Theory for the Spatiotemporal Dynamics of Domain Walls close to a Nonequilibrium Ising-Bloch Transition

Abstract

We derive a generic model for the interaction of domain walls close to a nonequilibrium-Bloch transition. The universal scenario predicted by the model includes stationary Ising and Bloch localized structures (dissipative solitons), as well as drifting and oscillating Bloch structures. Our theory also explains the behavior of Bloch walls during a collision. The results are confirmed by numerical simulations of the Ginzburg-Landau equation forced at twice its natural frequency and are in agreement with previous observations in several physical systems.

Localized structures in dispersive and doubly resonant optical parametric oscillators

“…In addition to the type-I LSs, a large variety of type-II LSs also formed through locking of DWs connecting the equivalent states −A + with A + . These states exist for a wider range of parameters in region II and III b and may undergo nonequilibrium Ising-Bloch transitions [58], resulting in complex dynamics [59]. We have shown that in region III b , every type-II LS becomes a hybrid state composed of two type-I LSs related by the symmetry A → −A and are separated by L/2.…”

“…For high values of ρ, the DWs may undergo nonequilibrium Ising-Bloch transitions [58], resulting in the drifting of LSs, domain oscillations, and complex dynamics that were studied in detail in Ref. [59]. Fig.…”

“…The LSs resulting from this interaction are referred to as type-II LSs, and have been largely studied in the context of diffractive cavities [22,23]. Domain walls of this type may undergo nonequilibrium Ising-Bloch transition, where DWs start to drift [58], and as result LSs may show very complex dynamics [59]. An example of such a state is shown in Fig.…”

Two-dimensional localized structures in harmonically forced oscillatory systems

“…When d = 1,ũ (j) (r = 0) vanishes for j odd and scales as u (j) (r = 0) ∼ exp (−λ r ρ) sin (λ i ρ + φ) (A. 13) for j even, where φ is a constant. Since H(r = 0) − H(u − ) is a quadratic form inũ (j) (r = 0), the higher order correction to M due to the oscillatory tail scales as…”

“…This PDE possesses an up-down symmetry A → −A and exhibits a parameter regime where the two "equivalent" equilibria ±A 0 related by this symmetry are simultaneously stable. In 1D a codimension-zero family of steady Ising fronts between ±A 0 is generically present, and may undergo a pitchfork bifurcation into traveling Bloch fronts [4]; near this Ising-Bloch transition the domain walls can exhibit rich spatiotemporal dynamics [13]. In Ref.…”