Invariants for Darboux transformations of Arbitrary Order for $D_x{D}_y+a{D}_x+b{D}_y+c$

Abstract

We develop the method of regularized moving frames of Fels and Olver to obtain explicit general formulas for the basis invariants that generate all the joint differential invariants, under gauge transformations, for the operators [ø{L}=D_xD_y +a(x,y) D_x + b(x,y) D_y +c(x,y)] and an operator of arbitrary order. The problem appeared in connection with invariant construction of Darboux transformations for $øL$.