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We study the relationship between the partially synchronous state and the coupling structure in general dynamical systems. Our results show that, on the contrary to the widely accepted concept, topological symmetry in a coupling structure is the sufficient condition but not the necessary condition. Furthermore, we find the necessary and sufficient condition for the existence of the partial synchronization and develop a method to obtain all of the existing partially synchronous solutions for all nonspecific dynamics from a very large number of possible candidates.
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We consider propagating, spatially localised waves in a class of equations that contain variational and non-variational terms. The dynamics of the waves is analysed through a collective coordinate approach. Motivated by the variational approximation, we show that there is a natural choice of projection onto collective variables for reducing the governing (nonlinear) partial differential equation (PDE) to coupled ordinary differential equations (ODEs). This projection produces ODEs whose solutions are exactly the stationary states of the effective Lagrangian that would be considered from applying the variational approximation method. We illustrate our approach by applying it to a modified Fisher equation for a travelling front, containing a non-constant-coefficient nonlinear term. We present numerical results that show that our proposed projection captures both the equilibria and the dynamics of the PDE much more closely than previously proposed projections.
We determine the functional behavior near the discrete rotational symmetry axis of discrete vortices of the nonlinear Schrodinger equation. We show that these solutions present a central phase singularity whose charge is restricted by symmetry arguments. Consequently, we demonstrate that the existence of high-charged discrete vortices is related to the presence of other off-axis phase singularities, whose positions and charges are also restricted by symmetry arguments. To illustrate our theoretical results, we offer two numerical examples of high-charged discrete vortices in photonic crystal fibers showing hexagonal discrete rotational invariance.
The hidden local symmetry is a successful model to describe the properties of the vector mesons in QCD. We point out that if we identify this hidden gauge theory as the magnetic picture of QCD, a linearized version of the model simultaneously describes color confinement and chiral symmetry breaking. We demonstrate that such a structure can be seen in the Seiberg dual picture of a softly broken supersymmetric QCD. The model possesses exact chiral symmetry and reduces to QCD when mass parameters are taken to be large. Working in the regime of the small mass parameters, we show that there is a vacuum where chiral symmetry is spontaneously broken and simultaneously the magnetic gauge group is Higgsed. If the vacuum we find persists in the limit of large mass parameters, one can identify the rho meson as the massive magnetic gauge boson, that is an essential ingredient for color confinement.
Solvability of the ubiquitous quantum harmonic oscillator relies on a spectrum generating osp(1|2) superconformal symmetry. We study the problem of constructing all quantum mechanical models with a hidden osp(1|2) symmetry on a given space of states. This problem stems from interacting higher spin models coupled to gravity. In one dimension, we show that the solution to this problem is the Plyushchay family of quantum mechanical models with hidden superconformal symmetry obtained by viewing the harmonic oscillator as a one dimensional Dirac system, so that Grassmann parity equals wavefunction parity. These models--both oscillator and particle-like--realize all possible unitary irreducible representations of osp(1|2).
The presence of correlations between particles significantly separated in pseudorapidity in proton-proton and proton-nucleus collisions has raised questions about whether collective effects are observed in small collision systems as well as in heavy-ion collisions. The quantification of these long-range correlations by $v_n$ coefficients is of particular interest. A selection of the latest $v_n$ measurements is presented, including results from the recent $d$+Au beam energy scan at RHIC where a significant non-zero $v_2$ is measured down to low center-of-mass energies ($sqrt{s_{rm{NN}}}$ = 39 GeV). Results from a collision system scan - comprising $p$+Au, $d$+Au, and $^3$He+Au collisions - are also shown to address the role of the initial nuclear geometry in the final state anisotropy. Finally, the challenge of measuring multi-particle cumulants, particularly $c_2{4}$, in $p$+$p$ collisions is discussed, and new methods for reducing the effects of non-flow are shown to produce a more robust measurement of $v_2{4}$ in $p$+$p$ collisions.
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