Spectral stability and time evolution of N-solitons in the KdV hierarchy

This paper concerns spectral stability and time evolution of N-solitons in the Korteweg-de Vries (KdV) hierarchy with mixed commuting time flows. Spectral stability problem is analysed 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.