  The gauge-Miura correspondence establishes a map between the entire KdV and
mKdV hierarchies, including positive and also negative flows, from which new
relations besides the standard Miura transformation arise. We use this
correspondence to classify solutions of the KdV hierarchy in terms of
elementary tau functions of the mKdV hierarchy under both zero and nonzero
vacua. We illustrate how interesting nonlinear phenomena can be described
analytically from this construction, such as ``rogue waves'' of a complex KdV
system that corresponds to a limit of a vector nonlinear Schr\"odinger
equation.
