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This paper concerns spectral stability and time evolution of $N$-solitons inthe KdV hierarchy with mixed commuting time flows. Spectral stability problemis analyzed by using a pair of self-adjoint operators with finite numbers ofnegative eigenvalues. We show that the absence of unstable eigenvalues in thestability problem is related to the absence of negative eigenvalues of theseoperators in the constrained function spaces. Time evolution of $N$-solitons isuniquely characterized from the inverse scattering transform technique.Comment: 13 page

Spectral stability and time evolution ofN-solitons in the KdV hierarchy