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The Swift-Hohenberg equation with quadratic and cubic nonlinearities exhibits a remarkable wealth of stable spatially localized states. presence these states is related to phenomenon called homoclinic snaking. Numerical computations are used illustrate the changes in solution as it grows spatial extent determine stability properties resulting evolution once they lose illustrated using direct simulations time.

The bistable Swift-Hohenberg equation exhibits multiple stable and unstable spatially localized states of arbitrary length in the vicinity Maxwell point between homogeneous periodic states. These are organized a characteristic snakes-and-ladders structure. origin this structure one spatial dimension is reviewed, stability properties resulting with respect to perturbations both two dimensions described. relevance results several different physical systems discussed.

Abstract Rapidly rotating Rayleigh–Bénard convection is studied using an asymptotically reduced equation set valid in the limit of low Rossby numbers. Four distinct dynamical regimes are identified: a disordered cellular regime near threshold, weakly interacting convective Taylor columns at larger Rayleigh numbers, followed for yet numbers by breakdown into plume characterized efficiency and finally geostrophic turbulence. The transitions quantified examining properties horizontally...

We study two examples of two-dimensional nonlinear double-diffusive convection (thermohaline convection, and in an imposed vertical magnetic field) the limit where onset marginal overstability just precedes exchange stabilities. In this solutions can be found analytically. The branch oscillatory always terminates on steady solution branch. If is subcritical occurs when period oscillation becomes infinite, while if it supercritical, via a Hopf bifurcation. A detailed discussion stability...

Two-dimensional oscillatory convection in a binary fluid mixture an infinite plane porous layer heated from below is studied. Small-amplitude nonlinear solutions the form of standing and traveling waves are found their relative stability established. Stable preferred near onset. The interaction two types wave with steady overturning also As Rayleigh number increased period each type approaches infinity, as -ln(${R}_{c}^{\mathrm{SW}\mathrm{\ensuremath{-}}\mathrm{R}}$)...

Spatial localization is a common feature of physical systems, occurring in both conservative and dissipative systems. This article reviews the theoretical foundations our understanding spatial forced from mathematical point view physics perspective. It explains origin large multiplicity simultaneously stable spatially localized states present parameter region called pinning its relation to notion homoclinic snaking. The are described as bound fronts, notions front pinning, self-pinning,...

For rotationally constrained convection, the Taylor–Proudman theorem enforces an organization of nonlinear flows into tall columnar or compact plume structures. While coherent structures in convection under moderate rotation are exclusively cyclonic, recent experiments for rapid have revealed a transition to equal populations cyclonic and anticyclonic Direct numerical simulation (DNS) this regime is expensive, however, existing simulations yet reveal vortical In paper, we use reduced system...

Stationary spatially localized structures, sometimes called dissipative solitons, arise in many interesting and important applications, including buckling of slender structures under compression, nonlinear optics, fluid flow, surface catalysis, neurobiology more. The recent resurgence interest these has led to significant advances our understanding the origin properties states, turn suggest new questions, both general system-specific. This paper surveys results focusing on open problems,...

We demonstrate, via simulations of asymptotically reduced equations describing rotationally constrained Rayleigh-Bénard convection, that the efficiency turbulent motion in fluid bulk limits overall heat transport and determines scaling nondimensional Nusselt number Nu with Rayleigh Ra, Ekman E, Prandtl σ. For E << 1 inviscid theory predicts confirm large Ra law Nu-1 ≈ C(1)σ(-1/2)Ra(3/2)E(2), where C(1) is a constant, estimated as 0.04 ± 0.0025. In contrast, corresponding result for...

Chimera states consisting of domains coherently and incoherently oscillating identical oscillators with nonlocal coupling are studied. These usually coexist the fully synchronized state have a small basin attraction. We propose phase-coupled model in which chimera develop from random initial conditions. Several classes been found: (a) stationary multicluster evenly distributed coherent clusters, (b) unevenly (c) single cluster traveling constant speed across system. Traveling also...

