Abstract
We describe a broad new class of exact solutions of the KdV hierarchy. In general, these solutions do not vanish at infinity, and are neither periodic nor quasi-periodic. This class includes algebro-geometric finite-gap solutions as a particular case. The spectra of the corresponding Schrodinger operators have the same structure as those of N-gap periodic potentials, except that the reflectionless property holds only in the infinite band. These potentials are given, in a non-unique way, by 2N real positive functions defined on the allowed bands. In this letter we restrict ourselves to potentials with one allowed band on the negative semi-axis; however, our results apply in general. We support our results with numerical calculations.
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