Solutions and linearization of the nonlinear dynamics of radiation damping

Abstract

The techniques of Painleve analysis and Lie algebra analysis were applied to the nonlinear Bloch equations with radiation damping. Painleve analysis is useful in finding when explicit solutions exist to a nonlinear system. It was applied to the radiation-damped system with damping time Tr, and with T1 and T2 relaxation, but with no externally applied radiofrequency (RF) pulse. Two cases were identified where explicit solutions could be found. The first case ( 1/T1:0) is well known, the second case ( 1/T1=1/Tγ+1/T2) is apparently not previously known. Lie algebra analysis was used to show that the system with no relaxation, but with an externally applied RF pulse, could be transformed into a linear system. This simplifies the forward problem of finding the magnetization response to a given pulse. It also allows the inverse problem to be solved, where the pulse is calculated to result in a given magnetization response as functions of both resonance offset and radiation damping time.