One of the most fascinating topics in current quantum physics are hybridised systems, in which different quantum resonators are strongly coupled. Prominent examples are circular resonators with high quality factors that allow the coupling of optical whispering gallery modes 1-5 to microwave cavities 6 or magnon resonances in optomagnonics 7-9 . Whispering gallery modes play a special role in this endeavour because of their high quality factor and strong localisation, which ultimately increases the overlap of the wavefunctions of quantum particles in hybridised systems. The hybridisation with magnons, the collective quantum excitations of the electron spins in a magnetically ordered material, is of particular interest because magnons can take over two functionalities: due to their collective nature they are robust and can serve as a quantum memory 10 and, moreover, they can act as a wavelength converter between microwave and THz photons 9 . However, the observation of whispering gallery magnons has not yet been achieved due to the lack of efficient excitation schemes for magnons with large wave vectors in a circular geometry. To tackle this problem, we studied nonlinear 3-magnon scattering 11-14 as a means to generate whispering gallery magnons. This Letter discusses the basics of this non-linear mechanism in a confined, circular geometry from experimental and theoretical point of view.Whispering gallery magnons can only live in systems with rotational symmetry. This not only applies to the geometry of the magnetic element but also to the magnetisation texture therein. For that reason, we study a Ni 81 Fe 19 disc that inherently exhibits a magnetic vortex structure [15][16][17][18][19] . The arrows in Fig. 1a depict the generic features of such a vortex state: the magnetic moments curl in plane along circular lines around the vortex core, a nanoscopic region in the center of the disc where the magnetisation tilts out of plane. According to this rotational symmetry, the magnon eigenmodes in a vortex are characterised by mode numbers (n, m), with n = 0, 1, 2, ... counting the number of nodes across the disc radius and m = 0, ±1, ±2, ... counting the number of nodes in azimuthal direction over half the disc 20,21 .Other than commonly known waves, like sound, water or electromagnetic waves, magnons exhibit a strongly anisotropic dispersion relation in in-plane magnetised thin films 20 . In a vortex, this results in increasing (decreasing) mode energies for increasing n (m) as shown by the analytic calculations in Fig. 1b. The four exemplary intensity profiles for the eigenmodes (0, 0), (0, 10), (0, 20), and (0, 30), that are shown in Fig. 1c, reveal the character of whispering gallery magnons: the larger m, the more the magnon intensity is pushed toward the perimeter of the disc which can be understood intuitively by the reduction of exchange energy: Leaving an extended area around the vortex core with zero amplitude avoids a strong tilt of neighbouring spins close to the vortex core and, therefore, reduces the total ...

In the last decade, two revolutionary concepts in nano magnetism emerged from research for storage technologies and advanced information processing. The first suggests the use of magnetic domain walls (DWs) in ferromagnetic nanowires to permanently store information in DW racetrack memories 1 . The second proposes a hardware realisation of neuromorphic computing in nanomagnets using nonlinear magnetic oscillations in the GHz range 2,3 . Both ideas originate from the transfer of angular momentum from conduction electrons to localised spins in ferromagnets 4,5 , either to push data encoded in DWs along nanowires or to sustain magnetic oscillations in artificial neurones. Even though both concepts share a common ground, they live on very different time scales which rendered them incompatible so far. Here, we bridge both ideas by demonstrating the excitation of magnetic autooscillations inside nano-scale DWs using pure spin currents.The spin-tranfer-torque (STT) effect discovered in 1996 by Slonczewski and Berger 4,5 allows the manipulation of localised magnetic moments in a ferromagnet by the transfer of spin angular momentum from spin polarised conduction electrons. The direction of the magnetisation can either be switched permanently 6 or can be forced to oscillate at radio frequencies 7 . Quite soon thereafter it was recognised that a charge current in a ferromagnet, which is intrinsically spin polarised, can move magnetic domain walls (DWs) in nanowires 8 . This gave rise to the idea of the magnetic racetrack memory 1 and, quite recently, current induced skyrmion motion 9-11 . While these schemes target nonvolatile, long term data storage, the STT effect in spin-torque nano-oscillators can be exploited to drive magnetic auto-oscillations 12 and, eventually, to radiate spin waves 13 by compensating the intrinsic magnetic damping. Another leap was the development of spin-Hall nano-oscillators (SHNO) 26,27 in which pure spin currents are generated via the spin-Hall effect (SHE) 14-17 and by which even propagating spin waves are excited 18,19 . This puts SHNOs at the heart of magnonics 20-22 which proposes a novel type of low energy, non boolean computing based on magnons, the quanta of spin waves, as carriers of information 23 or even neuromorphic computing 3 based on the nonlinear character of magnonic auto-oscillations. In a previous work, we demonstrated that DWs can channel magnons in an effective magnetic potential well 24 . This raised the question if a magnetic DW can potentially be a self-organized, movable SHNO.In order for the STT effect to counteract the intrinsic magnetic damping, the magnetisation M has to have a component antiparallel to the polarisation P of the injected spin current I s with density J s . Due to spindependent scattering associated with the SHE, a charge current I c with density J c flowing in a heavy metal is converted into a transverse pure spin current with J s ⊥ J c ⊥ P (Fig. 1a). In a conventional nanowire, therefore, an exa b 4 µ m C r ( 1 . 5 ) / P t ( 7 ) / C o 4 0

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