Symmetry-breaking transitions in quiescent and moving solitons in fractional couplers

We consider phase transitions, in the form of spontaneous symmetry breaking (SSB) bifurcations of solitons, in dual-core couplers with fractional diffraction and cubic self-focusing acting in each core, characterized by Levy index $\alpha$. The system represents linearly-coupled optical waveguides with the fractional paraxial diffraction or group-velocity dispersion (the latter system was used in a recent experiment, which demonstrated the first observation of the wave propagation in an effectively fractional setup). By dint of numerical computations and variational approximation (VA), we identify the SSB in the fractional coupler as the bifurcation of the subcritical type (i.e., the symmetry-breaking phase transition of the first kind), whose subcriticality becomes stronger with the increase of fractionality $2 - \alpha$, in comparison with very weak subcriticality in the case of the non-fractional diffraction, $\alpha = 2$. In the Cauchy limit of $\alpha = 1$, it carries over into the extreme subcritical bifurcation, manifesting backward-going branches of asymmetric solitons which never turn forward. The analysis of the SSB bifurcation is extended for moving (tilted) solitons, which is a nontrivial problem because the fractional diffraction does not admit Galilean invariance. Collisions between moving solitons are studied too, featuring a two-soliton symmetry-breaking effect and merger of the solitons.