superpose them as before to form the azimuthon-like object $$U(x,y,z=0)=A\left(D_1+iBD_2\right).
\label{squareansatz}$$ The field Eq. (\[squareansatz\]) is then used as an initial condition to the nonlinear Schr[ö]{}dinger equation Eq. (\[partial\]). However, in contrast to the circular waveguide, the orientation of the dipoles $D_1$ and $D_2$ is important in nonlinear regime. If the two orthogonal dipoles $D_1$ and $D_2$ are oriented along the diagonals of the waveguide cross-section (see first subplot of Fig. \[square\_r\_evolution\]), for a given amplitude $A$ rotation occurs only if the modulation parameter $B$ exceeds a critical value $B=B_{\rm cr}$. Moreover, the rotation is no longer constant as in the case of cylindrical waveguide but fluctuates and hence in what follows we use its average value (termed “average frequency”) $\bar \omega=1/L\int_0^L\omega(z)~\mathrm{d}z$ with $L$ being the propagation distance corresponding to one full $2\pi$ rotation. This threshold value $B_{\rm cr}$ decreases if we choose different initial dipole orientations [^1], and appears to be zero (within our numerical accuracy) for parallel orientation with respect to the waveguide boundaries.

![The propagation of the azimuthon-like dipole, that rotates in the square waveguide. Top row: intensity (left) and phase (right) distribution corresponding to the input and soliton rotation by $\pi/4$, respectively. The white line indicates 