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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 examine the impact of small parity-time () symmetric perturbations on nonlinear optical honeycomb lattices in the tight-binding limit. We show for strained lattices that complex dispersion relationships do not form under perturbation, and we find a variety of nonlinear wave equations which describe the effective dynamics in this regime. The existence of semilocalized gap solitons in this case is also shown, though we numerically demonstrate these solitons are likely unstable. We show for unstrained lattices under the effect of a restricted class of perturbations, which prevent complex dispersion relationships from appearing, that nontrivial phase dynamics emerge as a result of the perturbation. This phase can be understood as momentum imparted to optical beams by the lattice, thus showing perturbations offer potentially novel means for the control of light in honeycomb lattices.

We formulate a theory for the collective phase description of globally coupled noisy limit-cycle oscillators exhibiting macroscopic rhythms. Collective phase equations describing such macroscopic rhythms are derived by means of a two-step phase reduction. The collective phase sensitivity and collective phase coupling functions, which quantitatively characterize the macroscopic rhythms, are illustrated using three representative models of limit-cycle oscillators. As an important result of the theory, we demonstrate noise-induced anti-phase synchronization between macroscopic rhythms by direct numerical simulations of the three models., Comment: 19 pages, 11 figures

Based on the model system undergoing phase separation and chemical reactions, we investigate the dynamics of propagating dissipative waves under external forcing which is periodic both in space and time. A phase diagram for the entrained and non-entrained states under the external forcing is obtained numerically. Theoretical analysis in terms of phase description of the traveling waves is carried out to show that the transition between the entrained and the non-entrained states by changing the external frequency occurs either through a saddle-node bifurcation or through a Hopf bifurcation and that these two bifurcation lines are connected at a Bogdanov-Takens bifurcation point.

A vector equation of motion in the Lagrangian frame of reference is rearranged into a set of two scalar equations to determine the relative-velocity magnitude of a particle and its direction. The resulting system, written in terms of the Lame coefficients, is solved jointly with continuum equations of motion in the Eulerian frame of reference. The method is exemplified by the flow of a heterogeneous mixture between permeable parallel walls and the flow of a heterogeneous non-Newtonian fluid over the surface of a sedimentation centrifuge rotor.

New types of stable discrete solitons are discovered. They represent the first example of asymmetric dark solitons and shock waves with a nonzero background. Both types of solutions exhibit a strong intrinsic phase dynamics. Their domains of existence and criteria of stability are identified. Numerical experiments support the analytical findings.

The theory of coupled phase oscillators provides a framework to understand the emergent properties of networks of neuronal oscillators. When the architecture of the network is dominated by short-range connections, the pattern of electrical output is predicted to correspond to traveling plane and rotating waves, in addition to synchronized output. We argue that this theory provides the foundation for understanding the traveling electrical waves that are observed across olfactory, visual, and visuomotor areas of cortex in a variety of species. The waves are typically present during periods outside of stimulation, while synchronous activity typically dominates in the presence of a strong stimulus. We suggest that the continuum of phase shifts during epochs with traveling waves provides a means to scan the incoming sensory stream for novel features. Experiments to test our theoretical approach are presented.

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.

The idea of phase dynamics originally considered for regular non-equilibrium patterns is applied to irregular patterns. In the dissipation-dominated regime, a slow modulation of irregular pattern is shown to decay in time according to the diffusion law. Possible relevance of this result for the ultra-slow diffusion modes observed in glass-forming systems is discussed. In this connection we speculate about a possibility of appearance of some sorts of soft structures resembling quasi-periodic microcrystalline states in glass-forming systems.

We carry out a thorough analysis on a class of cosmological spacetimes which admit three space-like Killing vectors of Bianchi class B and contain electromagnetic fields. Using dynamical system analysis, we show that a family of vacuum plane-wave solutions of the Einstein-Maxwell equations is the stable attractor for expanding universes. Phase dynamics are investigated in detail for particular symmetric models. We integrate the system exactly for some special cases to confirm the qualitative features. Some of the obtained solutions have not been presented previously to the best of our knowledge. Finally, based on those solutions, we discuss the relation between those homogeneous models and perturbations of open FLRW universes. We argue that the vacuum plane-wave modes correspond to a certain long-wavelength limit of electromagnetic perturbations., Revised for publication. 25 pages

We consider a three-component model of an active medium where local oscillations are coupled to a slowly propagating field. Using the Kuramoto approximation of phase dynamics, we show that this uniform medium can support steady autonomous pacemakers. These were patterns spontaneously emerge from the background of uniform bulk oscillations and are stable with respect to small perturbations.

