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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. [ABSTRACT FROM AUTHOR]

We analyze the effect of adding a weak, localized, inhomogeneity to a two dimensional array of oscillators with nonlocal coupling. We propose and also justify a model for the phase dynamics in this system. Our model is a generalization of a viscous eikonal equation that is known to describe the phase modulation of traveling waves in reaction–diffusion systems. We show the existence of a branch of target pattern solutions that bifurcates from the spatially homogeneous state when , the strength of the inhomogeneity, is nonzero and we also show that these target patterns have an asymptotic wavenumber that is small beyond all orders in . The strategy of our proof is to pose a good ansatz for an approximate form of the solution and use the implicit function theorem to prove the existence of a solution in its vicinity. The analysis presents two challenges. First, the linearization about the homogeneous state is a convolution operator of diffusive type and hence not invertible on the usual Sobolev spaces. Second, a regular perturbation expansion in does not provide a good ansatz for applying the implicit function theorem since the nonlinearities play a major role in determining the relevant approximation, which also needs to be ‘correct’ to all orders in . We overcome these two points by proving Fredholm properties for the linearization in appropriate Kondratiev spaces and using a refined ansatz for the approximate solution which was obtained using matched asymptotics. [ABSTRACT FROM AUTHOR]

We study the emergence of asymptotic patterns in Winfree ensemble such as the partial/complete phase-locking and bump states under the effect of heterogeneous frustrations. Although the Winfree model is the first model for the synchronization of limit-cycle oscillators, there is little literature on the mathematical validity of asymptotic patterns compared to the vast literature of the well-studied Kuramoto model. Recently, it has received a renewed attention in nonlinear dynamics and statistical physics communities due to its diverse asymptotic patterns that it can generate. In particular, we provide a rigorous result on the existence of bump states in a homogeneous ensemble with the same natural frequency. Our provided results exhibit the robustness of emerging asymptotic patterns with respect to frustrations. We derive several sufficient frameworks for the unique existence of an equilibrium state, bump states and uniform stability with respect to initial data. [ABSTRACT FROM AUTHOR]

This paper extends the Madelung–Bohm formulation of quantum mechanics to describe the time-reversible interaction of classical and quantum systems. The symplectic geometry of the Madelung transform leads to identifying hybrid quantum–classical Lagrangian paths extending the Bohmian trajectories from standard quantum theory. As the classical symplectic form is no longer preserved, the nontrivial evolution of the Poincaré integral is presented explicitly. Nevertheless, the classical phase-space components of the hybrid Bohmian trajectory identify a Hamiltonian flow parameterized by the quantum coordinate and this flow is associated to the motion of the classical subsystem. In addition, the continuity equation of the joint quantum–classical density is presented explicitly. While the von Neumann density operator of the quantum subsystem is always positive-definite by construction, the hybrid density is generally allowed to be unsigned. However, the paper concludes by presenting an infinite family of hybrid Hamiltonians whose corresponding evolution preserves the sign of the probability density for the classical subsystem. [ABSTRACT FROM AUTHOR]

We consider the Cahn–Hilliard (CH) equation with a Burgers-type convective term that is used as a model of coarsening dynamics in laterally driven phase-separating systems. In the absence of driving, it is known that solutions to the standard CH equation are characterized by an initial stage of phase separation into regions of one phase surrounded by the other phase (i.e. clusters or drops/holes or islands are obtained) followed by the coarsening process, where the average size of the structures grows in time and their number decreases. Moreover, two main coarsening modes have been identified in the literature, namely, coarsening due to volume transfer and due to translation. In the opposite limit of strong driving, the well-known Kuramoto–Sivashinsky equation is recovered, which may produce complicated chaotic spatio-temporal oscillations. The primary aim of the present work is to perform a detailed and systematic investigation of the transitions in the solutions of the convective CH equation for a wide range of parameter values, and, in particular, to understand in detail how the coarsening dynamics is affected by an increase of the strength of the lateral driving force. Considering symmetric two-drop states, we find that one of the coarsening modes is stabilized at relatively weak driving, and the type of the remaining mode may change as driving increases. Furthermore, there exist intervals in the driving strength where coarsening is completely stabilized. In the intervals where the symmetric two-drop states are unstable they can evolve, for example, into one-drop states, two-drop states of broken symmetry or even time-periodic two-drop states that consist of two traveling drops that periodically exchange mass. We present detailed stability diagrams for symmetric two-drop states in various parameter planes and corroborate our findings by selected time simulations. [ABSTRACT FROM AUTHOR]

