Abstract
An extended Harry Dym hierarchy is constructed by using eigenfunctions and adjoint eigenfunctions of the spectral problems of the Harry Dym hierarchy associated with the pseudo-differential operator L = u partial derivative + u(0) + u(1)partial derivative(-1) + center dot center dot center dot. The corresponding Lax presentation possesses a self-consistent source involving squared eigenfunctions. The resulting extended Harry Dym hierarchy is reduced to the Harry Dym hierarchy with self-consistent sources under the n-reduction, L-n = (L-n)(>= 2), and the k-constrained Harry Dym hierarchy under the k-constraint, L-k = (L-k)(>= 2) + Sigma(N)(i=1) q(i)partial derivative(-1)r(i)partial derivative(2). A few particular examples are computed, together with their Lax pairs.