Based on the complex Ginzburg-Landau equation (CGLE), a new mapping model of oscillatory media is proposed. The present dynamics is fully determined by an effective phase field renormalized by amplitude. The model exhibits phase turbulence, amplitude turbulence, and a frozen state reported in the CGLE. In addition, we find a state in which the phase and amplitude have spiral structures with opposite rotational directions. This state is found to be observed also in the CGLE. Thus, one concludes that the behaviors observed in the CGLE can be described by only the phase dynamics appropriately constructed.

We experimentally address long-time dynamics of an artificially curved growth interface in directional solidification. Repetitive cell nucleations are found to appear in a disordered way, but to eventually organize themselves in a coherent way, for long times. This behavior is recovered by simulation of a nonlinear advection-diffusion model for phase dynamics. The existence of a periodic attractor is supported by the derivation of a Lyapunov functional for this model.

We study numerically the effects of noise and periodic forcings on cluster synchronization in a chain of Van der Pol oscillators. We generalize the notion of effective synchronization to the case of a spatially extended system. It is shown that the structure of synchronized clusters can be effectively controlled by applying local external forcings. The effect of amplitude relations on the phase dynamics is also explored.

The slow dynamics of nearly stationary patterns in a FitzHugh-Nagumo model are studied using a phase dynamics approach. A Cross-Newell phase equation describing slow and weak modulations of periodic stationary solutions is derived. The derivation applies to the bistable, excitable, and the Turing unstable regimes. In the bistable case stability thresholds are obtained for the Eckhaus and the zigzag instabilities and for the transition to traveling waves. Neutral stability curves demonstrate the destabilization of stationary planar patterns at low wavenumbers to zigzag and traveling modes. Numerical solutions of the model system support the theoretical findings.

Study of weakly non-linear dynamics of a reaction–super-diffusion system near a Hopf bifurcation by means of fractional analogues of complex Ginzburg-Landau and Kuramoto-Sivashinsky equations is presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We construct exact ring soliton-like solutions of the cylindrically symmetric (i.e., radial) Gross- Pitaevskii equation with a potential, using the similarity transformation method. Depending on the choice of the allowed free functions, the solutions can take the form of stationary dark or bright rings whose time dependence is in the phase dynamics only, or oscillating and bouncing solutions, related to the second Painlev\'e transcendent. In each case the potential can be chosen to be time-independent., Comment: 8 pages, 7 figures. Version 2: stability analysis of the dark solution

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.

In this note we will introduce in the first section the concept of phase dynamics, followed by an outline of some recent developments associated with the properties of nonlinear phase equations, and we will conclude with an extensive list of references on phase dynamics and closely related subjects including envelope equations, the motion of defects, and the coupling between amplitude and phase equations.

The phase dynamic equation of motion for a propagating periodic structure is derived in a euclidean invariant manner. The formulation is applied to the generalized Swift-Hohenberg model where the coefficients are complex numbers.

Des equations de champ non lineaires dissipatives derivees de fonctionnelles de potentiel avec minima degeneres sont reduites a un systeme couple d'equations du mouvement des defauts topologiques (dislocations) et de la variable de phase. La theorie est appliquee a l'equation de thermoconvection de Swift-Hohenberg

The effects of the amplitude nonuniformity of the magnetic field in a betatron on the forced and betatron radial oscillations of electrons during extraction (or discharge toward the target) at the end of the acceleration cycle are examined. The amplitude and phase dynamics of radial betatron oscillations are studied by means of an analog computer.

We discuss the long-wavelength, low-frequency excitations of non- equilibrium systems with a helical structure as observed in spiral vortex flow in the Taylor instability in long cylinders. We find that a single-phase equation can give rise to a propagating mode, a situation unknown from hydrodynamics in local thermodynamic equilibrium.

We present the nonlinear phase equations describing the stability of a one-dimensional periodic pattern, with mean flow effect due to Galilean invariance. We show that the phase dynamics is second order in time, and could lead to an oscillatory instability of the pattern.

We present dynamic equations for the slow macroscopic variables of the wavy vortex state. The static solutions explain in a qualitative way recent experimental results of Ahlers et al. on the variation of the vortex diameter throughout the cell. In addition, we predict the existence of a pair of propagating or overdamped normal modes (depending on the wave vector of the disturbance) formed by the slow variables.

The equation of motion for the slow time dependence of convective rolls due to long-wavelength inhomogeneities is shown to have a singular dependence on the wave vector of the disturbance. Consequences for the skew varicose instability and the wave-number selection principle in textures of curved rolls suggested by Pomeau and Manneville are discussed.

Kinetics of phase changes often involves states with broken translational symmetry typified by domain walls which may generally be called patterns. Here we present a brief review of our works in this area emphasizing pattern evolution in incommensurate modulated systems. We show that the idea of phase dynamics introduced recently in hydrodynamics combined with defect dynamics is very useful in describing slightly disturbed modulated states. Otherwise, the equations of motion of discommensurations and dislocations must be solved to obtain pattern evolution.