# gst: route from avalanches with random fields to stochastic branching events

<< Many complex systems with discrete symmetry breaking exhibit avalanche dynamics. Under quasistatically slow external driving, the temporal evolution v(t) of physical observables appears to be split between long periods of quiescence and well-delimited fast transformation events, the so-called avalanches. >>

<< Due to their fast nature, such avalanches can be regarded as instantaneous and labeled as point events k in time with a branching parameter. (..) Many physical systems exhibit criticality in the form of scale-free avalanches rendering power-law distributions of sizes and durations. >>️

<< Avalanches in mean-field models can be mapped to memoryless branching processes defining a universality class. (AA) present a reduced expression mapping a broad family of critical and subcriticial avalanches in mean-field models at the thermodynamic limit to rooted trees in a memoryless Poisson branching processes with random occurrence times. (They) derive the exact mapping for the athermal random field Ising model and the democratic fiber bundle model, where avalanche statistics progress towards criticality, and as an approximation for the self-organized criticality in slip mean-field theory. Avalanche dynamics and statistics in the three models differ only on the evolution of the field density, interaction strength, and the product of both terms determining the branching number. >>

️

Jordi Baró, Álvaro Corral. A universal route from avalanches in mean-field models with random fields to stochastic Poisson branching events. arXiv: 2502.08526v3 [cond-mat.dis-nn]. Feb 15, 2025. 

https://arxiv.org/abs/2502.08526

Also: waves, instability, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, waves, instability, transition

# gst: scattered waves fuel emergent activity.

<< Active matter taps into external energy sources to power its own processes. Systems of passive particles ordinarily lack this capacity, but can become active if the constituent particles interact with each other nonreciprocally. By reformulating the theory of classical wave-matter interactions, (AA) demonstrate that interactions mediated by scattered waves generally are not constrained by Newton's third law. The resulting center-of-mass forces propel clusters of scatterers, enabling them to extract energy from the wave and rendering them active. This form of activity is an emergent property of the scatterers' state of organization and can arise in any system where mobile objects scatter waves. Emergent activity flips the script on conventional active matter whose nonreciprocity emerges from its activity, and not the other way around. >>

AA << combine theory, experiment, and simulation to illustrate how emergent activity arises in wave-matter composite systems and to explore the phenomenology of emergent activity in experimentally accessible models. These preliminary studies suggest that heterogeneity is a singular perturbation to the dynamics of wave-matter composite systems, and induces emergent activity under all but the most limited circumstances. >>️

Ella M. King, Mia C. Morrell, et al. Scattered waves fuel emergent activity. Phys. Rev. Research 7, 013055. Jan 15, 2025

https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.7.013055

Also: waves, particle, self-assembly, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, waves, particle, self-assembly

# gst: chaotic dynamics creates and destroys branched flow.

<< The phenomenon of branched flow, visualized as a chaotic arborescent pattern of propagating particles, waves, or rays, has been identified in disparate physical systems ranging from electrons to tsunamis, with periodic systems only recently being added to this list. >>

Here, AA << explore the laws governing the evolution of the branches in periodic potentials. On one hand, (They) observe that branch formation follows a similar pattern in all nonintegrable potentials, no matter whether the potentials are periodic or completely irregular. Chaotic dynamics ultimately drives the birth of the branches. >>

<< On the other hand, (AA) results reveal that for periodic potentials the decay of the branches exhibits new characteristics due to the presence of infinitely stable branches known as superwires. Again, the interplay between branched flow and superwires is deeply connected to Hamiltonian chaos. >>

Alexandre Wagemakers, Aleksi Hartikainen, et al. Chaotic dynamics creates and destroys branched flow. Phys. Rev. E 111, 014214. Jan 7, 2025.

