# 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: apropos of bubbles, the case of bubbles collapsing near a wall.

AA << study examines the pressure exerted by a cavitation bubble collapsing near a rigid wall. A laser-generated bubble in a water basin undergoes growth, collapse, second growth, and final collapse. Shock waves and liquid jets from non-spherical collapses are influenced by the stand-off ratio γ, defined as the bubble centroid distance from the wall divided by the bubble radius. (AA) detail shock mechanisms, such as tip or torus collapse, for various γ values. High-speed and Schlieren imaging visualize the microjet and shock waves. The microjet's evolution is tracked for large γ, while shock waves are captured in composite images showing multiple shock positions. Quantitative analyses of the microjet interface, shock wave velocities, and impact times are reported. Wall-mounted sensors and a needle hydrophone measure pressure and compare with high-speed observations to assess the dominant contributions to pressure changes with γ, revealing implications for cavitation erosion mechanisms. >>️

Roshan Kumar Subramanian, Zhidian Yang, et al. Bubble collapse near a wall. Part 1: An experimental study on the impact of shock waves and microjet on the wall pressure. arXiv: 2408.03479v2 [physics.flu-dyn]. Aug 8, 2024. 

https://arxiv.org/abs/2408.03479

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

Keywords: gst, bubble, bubble collapse 

# gst: apropos of breaking mechanisms, crack of a floating particle raft caused by waves.

<< When particles of a few tens of microns are spread on the surface of water, they aggregate under the action of capillary forces and form a thin floating membrane, a particle raft. >>

<< For a sufficiently strong wave amplitude, the raft breaks up progressively by developing cracks and producing fragments whose sizes decrease on a timescale long compared to the period of the wave. (AA)  characterize the breaking mechanisms.  >>️️

<< The visual appearance of the fragments distributed in size and surrounded by open water bears a notable resemblance to the floes produced by the fracturing of sea ice by waves in the polar oceans. Fragmentation concepts and morphological tools built for sea ice floes can be applied to (AA) macroscopic analog, on which the entire dynamic evolution is accessible. However, the mechanics of the two systems differ, as (AA) particle raft breaks due to the viscous stresses, whereas the sea ice fractures due to its bending by the waves. >>️

Louis Saddier, Ambre Palotai, et al. Breaking of a floating particle raft by water waves. Phys. Rev. Fluids 9, 094302. Sep 27, 2024.

https://journals.aps.org/prfluids/abstract/10.1103/PhysRevFluids.9.094302 

Also: 'crack', 'particle', in  https://www.inkgmr.net/kwrds.html 

Also: 'break' in  https://flashontrack.blogspot.com/search?q=break

Also: 'raft' in  https://flashontrack.blogspot.com/search?q=raft

Also: 'zattera' (quasi-stochastic poetry) in  https://inkpi.blogspot.com/search?q=zattera  'randa' (quasi-stochastic poetry) in https://inkpi.blogspot.com/search?q=randa

Keywords: gst, crack, break, particle, raft, particle raft

# 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: apropos of synchronization of large bubble clouds, self-oscillating acoustic bubbles that bounce

<< Self-oscillations underlie many natural phenomena such as heartbeat, ocean waves, and the pulsation of variable stars. (..) In this study, (AA) consider two closely spaced bubbles pulsating in the kHz range in response to ultrasonic excitation. A translational bouncing motion emerges from their interaction with a much lower frequency than the bubble pulsation frequency. >>

Their analysis << reveals that the observed bubble bouncing exhibits the main features of self-oscillation, such as negative damping and the emergence of a limit cycle. These results highlight unexpected nonlinear effects in the field of microbubbles and give insights into the understanding of synchronization in large bubble clouds. >>️

Gabriel Regnault, Alexander A Doinikov, et al. Phenomenon of self-oscillation in bubble dynamics: Bouncing acoustic bubbles. arXiv: 2404.05822v1 [physics.flu-dyn]. Apr 8, 2024.

https://arxiv.org/abs/2404.05822

Also: bubble, bouncing, behav, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, bubble, self-oscillation, bouncing, behavior, collective behavior, bouncing motion, synchronization

# 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: exploring the on-demand dynamical generation of a plethora of dispersive shock waves arising in attractive one-dimensional droplet-bearing environment.

