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The study of hyperbolic waves involves various notions which help characterise how these structures evolve. One important facet is the notion of genuine nonlinearity, namely the ability for shocks and rarefactions to form instead of contact discontinuities. In the context of the Whitham modulation equations, this paper demonstrate that a loss of genuine nonlinearity leads to the appearance of a dispersive set of dynamics in the form of the modified Korteweg de-Vries equation governing the evolution of the waves instead. Its form is universal in the sense that its coefficients can be written entirely using linear properties of the underlying waves such as the conservation laws and linear dispersion relation. This insight is applied to two systems of physical interest, one an optical model and the other a stratified hydrodynamics experiment, to demonstrate how it can be used to provide insight into how waves in these systems evolve when genuine nonlinearity is lost.

We investigate nonlinear phase dynamics of an ideal kink mode, induced by E × B flow. Here the phase is the cross phase (θ c) between perturbed stream function of velocity ( ϕ ˜ ) and magnetic field ( ψ ˜ ), i.e. θ c = θ ϕ − θ ψ . A dimensionless parameter, analogous to the Richardson number, R i = 16 γ kink 2 / ω ˆ E 2 (γ kink: the normalized growth rate of the pure kink mode; ω ˆ E : normalized E × B shearing rate) is defined to measure the competition between phase pinning by the current density and phase detuning by the flow shear. When R i > 1, θ c is locked to a fixed value, corresponding to the conventional eigenmode solution. When R i ≤ 1, θ c enters a phase slipping or oscillating state, corresponding to a nonmodal solution. The nonlinear phase dynamics method provides a more intuitive explanation of the complex dynamical behavior of the kink mode in the presence of E × B shear flow.

Driven mean-field threshold systems demonstrate complex observable space-time patterns of behavior that are difficult to understand or predict without knowledge of the underlying dynamics, which are typically unobservable. Here we describe a new method based on phase dynamics techniques to analyze and forecast the space-time patterns of activity in these systems. Application to earthquake data from a typical, seismically active region shows that the method holds considerable promise for forecasting the temporal occurrence of the largest future events. We demonstrate the power of our technique via an application to the difficult problem of earthquake forecasting in southern California.

Transient phase dynamics, synchronization and desynchronization which are stimulus-locked (i.e. tightly time-locked to a repetitively administered stimulus) are studied in two coupled phase oscillators in the presence of noise. The presented method makes it possible to detect such processes in numerical and experimental signals. The time resolution is enormous, since it is only restricted by the sampling rate. Stochastic stimulus locking of the phases or the n:m phase difference at a particular time t relative to stimulus onset is defined by the presence of one or more prominent peaks in the cross-trial distribution of the phases or the n:m phase difference at time t relative to stimulus onset in an ensemble of post-stimulus responses. Unlike the presented approach, both triggered averaging (where an ensemble of post-stimulus responses is simply averaged) and cross-trial cross correlation (CTCC) lead to severe misinterpretations: The oscillators' coupling may cause a transient anti-phase clustering of the post-stimulus responses. Triggered averaging cannot distinguish such a response decorrelation from a mean amplitude decrease of the single responses. CTCC not only depends on the oscillators' phase difference but also on their phases and, thus, inevitably displays "artificial" oscillations that are not related to synchronization or desynchronization.

We discuss the phase dynamics in the system of a set of nonlinear coupled oscillators. We find a direct and simple geometric method to observe the geometric phase for this system, and it provides a possibility for the experimental measurements.

Physico-chemical systems driven away from thermal equilibrium usually undergo various types of instabilities leading to the formation of spatio-temporal patterns on macroscopic time and space scales. In two-dimensional geometries, patterns of different symmetries may be simultaneously stable. We study here the phase stability of hexagonal planforms in the framework of amplitude equations and reduced dynamical models close to the instability points.

Hodgkin-Huxley model neurons coupled by weak excitatory interactions are studied by a phase reduction technique. All the information about the coupling between the neurons and their synchronization is then contained in an effective interaction between their phases. One shows analytically that an excitatory coupling can result in an effective inhibition between the neurons reducing their firing rates. Systems of two neurons exhibit bistability and out-of-phase locking. It is suggested that these features may have significant consequences for networks.