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.111.014214

arXiv: 2406.12922v2 [nlin.PS]. 

https://arxiv.org/abs/2406.12922

Also: chaos, waves, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, chaos, waves, branched flows, superwires, transitions

# gst: floating droplets excited with Faraday waves

<< The Faraday instability has been extensively studied in bounded containers but only recently has research on this phenomenon in flexible domains been conducted. (AA) study floating liquid droplets with Faraday waves excited on their surface, which undergo a slow time evolution toward a stable noncircular shape. (AA) develop a theoretical model for the evolution of the boundary of the droplet, thus allowing to simulate its full transient motion toward steady state. >>

<< By changing the forcing frequency and amplitude of (the) system, (They) observe a variety of stable droplet shapes. (..) Interesting transient behavior such as hysteresis is also discussed, where the final droplet shape depends on its previous shape. Finally, (They) touch upon droplets that do not reach a steady state shape, instead oscillating periodically in time or rotating at a constant angular velocity. >>️

L. Mazereeuw. Theoretical and experimental investigation of the shapes formed by floating droplets excited with Faraday waves. Phys. Rev. Fluids 9, 124404. Dec 19, 2024.

https://journals.aps.org/prfluids/abstract/10.1103/PhysRevFluids.9.124404

Also: drop, waves, instability, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: drops, droplets, droploids, waves, instability, Faraday instability, transitions   

# gst: soliton dynamics over a disordered topography

AA << report on the dynamics of a soliton propagating on the surface of a fluid in a 4-m-long canal with a random or periodic bottom topography. Using a full space-and-time resolved wave field measurement, (They) evidence, for the first time experimentally, how the soliton is affected by the disorder, in the context of Anderson localization, and how localization depends on nonlinearity. >>

<< For weak soliton amplitudes, the localization length is found in quantitative agreement with a linear shallow-water theory. For higher amplitudes, this spatial attenuation of the soliton amplitude is found to be enhanced. >>

<< Behind the leading soliton slowed down by the topography, different experimentally unreported dynamics occur: fission into backward and forward nondispersive pulses for the periodic case, and scattering into dispersive waves for the random case. >>

Guillaume Ricard, Eric Falcon. Soliton Dynamics over a Disordered Topography. Phys. Rev. Lett. 133, 264002. Dec 27, 2024.

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.133.264002

Also: soliton, waves, disorder, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, soliton, waves, disorder

# gst: nascent water waves induced by the impulsive motion of a solid wall; an unsteady hydraulic jump theory

Fig. 2: Representative cases of the different regimes of impulse waves observed: (a) dispersive wave; (b) solitary-like wave; (c) plunging breaking bore; (d) water jet 

AA << investigated the generation phase of laboratory-scale water waves induced by the impulsive motion of a rigid piston, whose maximum velocity U and total stroke L are independently varied, as well as the initial liquid depth h. By doing so, the influence of two dimensionless numbers is studied: the Froude number Frp (..), with g the gravitational acceleration, and the relative stroke Λp (..) of the piston. >>️

<< For large Froude numbers, an unsteady hydraulic jump theory is proposed, which accurately predicts the time evolution of the wave amplitude at the contact with the piston throughout the generation phase. At the end of the formation process, the dimensionless volume of the bump evolves linearly with Λp and the wave aspect ratio is found to be governed by the relative acceleration γ/g. As the piston begins its constant deceleration, the water bump evolves into a propagating wave and several regimes (such as dispersive, solitary-like and bore waves, as well as water jets) are then reported and mapped in a phase diagram in the (Frp, Λp) plane. While the transition from waves to water jets is observed if the typical acceleration of the piston is close enough to the gravitational acceleration g, the wave regimes are found to be mainly selected by the relative piston stroke Λp while the Froude number determines whether the generated wave breaks or not. >>️

️

Wladimir Sarlin, Zhaodong Niu, et al. Nascent water waves induced by the impulsive motion of a solid wall. arXiv: 2412.08216v1 [physics.flu-dyn]. Dec 11, 2024. 

https://arxiv.org/abs/2412.08216

Also: waves, soliton, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, waves, soliton

