On the permutability of Backlund transformations

Let (bar u = B_eta u) : $$begin{gathered} 2(q - bar q) + (Delta (bar q - 2q) + (2q_x + bar q_x ))eta = 0, hfill 2(r - bar r) + (Delta (2bar r - r) + (r_x + 2bar r_x ))eta = 0,u = u(q,r)r hfill end{gathered} $$ be the Backlund transformation (BT) of the hierarchy of AKNS equations, where η is a parameter and (Delta = int_{ - infty }^x {(qr - bar qbar r)} dx') . It is shown in this paper that the infinitesimal BTB _η+s B _η ^?1 admits the following expansion $$B_{eta + varepsilon } B_eta ^{ - 1} u = u + varepsilon sumlimits_{n = 0}^infty {beta _n (JL^{n + 1} u)eta ^n ,beta _n = 1 + ( - 1)^n 2^{ - n - 1} } $$ whereL is the recurrence operator of the hierarchy and ? is an infinitesimal parameter. This expansion implies the equivalence between the permutability of BTs and the involution in pairs of conserved densities.