Fig. 1. (a) Basic modes, (b) degenerate dipole modes, (c) degenerate quadrupole modes, (d) degenerate hexapole modes, and (e) degenerate octopole modes. First row: first-order modes, second row: second-order modes, and third row: third-order modes. The panels are shown in the window −2≤x≤2 and −2≤y≤2. Other parameters: V0=500 and w=1.

Fig. 2. Amplitude and phase of (a)–(d) the first-order and (e)–(h) the second-order azimuthons constructed from (a) and (e) the degenerate dipoles, (b) and (f) the quadrupoles, (c) and (g) the hexapoles, and (d) and (h) the octopoles. The panels are shown in the window −2≤x≤2 and −2≤y≤2. Other parameters: A=0.4 and B=0.5.

Fig. 3. Rabi transition of (a) a dipole and (b) a hexapole. In each case, the propagation is shown by the isosurface plot above which amplitude distributions at selected distances are shown. In both cases, the weak longitudinally periodic modulation exists in the region 30≤z≤90, with d≈25.2 and μ≈0.031 in (a) and d≈36.36 and μ≈0.014 in (b). (c) The Rabi oscillation period zR versus the frequency detuning ℓ.

Fig. 4. (a) Rabi transition of a deformed dipole. (b) The amplitude and phase of the azimuthon based on the dipole in (a). (c) The transition of a deformed hexapole. (d) The amplitude and phase of the azimuthon based on the hexapole in (c). (e) The transition of a deformed higher-order dipole. (f) The amplitude and phase of the azimuthon based on the higher-order dipole in (e). The panels are shown in the window −2≤x≤2 and −2≤y≤2. Other parameters: A=0.4 and B=0.5.

Fig. 5. (a) Transition between dipole and hexapole azimuthons with d≈25.2 and μ≈0.085. (b) Transition between dipole and hexapole azimuthons with d≈22.1 and μ≈0.034. The weak longitudinally periodic modulation has to always exist during propagation.