# gst: bent and reverse bent solitons; obliquely interacting solitary waves and wave wakes in free-surface flows.

This AA paper << investigates the weakly nonlinear isotropic bi-directional Benney-Luke (BL) equation, which is used to describe oceanic surface and internal waves in shallow water, with a particular focus on soliton dynamics. Using the Whitham modulation theory, (AA) derive the modulation equations associated with the BL equation that describe the evolution of soliton amplitude and slope. By analyzing rarefaction waves and shock waves within these modulation equations, (AA) derive the Riemann invariants and modified Rankine-Hugoniot conditions. These expressions help characterize the Mach expansion and Mach reflection phenomena of bent and reverse bent solitons. >>

<< Furthermore, as a far-field approximation for the forced BL equation - which models wave and flow interactions with local topography - the modulation equations yield a slowly varying similarity solution. This solution indicates that the precursor wavefronts created by topography moving at subcritical or critical speeds take the shape of a circular arc, in contrast to the parabolic wavefronts observed in the forced KP equation. >>

Lei Hu, Xudan Luo, Zhan Wang. Obliquely interacting solitary waves and wave wakes in free-surface flows. arXiv: 2412.05034v1 [nlin.PS]. Dec 6, 2024. 

https://arxiv.org/abs/2412.05034

Also: waves, soliton, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, waves, solitons

# gst: fluid stretching and compression to control (chemical) waves

<< Oscillatory kinetics coupled to diffusion can produce traveling waves as observed in physical, chemical, and biological systems. (AA) show experimentally that the properties of such waves can be controlled by fluid stretching and compression in a hyperbolic flow. >>

️

<< Localized packet waves consisting in a train of parallel waves can develop due to a balance between diffusive broadening and advective compression along the unstable manifold. At a given distance from the stagnation point, the parallel waves transform into planelike waves and smeared waves where the transverse parabolic flow profile disturbs the patterns in the gap width. >> 

<< Once a wave packet has been obtained, it imprints a privileged direction that is maintained even if the compression rate is decreased. The width of the wave packet then scales inversely with the compression rate. >>️

S. Izumoto, D. M. Escala, et al. Control of Chemical Waves by Fluid Stretching and Compression. Phys. Rev. Lett. 133, 218001. Nov 21, 2024. 

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.133.218001

Also: waves, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, waves

# gst: self-organized target wave chimeras from asynchronous oscillators in reaction-diffusion media

<< An important development in nonlinear dynamics is the discovery of chimera states that represent the coexistence of synchronized and desynchronized activity in populations of identically coupled oscillators. >>

 Here, AA << unveil a novel chimera state called “self-organized target wave chimera” in reaction-diffusion media where synchronized target waves spontaneously emerge from a pacemaker composed of asynchronous oscillators. This regime contrasts with a widely accepted perspective that synchronized target waves can be generated only by the individuals, which comprise the pacemaker, behaving in a synchronized manner. >>

AA << characterize the features of self-organized target wave chimeras and present a phase diagram of existence of such a regime. >>

Bing-Wei Li, Jie Xiao, et al. Self-Organized Target Wave Chimeras in Reaction-Diffusion Media. Phys. Rev. Lett. 133, 207203. Nov 15, 2024. 

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.133.207203

Also: chimera, self-assembly, waves, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, chimera, self-assembly, waves, self-organized target wave chimera, asynchronous oscillators

# gst: vortex structures under dimples and scars in turbulent free-surface flows

<< Turbulence beneath a free surface leaves characteristic long-lived signatures on the surface, such as upwelling 'boils', near-circular 'dimples' and elongated 'scars', easily identifiable by eye, e.g., in riverine flows. >>️

AA << explore the connection between these surface signatures and the underlying vortical structures. We investigate dimples, known to be imprints of surface-attached vortices, and scars, which have yet to be extensively studied, by analysing the conditional probabilities that a point beneath a signature is within a vortex core as well as the inclination angles of sub-signature vorticity. >>️

<< The analysis shows that the likelihood of vortex presence beneath a dimple decreases from the surface down through the viscous and blockage layers in a near-Gaussian manner, influenced by the dimple's size and the bulk turbulence. When expressed as a function of depth over the Taylor microscale λT, this probability is independent of Reynolds and Weber number. >>️

