of the Hamiltonian of the parallel dipole $\sim D_p$ at the same power level. As the red curve monotonically increases with modulation parameter $B$, the intersection of the two curves gives an estimate of the critical value of the azimuthon modulation $B_{\rm cr}$, which is shown in the right panel in Fig. \[squarerelations\] by a black square. In other words, in order to rotate from the diagonal to vertical position the azimuthon must overcome some kind of energy barrier represented by the difference of the values of the Hamiltonian in those two basic states. As Fig. \[angularmomentumcurve\] shows this can be achieved by increasing the value of $B$ in the diagonal state to $B_1=0.342$, which is very close to the value found numerically (see right panel in Fig. \[squarerelations\]). Similar phenomena have been identified, for instance, in discrete nonlinear systems where the discreteness introduces the so called Peierls-Nabarro potential which has to be overcome by a soliton to become mobile [@Kivshar_Campbell].

![ Left panel: Hamiltonian of $A(D_1+iBD_2)$ (red curve) as a function of $B$ and Hamiltonian of $D_3$ (blue line) (for the same power $P$). The intersection of the two curves defines $B_{\rm cr}$ (depicted by the black square). Right 