Exact half pulse synthesis via the inverse scattering transform

In a paper of Nielson et al. it is shown, using the linear approximation, that it might be possible to create a pair of RF-pulses, which, after summation of the unrephased signals achieve a specified transverse magnetization. Such pulses, designed using the linear approximation, show rather poor slice selectivity. Using the inverse scattering transform formalism we give an algorithm to exactly achieve a specified "summed" transverse magnetization profile. Indeed for a constant phase transverse profile, our algorithm produces infinitely many solutions.