Two parallel stacks of coupled Josephson junctions (JJs) are investigated to clarify the physics of transitions between the rotating and oscillating states and their effect on the IV-characteristics of the system. The detailed study of phase dynamics and bias dependence of the superconducting and diffusion currents allows one to explain all features of simulated IV-characteristics and demonstrate the correspondence in their behavior. The coupling between JJ in the stacks leads to the branching of IV-characteristics and a decrease in the hysteretic region. The crucial role of the diffusion current in the formation of the IV-characteristic of the parallel stacks of coupled JJs is demonstrated. We discuss the effect of symmetry in a number of junctions in the stacks and show a decrease of the branching in the symmetrical stacks. The observed effects might be useful for development of superconducting electronic devices based on intrinsic JJs.

The dynamics of the phase difference of counterpropagating waves is studied theoretically and experimentally in the self-modulation oscillation regime in a solid-state ring laser. It is found that in the case of a small enough frequency nonreciprocity of the ring resonator, the phase difference of counterpropagating waves changes within a limited range, performing periodic oscillations with the intensity self-modulation frequency. The instant frequency difference of counterpropagating waves also changes periodically in time; however, its mean value is zero (the frequency locking for counterpropagating waves takes place). The width of the frequency locking region is measured. It is shown that the phase difference of the coupling coefficients considerably affects the phase dynamics. This opens up new possibilities for determining the phase difference of coupling coefficients of counterpropagating waves.

The ultrafast phase dynamics of carriers in Ge quantum dots (QDs) is studied by using photocurrent correlation phase spectroscopy (PCPS) at room temperature. For better understanding the ultrafast phase dynamics of carriers on the sublevels of Ge QDs, the sublevels in Ge QDs are first investigated by Fourier transform infrared photocurrent spectroscopy, and the result shows that there are two sublevels (0.76 and 0.84 eV) in Ge QDs. The ultrafast phase dynamics signals measured by PCPS are well behaved with a smooth wave line and oscillating trailing edge. The dephasing time has been obtained by numerically simulating the PCPS data in terms of optical Bloch equations. The dephasing time () originating in the transition from the sublevel of the Ge QDs to the conduction band is slightly longer than that () originating in the Si band-to-band transition.

The radiation phase dynamics of a solid-state ring laser with a periodic pump modulation operating in a dynamic chaotic regime is studied theoretically and experimentally. It is found that, in the regime of synchronised dynamic chaos, the optical phases of counterpropagating waves change by {pi} in intervals between two adjacent chaotic radiation pulses. It is shown that spontaneous radiation has a significant effect on the chaotic dynamics of the counterpropagating wave intensities, but its effect on the phase dynamics in a solid-state ring laser is negligible. (laser applications and other topics in quantum electronics)

The phase dynamics of the coupled Josephson junctions in the framework of CCJJ and CCJJ+DC models is studied. The current voltage characteristics (CVC) are numerically calculated for the stacks with different number of junctions at different model parameters. We manifest the difference of these models for the branching at I = Ic and in the hysteretic region. The essential difference is observed in the breakpoint region, where the longitudinal plasma wave is created. We discuss the main features of both models, related with the role of the diffusion current between the superconducting layers.

The peculiarities of the phase dynamics are studied in a ring solid-state laser operating in transient quasi-sinusoidal oscillation regimes of the first and second kinds (QS-1 and QS-2) appearing upon periodic modulation of the pump power. It is shown that recording of a change in the phase difference of counterpropagating waves in the QS-2 regime under certain conditions makes it possible to determine directly the mutual nonreciprocity of the laser resonator.

The effect of synchronization has been studied in a system of two coupled Van der Pol oscillators under external harmonic force. The analysis has been carried out using the phase approach. The mechanisms of complete and partial synchronization have been established. The main type of bifurcation described in this paper is the saddle-node bifurcation of invariant curves that corresponds to the saddle-node bifurcation of two-dimensional tori in the complete system of differential equations for the dynamical system under study.

Edge transport bifurcation induced by external biasing is explored with self-consistent turbulence simulations in a flux-driven system with both closed and open magnetic field lines. Without bias, the nonlinear evolution of interchange turbulence produces large-scale turbulent eddies, leading to the high levels of radial transport in the edge region. With sufficiently strong biasing, a strong suppression of turbulence is found. The plasma potential structures are strongly modified with the generation of sheared mean flows at the plasma edge. Consequently, the turbulence-driven flux is decreased to a much lower level, indicating a transition to a state of reduced transport. The simulations show that the dynamics of the phase and amplitude of fluctuations play a crucial role in the mechanism of transport suppression driven by biasing.