AA << demonstrate the controllable generation of distinct types of dispersive shock-waves emerging in a quantum droplet bearing environment with the aid of step-like initial conditions. Dispersive regularization of the ensuing hydrodynamic singularities occurs due to the competition between meanfield repulsion and attractive quantum fluctuations. This interplay delineates the dominance of defocusing (hyperbolic) and focusing (elliptic) hydrodynamic phenomena respectively being designated by real and imaginary speed of sound. >>

<< Surprisingly, dispersive shock waves persist across the hyperbolic-to-elliptic threshold, while a plethora of additional wave patterns arise, such as rarefaction waves, traveling dispersive shock waves, (anti)kinks and droplet wavetrains. >>

AA << results pave the way for unveiling a multitude of unexplored coherently propagating waveforms in such attractively interacting mixtures. >>

Sathyanarayanan Chandramouli, Simeon I. Mistakidis, Garyfallia C. Katsimiga, Panayotis G. Kevrekidis. 

Dispersive shock waves in a one-dimensional droplet-bearing environment. arXiv: 2404.02998v2 [nlin.PS]. Apr 5, 2024. 

https://arxiv.org/abs/2404.02998

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

Keywords: gst, waves, drops 

# 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: droplets scoot like caterpillars.

<< From swells in an ocean to ripples in a puddle, the shearing effect of wind blowing over a liquid is visible at all scales. This shear determines the interactions between Earth’s atmosphere and its surface water and, researchers now explain, the movement of liquid droplets that crawl up and down the window of a moving car in the rain. In a series of experiments, (AA) show that airflow triggers surface waves that cause such droplets to crawl like caterpillars before they break apart. >>️

<< At first, the airflow across the droplet’s surface caused the droplet to extend into an oval shape. The droplet also began to tilt, with the liquid piling up at the droplet’s downwind edge. When the drag force exerted by the airflow overcame the capillary force between the glycerin and the glass, the droplet began to slide and to stretch out even more. Surface waves then developed on the elongated droplet and traveled toward its leading edge. The waves induced a stable caterpillar-like motion, with the droplet stretching and contracting along its length. Eventually, beyond a threshold length that depended on the droplet’s volume, the caterpillar was no longer able to withstand the shearing force and broke into several droplets. >>️

AA << say that the behavior follows the same pattern as that of an elongated droplet sliding along an incline. >>

️

Rachel Berkowitz. Droplets Scoot Like Caterpillars. Physics 16, s110. Sep 1, 2023.

https://physics.aps.org/articles/v16/s110  

A. Chahine, J. Sebilleau, R. Mathis, D. Legendre. Caterpillar like motion of droplet in a shear flow. Phys. Rev. Fluids 8, 093601. Sep 1, 2023.

https://journals.aps.org/prfluids/abstract/10.1103/PhysRevFluids.8.093601

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

Keywords: gst, drop, droplet, droploid, bubble, transition

# 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

# gst: apropos of collisions of a vortex dipoles, vortex unbinding, vortex pinning to the edge, and emission of rarefaction pulses.

<< Vortices and dark solitons are fundamental defects that appear in nonlinear physics at all scales, from nonlinear optics to cosmic fluids. In two or three dimensions, dark solitons are fundamentally connected to highly ordered vortex states through an intrinsic modulation instability in the surrounding matter. >>️

<< Here, [AA] report an experimental realization of ring dark soliton (RDS)  generation in a two-dimensional atomic superfluid trapped in a circular box. By quenching the confining box potential, [they] observe an RDS emitted from the edge and its peculiar signature in the radial motion. As an RDS evolves, [they] observe transverse modulations at discrete azimuthal angles, which clearly result in a patterned formation of a circular vortex dipole array. Through collisions of the vortex dipoles with the box trap, [they] observe vortex unbinding, vortex pinning to the edge, and emission of rarefaction pulses. >>

Hikaru Tamura, Cheng-An Chen, Chen-Lung Hung. Observation of Self-Patterned Defect Formation in Atomic Superfluids–from Ring Dark Solitons to Vortex Dipole Necklaces. Phys. Rev. X 13, 031029. Sep 14, 2023. 

https://journals.aps.org/prx/abstract/10.1103/PhysRevX.13.031029

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

Keywords: gst, soliton, dark soliton, vortex, turbulence, waves. 

# gst: dynamics and interactions of multiple bright droplets and bubbles, and interactions of kinks with droplets and with antikinks.

<< Droplets bearing different chemical potentials experience mass-exchange phenomena. Individual bubbles exhibit core expansion and mutual attraction prior to their destabilization. Droplets interacting with kinks are absorbed by them, a process accompanied by the emission of dispersive shock waves and gray solitons. Kink-antikink interactions are repulsive, generating counter-propagating shock waves. >> 

 

G. C. Katsimiga, S. I. Mistakidis, et al. Interactions and dynamics of one-dimensional droplets, bubbles and kinks. arXiv: 2306.07055v2 [cond-mat.quant-gas]. Jul 26, 2023.

https://arxiv.org/abs/2306.07055

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

Keywords: gst, drop, bubble, kink, wave, soliton, homoclinic orbits