In an appropriate moving coordinate frame, source defects are time-periodic solutions to reaction–diffusion equations that are spatially asymptotic to spatially periodic wave trains whose group velocities point away from the core of the defect. In this paper, we rigorously establish nonlinear stability of spectrally stable source defects in the complex Ginzburg–Landau equation. Due to the outward transport at the far field, localized perturbations may lead to a highly non-localized response even on the linear level. To overcome this, we first investigate in detail the dynamics of the solution to the linearized equation. This allows us to determine an approximate solution that satisfies the full equation up to and including quadratic terms in the nonlinearity. This approximation utilizes the fact that the non-localized phase response, resulting from the embedded zero eigenvalues, can be captured, to leading order, by the nonlinear Burgers equation. The analysis is completed by obtaining detailed estimates for the resolvent kernel and pointwise estimates for Green's function, which allow one to close a nonlinear iteration scheme. [ABSTRACT FROM AUTHOR]

We perform a detailed analysis of the phase sensitivity of optical fibre soliton collision and energy exchange dynamics resulting from Raman scattering. We show that the relative phase of colliding solitons has a significant effect on the fraction of energy transferred for solitons with a large bandwidth and/or with a small centre frequency detuning. We consider the implications of our results from the particular perspective of optical rogue wave formation in noise-driven supercontinuum generation. A particular conclusion is that extreme soliton trajectories identified as a class of optical roguewave arise because of collisions between emerging solitons with a zero-phase difference. [ABSTRACT FROM AUTHOR]

Somitogenesis is a process for the development of somites which are transient, segmental structures that lie along the anterior-posterior axis of vertebrate embryos. The pattern of somites is governed by the segmentation clock and its timing is controlled by the clock genes which undergo synchronous oscillation over adjacent cells in the posterior presomitic mesoderm (PSM). In this paper, we analyze a mathematical model which depicts the kinetics of the zebrafish segmentation clock genes subject to direct autorepression by their own products under time delay, and cell-to-cell interaction through Delta-Notch signalling. Our goal is to elucidate how synchronous oscillations are generated for the cells in the posterior PSM, and how oscillations are arrested for the cells in the anterior PSM. For this system of delayed equations, an iteration technique is employed to derive the global convergence to the synchronous equilibrium, which corresponds to the oscillation-arrested. By applying the delay Hopf bifurcation theory and the center manifold theorem, we derive the criteria for the existence of stable synchronous oscillations for the cells at the tail bud of the PSM. Our analysis provides the basic parameter ranges and delay magnitudes for stable synchronous, asynchronous oscillation and oscillation-arrested. We exhibit how synchronous oscillations are affected by the degradation rates and delays. Extended from the analytic theory, further numerical findings linked to the segmentation process are presented. [ABSTRACT FROM AUTHOR]

We introduce the Schrödinger–Lohe model (S–L) with interaction frustration which is a toy model for quantum synchronization, and study its emergent collective dynamics. Our proposed model might not capture quantum entanglement which is one of genuine quantum phenomenon as it is, however it exhibits collective synchronous behaviors and can be reduced to the Kuramoto model with frustration in the case of spatial homogeneity and unit mass. For the emergent dynamics of the model, we present two sufficient frameworks leading to the complete and practical synchronizations for identical and non-identical oscillators, respectively, in terms of frustration function, system parameters and initial data. When the frustration functions admit real diagonal perturbations, complete synchronization emerges for identical ensemble with some class of initial data, in contrast, for non-identical ensemble, we can expect practical synchronization in which the diameter for wave functions can be controlled by tuning the coupling strength. For purely imaginary valued frustrations, we analytically show that complete or practical synchronization cannot occur for generic initial data. This illustrates that the S–L model exhibits some interesting dynamic patterns different from the Kuramoto synchronization. We provide several numerical examples which manifest periodic behaviors, reminiscent of the Kuramoto model with cosine coupling. [ABSTRACT FROM AUTHOR]

The effect of rapid oscillations, related to large dispersion terms, on the dynamics of dissipative evolution equations is studied for the model examples of the 1D complex Ginzburg–Landau and the Kuramoto–Sivashinsky equations. Three different scenarios of this effect are demonstrated. According to the first scenario, the dissipation mechanism is not affected and the diameter of the global attractor remains uniformly bounded with respect to the very large dispersion coefficient. However, the limit equation, as the dispersion parameter tends to infinity, becomes a gradient system. Therefore, adding the large dispersion term actually suppresses the non-trivial dynamics. According to the second scenario, neither the dissipation mechanism, nor the dynamics are essentially affected by the large dispersion and the limit dynamics remains complicated (chaotic). Finally, it is demonstrated in the third scenario that the dissipation mechanism is completely destroyed by the large dispersion, and that the diameter of the global attractor grows together with the growth of the dispersion parameter. [ABSTRACT FROM AUTHOR]

