## 3 Results
### Eigenmode of Isolated Elliptical Nanodots  
Let us now consider the more complicated case of elliptical nanodots, which, different from the wires discussed above, are laterally confined in both in-plane directions. For such case, we adopt the classification and the labelling scheme of the eigenmodes of magnetic nanodots that have been described in a recent review paper [30]: each mode is labelled with two integer indices (\(n_{x}\),\(n_{y}\)) whose values correspond to the number of nodal lines perpendicular or parallel to the direction of the magnetization (x axis), respectively. In Figure 3 one can see the results of our simulations relative to the smallest dots considered in this study, having lateral dimensions 100 \(\times\) 50 nm\({}^{2}\). The central panel shows the evolution of the mode frequencies as a function of \(D\), while in the left- and right-hand panels one finds the intensity spectra where each peak corresponds to the mode whose bidimensional spatial profile is shown in color scale. In this first approach we limit our analysis to the range of \(D\) between 0 and 2 mJ/m\({}^{2}\), i.e., a range of values that are easily accessible in real samples [19]. In absence of DMI, the (0,0) mode, characterized by the absence of any nodal line, is by far the most intense one. Therefore, it is called the \"fundamental\" mode, although magnetostatic effects appear in the fact that the mode amplitude is mainly localized at the edges, where the demagnetizing field is relatively strong and the dynamic amplitude is larger than elsewhere. Similar to the case of the stripes, the odd mode (0,1) is completely absent in the spectra calculated for \(D=0\), but it starts to appear for finite values of \(D\) and becomes very intense, i.e., comparable with the fundamental mode (0,0), for \(D=2\) mJ/m\({}^{2}\)Again, the reason for this marked modification of the intensity of different modes in the spectra is the fact that the presence of DMI leads to a progressive loose of stationary character along the y direction and the appearance of a finite net dynamical magnetic moment for odd modes. This can be appreciated also in the bottom panel of Figure 3, where we show the temporal evolution of the mode (2,0) at different times. For \(D=0\) it exhibits a clear stationary character, with two nodal lines perpendicular to the static magnetization. Instead for \(D=1\) and \(2\) mJ/m\({}^{2}\), it is visible a continuous shift of the profile along the y direction, i.e., a propagative character along y. Animations where one can follow in detail the evolution of the spatial profile of the various modes as a function of time, for different intensity of the DMI, are available in the Supplemental Material. Please note that, similarly to the case of the wires, in the case of elliptical dots, the antisymmetric mode (0,1) also exhibits a larger slope than the fundamental mode (0,0), and further on, in this same section, we will discuss its softening for large values of D.  
Let us now consider how the characteristics of the mode spectrum change when the lateral dimensions of the magnetic dot increase. In Figures 4 and 5 we report the results  
Figure 3: Top central panel: dependence of the frequencies of the spin wave eigenmodes on the DMI constant \(D\) for the elliptical dot of lateral dimensions 100 \(\times\) 50 nm\({}^{2}\). On the left- and right-hand sides, the intensity spectra corresponding to \(D=0\) and \(2\) mJ/m\({}^{2}\) are reported, respectively, together with the color insets that represent the spatial profile of the dynamical magnetization, expressed as the product of the modulus of the dynamical magnetization by the sign of its phase. Bottom panel: spatial profile of the mode (2,0) plotted at different times during the period T, for three different values of \(D\). The animations with the movies of the dynamics of all the modes for several values of \(D\) are available in the Supplemental Material. relative to dots of lateral dimensions 200 \(\times\) 100 nm\({}^{2}\) and 300 \(\times\) 150 nm\({}^{2}\), respectively.