatives, and are linear differential operators expressed in powers of the derivative operator , and is the spectral parameter. The first Lax pair, given by the operators is well known [4, 5]. The second Lax pair, given by the operators is new, in the sense that it appeared in [1] for the first time in the literature. Many experts, according to their private communications, noticed that the second Lax pair (4) is related to the first Lax pair (3) by the transformation where the dagger denotes the Hermitian conjugate. This transformation (5) always turns a Lax pair of an integrable equation into a Lax pair of the same equation, but usually the resulting Lax pair has essentially the same form as the original one (we believe that for this reason no second Lax pair was discovered in [1] for the Kaup-Kupershmidt equation, in particular). Let us note, however, that the Lax pairs (3) and (4) are different in form. Some other experts, also according to their private communications, noticed that the old Lax pair (3) and the new one (4) are related to each other by the transformation which corresponds to the transformation made in (2). Thus, there exist (at least) two different ways to relate the Lax pairs (3) and (4) to each other, and we believe that this point deserves further investigation using more general description of Lax pairs than their operator form. In the present paper, we study these two Lax pairs of the Sawada-Kotera equation (1)—the old one, (2) with (3), and the new one, (2) with (4)—in the matrix form or, what is the same, in the form of zero-curvature representations (ZCRs) where is a three-component column vector, and are matrices, and the square brackets denote the matrix commutator. In Section 2, we show that, for any nonzero value of the spectral parameter, the new Lax pair of the Sawada-Kotera equation and the old one are related to each other by a gauge transformation of ZCRs where