The high-velocity oxygen fuel (HVOF) thermal spray is used for a particulate deposition process in which micro-sized particles are propelled and heated in a supersonic combusting gas stream. It is characterized by high gas velocity and high density and is being used in an increasing variety of coating applications, such as ceramic and composite coatings, to improve wear and abrasion resistance. The particle temperature and velocity are two of the most important parameters in HVOF thermal spraying, which affect the quality of the coatings. To understand the particle dynamics, it is necessary to study, first, the thermal flow characteristics in the HVOF system. In this study, a numerical analysis is performed to predict the gas dynamic behaviors, and the effect of the geometrical parameter is studied to optimize the nozzle design.

We study the phase dynamics of turbulent Taylor vortex flow by applying a forced modulation to the upper boundary of a large-aspect-ratio system of concentric cylinders. The experiment shows that the axial phase variable of the turbulent Taylor vortex flow is a deterministic variable and its dynamics can be described by a simple diffusion model. The resultant diffusion coefficient of the turbulent Taylor vortex flow is about an order of magnitude larger than that of laminar/regular Taylor vortex flow. We argue that the approach of phase dynamics can also be applied to turbulent flows as long as there is, on sufficiently long-time scales, a coherent structure discernable in the system.

We study the spatial stability of systems at onset of the temporal intermittent transition to chaos. We show that the phase dynamics during laminar episodes, described as orbits near a saddle-node bifurcation of limit cycles, is second order in time and can lead to propagative behaviour.

Using the approach of low-energy effective field theory, the phase diagram is studied for a mixture of two species of pseudospin-$\1/2$ Bose atoms with interspecies spin-exchange. There are four mean-field regimes on the parameter plane of $g_e$ and $g_z$, where $g_e$ is the interspecies spin-exchange interaction strength, while $g_z$ is the difference between the interaction strength of interspecies scattering without spin-exchange of equal spins and that of unequal spins. Two regimes, with $|g_z| > |g_e|$, correspond to ground states with the total spins of the two species parallel or antiparallel along $z$ direction, and the low energy excitations are equivalent to those of two-component spinless Bosons. The other two regimes, with $|g_e| > |g_z|$, correspond to ground states with the total spins of the two species parallel or antiparallel on $xy$ plane, and the low energy excitations are described by a sine-Gordon model coupled with a free scalar field, where the effective fields are combinations of the phases of the original four Boson fields. In (1+1)-dimension, they are described by Kosterlitz-Thouless renormalization group (RG) equations, and there are three sectors in the phase plane of a scaling dimension and a dimensionless parameter proportional to the strength of the cosine interaction, both depending on the densities. The gaps of these elementary excitations are experimental probes of the underlying many-body ground states., 13 pages

We probe the physical mechanism behind the known phenomenon of power synchronization of two diode lasers that are mutually coupled via their delayed optical fields. In a diode laser, the amplitude and the phase of the optical field are coupled by the so-called linewidth enhancement factor, $\alpha$. In this work, we explore the role of optical phases of the electric fields in amplitude (and hence power) synchronization through $\alpha$ in such mutually delay-coupled diode laser systems. Our numerical results show that the synchronization of optical phases drives the powers of lasers to synchronized death regimes. We also find that as $\alpha$ varies for different diode lasers, the system goes through a sequence of in-phase amplitude-death states. Within the windows between successive amplitude-death regions, the cross-correlation between the field amplitudes exhibits a universal power-law behaviour with respect to $\alpha$., Comment: J. Phys. B: At. Mol. Opt. Phys. 44 (2011)

We study quantum phase fluctuations in planar junctions between high-Tc superconductors for a range of relative orientations of the a-b crystal axes. Specifically, attention is directed towards the fermionic subspace of low energy Andreev bound states, midgap states (MGS). This subspace, being responsible for most of the Josephson current, strongly couples to the phase degree of freedom. Low frequency phase fluctuations may introduce intrinsic dynamics in this subspace, which would lead to a strong dynamical modulation of the Josephson current, and thus affect the phase quantum dynamics. In this article it is found that transitions between pairs of MGS induced by the phase fluctuations are forbidden due to specific symmetry properties of the MGS, and the macroscopic quantum Hamiltonian remains essentially adiabatic within the assumption of a specular interface.

We propose a device based on a Q-switched self-sustained oscillator with two nonlinear delayed feedback loops. Due to the appropriate phase transformation of the signal that influences the generation of each successive pulse, the phase difference between the two neighboring pulses evolves according to the Bernoulli doubling map. It corresponds to a hyperbolic chaotic attractor yielding a robust, structurally stable chaos. We discuss possible experimental implementations of the scheme.