Integrable models for the dynamics of a longitudinal-transverse acoustic wave in a crystal with paramagnetic impurities

We theoretically study the evolution of longitudinal-transverse acoustic pulses propagating parallel to an external magnetic field in a system of resonant paramagnetic impurities with an effective spin S=1/2. For equal group velocities of the longitudinal and transverse waves, the pulse dynamics is shown to be described by evolution equations. In limiting cases, these equations reduce to equations integrable in terms of the inverse scattering transform method (ISTM). For the most general integrable system of equations that describes the dynamics of acoustic pulses outside the scope of the slow-envelope approximation, we derive the corresponding ISTM equations. These equations are used to find a soliton solution and a self-similar solution. The latter describes the leading edge of the packet of acoustic pulses generated when the initial unstable state of a spin system decays. Analysis of our solutions and models indicates that the presence of a longitudinal acoustic wave leads not only to a change in the amplitude and phase of the transverse wave but also to a qualitatively new dynamics of sound in such a medium.