Recent experiments performed in current-driven nano-contacts with strong perpendicular anisotropy have shown that spin-transfer torque can drive self-localized spin waves [1, 2] that above a certain threshold intensity can condense into a highly nonlinear magnetodynamic and nano-sized state known as a magnetic droplet soliton [3]. Here we demonstrate analytically, numerically, and experimentally that at sufficiently large driving currents, and for a spin polarization that is tilted away from the film normal, the circular droplet soliton can become unstable to periodic excitations of its perimeter. We furthermore show that these perimeter excitation modes (PEMs) are parametrically excited when the fundamental droplet soliton precession frequency is close to twice the frequency of one or more of the PEMs. As a consequence, for increasing applied fields, progressively higher PEMs can be excited. Quantitative agreement with experiment confirms this picture.
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