Half solitons as solutions to the Zakharov-Shabat eigenvalue problem for rational reflection coefficient with application in the design of selective pulses in nuclear magnetic resonance

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Abstract
It is shown how the Zakharov-Shabat (ZS) eigenvalue problem for rational reflection coefficient may be reduced to the ZS problem with zero reflection coefficient. The soliton solutions to this reduced problem are obtained using the B\"acklund transform. Hence the solutions to the original problem are shown to be half solitons. It is demonstrated how selective pulses in nuclear magnetic resonance may be calculated using this technique. In particular, almost perfect 90^\circ\else\textdegree\fi{} self-refocused and 180^\circ\else\textdegree\fi{} refocusing selective pulses are demonstrated.