Rotating Rayleigh-Bénard convection exhibits, in the limit of rapid rotation, a turbulent state known as geostrophic turbulence. This is present for sufficiently large Rayleigh numbers representing thermal forcing system, and characterized by leading order balance between Coriolis force pressure gradient. itself unstable to generation depth-independent or barotropic vortex structures ever larger scale through process spectral condensation. involves an inverse cascade mechanism with positive...

We analyze dark pulse Kerr frequency combs in optical resonators with normal group-velocity dispersion using the Lugiato-Lefever model. show that time domain these correspond to interlocked switching waves between upper and lower homogeneous states, explain how this fact accounts for many of their experimentally observed properties. Modulational instability does not play any role existence. Furthermore, we provide a detailed map indicating where stable can be found parameter space, they are...

Linear stability results for convection in binary fluid mixtures are given experimental boundary conditions and parameter values. Normal $^{3}\mathrm{\ensuremath{-}}^{4}$He ethanol-water considered with no-slip on the velocity no outward mass flux. Two cases detail: fixed temperature at top bottom, thermal flux bottom. The role played by Biot number is emphasized. presented a manner most useful to experimentalists. errors incurred determining separation ratio S from observations using scaled...

We have studied the transition between oscillatory and steady convection in a simplified model of two-dimensional thermosolutal convection. This is exact to second order amplitude motion qualitatively accurate for larger amplitudes. If ratio solutal diffusivity thermal sufficiently small Rayleigh number, R S , large, sets as overstable oscillations, these oscillations grow T increased. In addition this branch, there branch solutions that bifurcates from static equilibrium towards lower...

Multiple states of spatially localized steady convection are found in numerical simulations water–ethanol mixtures two dimensions. Realistic boundary conditions at the top and bottom used, with periodic horizontal. The form by a mechanism similar to pinning region around Maxwell point variational systems, but located parameter regime which conduction state is overstable. Despite this can be stable. properties described detail, their destruction increasing or decreasing Rayleigh number...

Experimental observations of azimuthally traveling waves in rotating Rayleigh-Bénard convection a circular container are presented and described terms the theory bifurcation with symmetry. The amplitude convective states varies as √ε traveling-wave frequency depends linearly on ε finite value at onset. Here = R/Rc - 1, where Rc is critical Rayleigh number. onset decreases to zero dimensionless rotation rate Ω zero. These experimental consistent presence Hopf from conduction state expected...

We investigate the bifurcation structure of stationary localized patterns two-dimensional Swift–Hohenberg equation on an infinitely long cylinder and plane. On cylinders, we find roll, square, stripe patches that exhibit snaking nonsnaking behavior same branch. Some these snake between four saddle-node limits; in this case, recent analytical results predict existence a rich to asymmetric solutions, trace out branches PDE spectra along branches. plane, study fully roll structures, which are...

The versal deformation of a vector field co-dimension two that is equivariant under representation the symmetry group O(2 ) and has nilpotent linearization at origin studied. An appropriate scaling allows us to formulate problem in terms central-force with small dissipative perturbation. We derive analyse averaged equations for angular momentum energy classical motion. unfolded system possesses four different types non-trivial solutions: steady-state three others, which are referred wave...

The partial differential equations governing two-dimensional thermosolutal convection in a Boussinesq fluid with free boundary conditions have been solved numerically regime where oscillatory solutions can be found. A systematic study of the transition from nonlinear periodic oscillations to temporal chaos has revealed sequences period-doubling bifurcations. Overstability occurs if ratio solutal thermal diffusivity τ < 1 and Rayleigh number RS is sufficiently large. Solutions obtained for...

The conserved Swift-Hohenberg equation with cubic nonlinearity provides the simplest microscopic description of thermodynamic transition from a fluid state to crystalline state. resulting phase field crystal model describes variety spatially localized structures, in addition different extended periodic structures. location these structures temperature versus mean order parameter plane is determined using combination numerical continuation one dimension and direct simulation two three...

A two-dimensional system of soft particles interacting via a two-length-scale potential is studied. Density functional theory and Brownian dynamics simulations reveal fluid phase two crystalline phases with different lattice spacing. Of these the larger spacing can form an exotic periodic state fraction highly mobile particles: crystal liquid. Near transition between this smaller phase, quasicrystalline structures may be created by competition linear instability at one scale nonlinear...