Solutions to the Time Dependent Schrödinger and the Kadomtsev-Petviashvili Equations

A method to obtain a new class of discrete eigenfunctions and associated real, nonsingular, decaying, ``reflectionless'' potentials to the time dependent Schr\"odinger equation is presented. Using the inverse scattering transform, related solutions of the Kadomtsev-Petviashvili equation are found. The eigenfunctions have poles of order $m$, $mg1$ in the complex plane and are also characterized by an index, or ``charge,'' which is obtained as a constraint in the theory.