<< Conversely, the probability of finding a vortex beneath a scar increases sharply from the surface to a peak at the edge of the viscous layer, at a depth of approximately λT/4. Distributions of vortical orientation also show a clear pattern: a strong preference for vertical alignment below dimples and an equally strong preference for horizontal alignment below scars. >>️

AA << findings suggest that scars can be defined as imprints of horizontal vortices approximately a quarter of the Taylor microscale beneath the surface, analogous to how dimples can be defined as imprints of surface-attached vertical vortex tubes. >>

Jørgen R. Aarnes, Omer Babiker, et al. Vortex structures under dimples and scars in turbulent free-surface flows. arXiv: 2409.05409v1 [physics.flu-dyn]. 

9 Sep 2024.

https://arxiv.org/abs/2409.05409

Also: vortex, turbulence, waves, bubble, drop, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, vortex, turbulence, waves, bubble, drop, transition

# gst: collapse of a toroidal bubble inducing shock waves

<< When bubbles collapse near a wall, they typically experience an asymmetric deformation. This collapse leads to the creation of a jet that strikes the bubble interface, causing the formation of a toroidal bubble and the subsequent release of a water-hammer shock. >>️

AA << findings demonstrate that shock waves emitted from the toroidal bubble consistently propagate toward the central axis of the torus, resulting in significant pressure shocks along the axis, similar to the water-hammer shock formed during the collapse of a spherical bubble. >>️

<< In contrast, weak pressure waves are generated in the transverse directions, leading to relatively weaker pressure peaks. Furthermore, the wall-pressure peak induced by the toroidal bubble is approximately three times higher than that induced by the spherical bubble. >>️

Cheng Liu, Xiaobin Yang, et al. Investigations on the shock wave induced by collapse of a toroidal bubble. Phys. Rev. E 110, 015103. Jul 16, 2024. 

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.110.015103

Also: bubble, drop, waves, collapse, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, bubble, drop, waves, collapse

# gst: helical instabilities from mixed mode transitions in boundary layers

<< Recent (..) direct numerical simulations (DNS) of adverse- and zero-pressure-gradient boundary layers beneath moderate levels of free stream turbulence (𝑇⁢𝑢≤2%) revealed a mixed mode transition regime, intermediate between orderly and bypass routes. >>️

<< In this regime, the amplitudes of the Klebanoff streaks and instability waves are similar, and both can contribute significantly as these interact. Three-dimensional visualizations of transitional eddies revealed a helical pattern, quite distinct from the sinuous and varicose forms seen in pure bypass transition. This raises the fundamental question of whether the helical pattern could be attributed to a previously unknown instability mode. >>️

In AA work << based on stability analyses, (they) show that it is indeed the case. Two-dimensional stability analyses are performed herein for base flows extracted from DNS flow fields. >>️

Rikhi Bose, Paul A. Durbin. Mixed mode transition in boundary layers: Helical instability. Phys. Rev. Fluids 9, 063905. Jun 12, 2024. 

https://journals.aps.org/prfluids/abstract/10.1103/PhysRevFluids.9.063905

Also: instability, transition, turbulence, waves, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, instability, transition, turbulence, waves

# gst: a case of noise-induced excitation wave (in coupled FH-N eq.)

<< There are various research topics such as stochastic resonance, coherent resonance, and neuroavalanche in excitable systems under external noises. >>️

AA << have studied the propagation of excitation waves in the coupled noisy FitzHugh-Nagumo equations with a one-dimensional pacemaker region and found that there is a phase-transition-like phenomenon from the short-range propagation to the whole-system propagation by changing the noise strength T.  >>️

Hidetsugu Sakaguchi. Noise-induced excitation wave and its size distribution in coupled FitzHugh-Nagumo equations on a square lattice. Phys. Rev. E 109, 044211. Apr 19, 2024.