The main observation of this paper is that the modified Korteweg–de Vries equation has its natural origin in phase modulation of a basic state such as a periodic travelling wave, or more generally, a family of relative equilibria. Extension to 2  +  1 suggests that a modified Kadomtsev–Petviashvili (or a Konopelchenko–Dubrovsky) equation should emerge, but our result shows that there is an additional term which has gone heretofore unnoticed. Thus, through the novel application of phase modulation a new equation appears as the 2  +  1 extension to a previously known one. To demonstrate the theory it is applied to the cubic-quintic nonlinear Schrödinger (CQNLS) equation, showing that there are relevant parameter values where a modified KP equation bifurcates from periodic travelling wave solutions of the 2  +  1 CQNLS equation. [ABSTRACT FROM AUTHOR]

We prove results that enable the efficient and natural realization of a large class of robust heteroclinic networks in coupled identical cell systems. We also propose some general conjectures that relate a natural and large class of robust heteroclinic networks that occur in networks modelled by equations of Lotka–Volterra type, and certain networks of symmetric systems, to robust heteroclinic networks in coupled cell networks. [ABSTRACT FROM AUTHOR]

Inhibitory circuits of relaxation oscillators are often-used models for dynamics of biological networks. We present a qualitative and quantitative stability analysis of such a circuit constituted by three generic oscillators (of a Fitzhugh–Nagumo type) as its nodes coupled reciprocally. Depending on inhibitory strengths, and parameters of individual oscillators, the circuit exhibits polyrhythmicity of up to five simultaneously stable rhythms. With methods of bifurcation analysis and phase reduction, we investigate qualitative changes in stability of these circuit rhythms for a wide range of parameters. Furthermore, we quantify robustness of the rhythms maintained under random perturbations by monitoring phase diffusion in the circuit. Our findings allow us to describe how circuit dynamics relate to dynamics of individual nodes. We also find that quantitative and qualitative stability properties of polyrhythmicity do not always align. [ABSTRACT FROM AUTHOR]

We consider a system , with increasing functions f1 and f2, which has at most one positive equilibrium. Here the values of the functions ri, Gi, fi are positive for positive arguments, the delays in the cooperative term can be distributed and unbounded, both systems with concentrated delays and integro-differential systems are a particular case of the considered system. Analyzing the relation of the functions f1 and f2, we obtain several possible scenarios of the global behaviour. They include the cases when all nontrivial positive solutions tend to the same attractor which can be the positive equilibrium, the origin or infinity. Another possibility is the dependency of asymptotics on the initial conditions: either solutions with large enough initial values tend to the equilibrium, while others tend to zero, or solutions with small enough initial values tend to the equilibrium, while others infinitely grow. In some sense solutions of the equation are intrinsically non-oscillatory: if both initial functions are less/greater than the equilibrium value, so is the solution for any positive time value. The paper continues the study of equations with monotone production functions initiated in Berezansky and Braverman (2013 Nonlinearity26 2833–49). [ABSTRACT FROM AUTHOR]

Travelling waves are an important class of signal propagation phenomena in extended systems with a preferred direction of information flow. We study the generation of travelling waves in unidirectional chains of coupled oscillators communicating via a phase-dependent pulse-response interaction borrowed from mathematical neuroscience. Within the context of such systems, we develop a widely applicable, jointly numerical and analytical methodology for deducing existence and stability of periodic travelling waves. We provide careful numerical studies that support the existence of a periodic travelling wave solution as well as the asymptotic relaxation of a single oscillator to the wave when it is forced with the wave profile. Using this evidence as an assumption, we analytically prove global stability of waves in the infinite chain, with respect to initial perturbations of downstream sites. This rigorous stability result suggests that asymptotic relaxation to the travelling wave occurs even when the forcing is perturbed from the wave profile, a property of the motivating system that is supported by previous work as well as the convergence of the more sophisticated numerical algorithm that we propose in order to compute a high-precision approximation to the solution. We provide additional numerical studies that show that the wave is part of a one-parameter family, and we illustrate the structural robustness of this family with respect to changes in the coupling strength. [ABSTRACT FROM AUTHOR]

We explore the bifurcation structure of a modified Cahn–Hilliard equation that describes a system that may undergo a first-order phase transition and is kept permanently out of equilibrium by a lateral driving. This forms a simple model, e.g., for the deposition of stripe patterns of different phases of surfactant molecules through Langmuir–Blodgett transfer. Employing continuation techniques the bifurcation structure is numerically investigated using the non-dimensional transfer velocity as the main control parameter. It is found that the snaking structure of steady front states is intertwined with a large number of branches of time-periodic solutions that emerge from Hopf or period-doubling bifurcations and end in global bifurcations (sniper and homoclinic). Overall the bifurcation diagram has a harp-like appearance. This is complemented by a two-parameter study in non-dimensional transfer velocity and domain size (as a measure of the distance to the phase transition threshold) that elucidates through which local and global codimension 2 bifurcations the entire harp-like structure emerges. [ABSTRACT FROM AUTHOR]