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At equilibrium, Bloch walls are chiral interfaces between domains with different magnetization. Far from equilibrium, a set of forced oscillators can exhibit walls between states with different phases. In this Letter, we show that when these walls become chiral, they move with a velocity simply related to their chirality. This surprising behavior is a straightforward consequence of nonvariational effects, which are typical of nonequilibrium systems.PACS numbers: 47.10.+g, 05.45.+b, 47.20.Ky Domain walls exhibit phase transitions, where Ising walls become Bloch walls,' and which have been related to spontaneous breaking of chirality, 2,3 in the frame of an anisotropic X-Y model. In this Letter, we investigate the nonequilibrium analog of these transitions. Namely, we consider the parametric forcing of an assembly of self-oscillators. In such a system, "phase anisotropy," induced by the forcing, provides a natural counterpart to crystalline anisotropy which allows the existence of walls. We show that spontaneous breaking of chirality is accompanied by the motion of chiral interfaces. A simple analysis allows us to describe this behavior. Since this motion is due to nonvariational effects, it is generic to nonequilibrium systems. The occurrence of such effects has been recently emphasized, 4 " 8 and here is another manifestation of these phenomena. In conclusion, we suggest an experiment where these phenomena are likely to be observed.We consider a one-dimensional system which undergoes a spatially homogeneous Hopf bifurcation, 9 with temporal frequency coo. Such a system can be seen as a continuous assembly of oscillators, which auto-oscillate above the Hopf bifurcation threshold. Now, we parametrically force this system at twice its natural frequency too. Let A be the complex order parameter which measures the amplitude of oscillations. It is assumed to vary slowly in space and time and obeys the following Ginzburg-Landau equation:where p measures the distance from the oscillatory instability threshold, v is the detuning parameter, V stands for d/dx, and y > 0 is the forcing amplitude.When the real parameters v, a, and j3 vanish, Eq. (1) can be cast into a variational form:bt 8Awhere 7 -J {-p CY 2 + v 2 ) + \vx\ 2 + |vr| 2 + HX 2 +Y 2 ) 2 -r (X 2 -Y 2 )}dx and A =X + iY. The quantity J turns out to be the free energy of the X-Y model in the presence of weak anisotropy whose amplitude is controlled by 7. In this limit, Eq.(1) exhibits spontaneous breaking of chirality. 2 Namely, Eq. (2) possesses stationary kinklike solutions 1,10 which connect stable homogeneous solutions (X=± VAI + 7, F=0). They read X I = ±Jp + ytanh{[{(p + y)] {/2 x}, r,=0, A-fl = ±v^T7tanh(V27x), Y B -±

We study the existence, the stability properties, and the bifurcation structure of static localized solutions in one dimension, near the robust existence of stable fronts between homogeneous solutions and periodic patterns.

Surface waves on a stationary flow of water are considered, in a linear model that includes the surface tension of the fluid. The resulting gravity-capillary waves experience a rich array of horizon effects when propagating against the flow. In some cases three horizons (points where the group velocity of the wave reverses) exist for waves with a single laboratory frequency. Some of these effects are familiar in fluid mechanics under the name of wave blocking, but other aspects, in particular waves with negative co-moving frequency and the Hawking effect, were overlooked until surface waves were investigated as examples of analogue gravity [Schützhold R and Unruh W G 2002 Phys. Rev. D 66 044019]. A comprehensive presentation of the various horizon effects for gravity-capillary waves is given, with emphasis on the deep water/short wavelength case kh 1 where many analytical results can be derived. A similarity of the state space of the waves to that of a thermodynamic system is pointed out.