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.109.044211

Also: waves, noise, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, waves, noise, transition

# gst: random walk model for dual cascades in wave turbulence.

<< Dual cascades in turbulent systems with two conserved quadratic quantities famously arise in both two-dimensional hydrodynamic turbulence and also in wave turbulence based on four-wave interactions. >>

<< in wave turbulence the systematic spectral fluxes observed in a dual cascade do not require an irreversible dynamical mechanism, rather, they arise as the inevitable outcome of blind chance. >>️️

Oliver Bühler. Random walk model for dual cascades in wave turbulence. Phys. Rev. E 109, 055102. May 1, 2024. 

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.109.055102

Also: waves, turbulence, random, weak, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, waves, turbulence, weak turbulence, random, random walks

# gst: evolving disorder and chaos induces acceleration of elastic waves.

<< Static or frozen disorder, characterised by spatial heterogeneities, influences diverse complex systems, encompassing many-body systems, equilibrium and nonequilibrium states of matter, intricate network topologies, biological systems, and wave-matter interactions. >>

AA << investigate elastic wave propagation in a one-dimensional heterogeneous medium with diagonal disorder. (They) examine two types of complex elastic materials: one with static disorder, where mass density randomly varies in space, and the other with evolving disorder, featuring random variations in both space and time. (AA) results indicate that evolving disorder enhances the propagation speed of Gaussian pulses compared to static disorder. Additionally, (They) demonstrate that the acceleration effect also occurs when the medium evolves chaotically rather than randomly over time. The latter establishes that evolving randomness is not a unique prerequisite for observing wavefront acceleration, introducing the concept of chaotic acceleration in complex media. >>️

M. Ahumada, L. Trujillo, J. F. Marín. Evolving disorder and chaos induces acceleration of elastic waves. arXiv: 2403.02113v1 [cond-mat.dis-nn]. Mar 4, 2024. 

https://arxiv.org/abs/2403.02113

Also: waves, elastic, chaos, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, waves, elastic, chaos, transition

# gst: self-trapped nonlinear waves with multiple phase singularities.

AA << investigate the existence of self-trapped nonlinear waves with multiple phase singularities. >>️

They << focus on configurations with an antivortex surrounded by a triangular arrangement of vortices within a hosting soliton. (AA) find stationary patterns that can be interpreted as stable self-trapped vortex crystals, constituting the first example of a configuration of this sort with space-independent potentials. Their stability is linked to their norm, transitioning from unstable to stable as their size increases, with an intermediate region where the structure is marginally unstable, undergoing a remarkable and puzzling self-reconstruction during its evolution. >>️

Angel Paredes and Humberto Michinel. Self-trapping of vortex crystals via competing nonlinearities.  Phys. Rev. E 109, 024216. Feb 22, 2024. 

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.109.024216

Also: waves, soliton, vortex, in https://www.inkgmr.net/kwrds.html

Keywords: gst, waves, soliton, self-trapping, vortex 

# gst: the hypothesis of a new type of rogue waves.

<< Much attention of researchers has been paid in the recent decades to the study of rogue waves. Various mechanisms of formation of these waves were suggested. The occurrence of rogue waves is most often investigated on the basis of the mechanisms of modulation instability and superposition of waves. In both cases, an evolution of rogue waves takes place against the background of a wave field, which is reflected in the definitions of such waves. In this report, the localized waves developed in the absence of the background wave fields are considered. At the same time, their dynamics corresponds to the dynamics of rogue waves that ”appear from nowhere and disappear without a trace”. >>️

<< The waves of this type are distinguished by the fact that their dynamics occur on the zero background. This implies that rogue waves presented here are formed solely due to the nonlinear focusing. >>️

N.V. Ustinov. New type of rogue waves. arXiv:2310.17254v1 [nlin.SI]. Oct 26, 2023.  

https://arxiv.org/abs/2310.17254

Chaos, Solitons & Fractals V. 179, Feb 2024, 114467. https://www.sciencedirect.com/science/article/abs/pii/S0960077924000183

Also: waves, in https://www.inkgmr.net/kwrds.html

Keywords: gst, waves, rogue waves 

# gst: pulsating active matter.

<< the mechanical pulsation of locally synchronized particles is a generic route to propagate deformation waves. >>

️

AA << consider a model of dense repulsive particles whose activity drives periodic change in size of each individual. >>️

They << show that the competition between repulsion and synchronization triggers an instability which promotes a wealth of dynamical patterns, ranging from spiral waves to defect turbulence. >>️

Yiwei Zhang, Etienne Fodor. Pulsating Active Matter. Phys. Rev. Lett. 131, 238302. Dec 8, 2023. 

