HALF-SOLITONS AS SOLUTIONS TO THE ZAKHAROV-SHABAT EIGENVALUE PROBLEM FOR RATIONAL REFLECTION COEFFICIENT .2. POTENTIALS ON INFINITE SUPPORT

It is shown how to invert the Zakharov-Shabat eigenvalue problem for r ational scattering coefficients, in order to produce potentials define d over the whole real line. The method reduces the problem to finding two semi-infinite potentials-which can be efficiently calculated using the soliton-lattice algorithm described in a previous paper. The inve rsion is usually unique for given rational scattering coefficients, si nce they usually specify the scattering data of the system uniquely. I n the case of nonunique inversion, it is possible to obtain a family o f potentials from the inversion. Each member of the family has the sam e scattering data, except for differing residues. Examples of the inve rsion include an 'adiabatic' inversion pulse for use in nuclear magnet ic resonance, and a demonstration of how the cubic nonlinear Schroding er equation may be integrated.