Application of the Pfaffian technique to the KR and mNVN equations

In this paper we discuss the differentiation of Pfaffians which is useful in Pfaffian technique. We apply the Pfaffian technique to two integrable systems; the Konopelchenko–Rogers (KR) equations and the modified Novikov–Veselov–Nizik (mNVN) equations and show that these equations in the bilinear form reduce to a Pfaffian identity. We also derive a new Lax pair for the mNVN equations which is gauge equivalent to a pair of operators.