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.131.238302

Also: waves, particle, transition, instability, turbulence, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, waves, particle, transition, instability, turbulence

# gst: collective patterns generated by capillary surfers.

<< Millimeter-sized “surfers” can self-propel across a vibrating liquid surface, interacting with other surfers to create collective patterns. >>

<< Self-propelled objects can move in mesmerizing patterns. The collective movements of groups of such objects typically occur in one of two flow regimes: the inertial regime—think swirling schools of fish in water—or the viscous regime—think swarming colonies of bacteria in mucus. Some self-propelled objects can travel in both flow regimes, a possibility that is less explored. >>️

AA << have studied the motion of a new system of self-propelled objects that move in this intermediate regime, finding that the objects organize into several distinct and tunable motion patterns. >>️️

<< Pairs of self-propelled surfers observed by the team move in one of seven different patterns (the video shows five). These include the “orbit,” where a pair of surfers rotate around a central point; the “tailgate,” where one surfer closely follows another, head to tail in a linear path; and the “jackknife,” where a pair of perpendicular surfers rotate stern to stern around their collision point. >>

<< When only one surfer was present, these mismatched amplitudes propelled the surfer in the direction of its bow. When there were two surfers close to each other, interactions among the waves caused the surfers to either repel each other so that they moved in opposite directions or to come together so that they collectively traced one of seven distinct patterns. >>️

️

Maggie Hudson. Synchronized Surfing of Self-Propelled Particles. Physics 16, s156. Nov 7, 2023. 

https://physics.aps.org/articles/v16/s156

Ian Ho, Giuseppe Pucci, Anand U. Oza, Daniel M. Harris. Capillary surfers: Wave-driven particles at a vibrating fluid interface. Phys. Rev. Fluids 8, L112001. Nov 7, 2023.

https://link.aps.org/doi/10.1103/PhysRevFluids.8.L112001

Anand U. Oza, Giuseppe Pucci, Ian Ho, Daniel M. Harris. Theoretical modeling of capillary surfer interactions on a vibrating fluid bath. Phys. Rev. Fluids 8, 114001. Nov 7, 2023.

https://link.aps.org/doi/10.1103/PhysRevFluids.8.114001

Also: waves, particle, swarm, in: https://www.inkgmr.net/kwrds.html

Keywords: gst, waves, wave-wave, capillary waves, particles, self-propelled particles, fluid-particle interactions, wave-particle interactions

# gst: apropos of transitions, three distinct new families of long-wave instabilities and potential new pathways to turbulence.

AA << reveal three previously unknown instabilities, distinct from the well-known Kelvin-Helmholtz Instability (KHI) and Holmboe Wave Instability (HWI), in that they have longer wavelengths (..) and often slower growth rates. >>

<< The circumstances under which turbulence can persist in strongly stratified flows remains a fascinating debate within the community. [AA] demonstrated that weakly unstable (very) long waves may trigger turbulence and mixing after long periods of time, even under initially very strongly stratified conditions. >>

Lu Zhu, Amir Atoufi, Adrien Lefauve, Rich R. Kerswell, P. F. Linden. Long-wave instabilities of sloping stratified exchange flows. arXiv:2309.10056v1 [physics.flu-dyn]. Sep 18, 2023.

https://doi.org/10.48550/arXiv.2309.10056

Also: waves, instability, transition, turbulence, chaos, in https://www.inkgmr.net/kwrds.html

Keywords: gst, waves, instability, long-wave instability, transition, turbulence, chaos