  This paper concerns spectral stability and time evolution of $N$-solitons in
the KdV hierarchy with mixed commuting time flows. Spectral stability problem
is analyzed by using a pair of self-adjoint operators with finite numbers of
negative eigenvalues. We show that the absence of unstable eigenvalues in the
stability problem is related to the absence of negative eigenvalues of these
operators in the constrained function spaces. Time evolution of $N$-solitons is
uniquely characterized from the inverse scattering transform technique.
