## 1 Introduction  
The study of chiral magnetism is attracting more and more attention in recent years, thanks to the discovery of the interfacial Dzyaloshinskii-Moriya interaction (DMI) [1, 2] in magnetic thin films and multilayers. This interaction is an antisymmetric exchange coupling between two nearest spin electrons mediated by an heavy metal under- or overlayer [3, 4, 5]. For future applications, an accurate measurement of its strength is indispensable, so that different methods have been developed to evaluate it, such as the study of the domain walls motion [6, 7, 8, 9] and the detection of spin waves (SW) by Brillouin light scattering (BLS) [10, 11, 12, 13, 14, 15, 16, 17]. Several investigations have been published in the last seven years concerning plain films and multilayers [18, 19], while there are only a few investigations concerning patterned nanostructures [20, 21, 22, 23, 24, 25]. In particular, it is important to achieve a detailed understanding of the modifications of the dynamics of isolated magnetic nanoelements, which are usually realized by e-beam lithography and can be exploited in magnetic memories [26].  
In a pioneering micromagnetic study, [PERSON] et al. [21] showed theoretically that Neel domain walls driven by the DMI can modify spin-wave propagation by an inducing nonreciprocal channeling along the center of the wall. Such channeling also occurs at the edges of perpendicularly magnetized wires and dots, where partial walls appear as a result of twisted boundary conditions.  
Subsequently, [PERSON] et al. [22] exploited both the frequency-domain method and micromagnetic simulations to investigate the impact of DMI on the ferromagnetic resonance (FMR) spectrum of isolated stripes magnetized in-plane, where DMI causes a red-shift of some peaks in the simulated FMR spectrum, as well as the appearance of new peaks.  
More recently, the first theoretical investigation of nanodots magnetized in-plane was performed by [PERSON] et al. [23] who proposed a model for quantized traveling waves in a confined geometry for the case of nonreciprocal propagation wherein waves at a given frequency have wavelengths that depend upon the direction of travel. They put in evidence that confinement in a nonreciprocal energy landscape produces a fixed nodal structure with wave amplitudes modulated in time and a phase velocity determined by the difference in wave- lengths between opposite propagation directions.  
In this work we aim at deepening the analysis of the effect of interfacial DMI on nanomagnetic elements, considering both isolated stripes of width 100 nm and elliptical nanodots with lateral dimensions 100 \(\times\) 50, 200 \(\times\) 100 and 300 \(\times\) 150 nm\({}^{2}\). All the nanostructures are magnetized in-plane along the easy direction and have a thickness of 1.5 nm. We systematically vary the DMI strength \(D\) between 0 and 4 mJ/m\({}^{2}\), analyzing the variation of both the eigenmodes spectrum and the spatial profile of the excited eigenmodes.  
Please note that the results that we find in this paper correspond to what is expected in a possible experiment of ferromagnetic resonance o of Brillouin light scattering (at perpendicular incidence) on real magnetic wires or dots with sizeable DMI, which are still lacking in the literature.