Spatially localized structures

2018

1. Phil. Trans. A

   Spatially localized structures in the Gray–Scott model

   Punit Gandhi, Yuval R. Zelnik, and Edgar Knobloch

   Philosophical Transactions of the Royal Society A, 2018

   Abs DOI Bib

   Spatially localized structures in the one-dimensional Gray–Scott reaction–diffusion model are studied using a combination of numerical continuation techniques and weakly nonlinear theory, focusing on the regime in which the activator and substrate diffusivities are different but comparable. Localized states arise in three different ways: in a subcritical Turing instability present in this regime, and from folds in the branch of spatially periodic Turing states. They also arise from the fold of spatially uniform states. These three solution branches interconnect in complex ways. We use numerical continuation techniques to explore their global behaviour within a formulation of the model that has been used to describe dryland vegetation patterns on a flat terrain. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)’.

   @article{gandhi2018spatially,
     title = {Spatially localized structures in the Gray--Scott model},
     author = {Gandhi, Punit and Zelnik, Yuval R. and Knobloch, Edgar},
     journal = {Philosophical Transactions of the Royal Society A},
     volume = {376},
     number = {2135},
     pages = {20170375},
     year = {2018},
     publisher = {The Royal Society Publishing},
     doi = {10.1098/rsta.2017.0375},
     url = {https://royalsocietypublishing.org/doi/abs/10.1098/rsta.2017.0375},
     eprint = {https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2017.0375}
   }

2. Chaos

   Implications of tristability in pattern-forming ecosystems

   Yuval R. Zelnik, Punit Gandhi, Edgar Knobloch, and Ehud Meron

   Chaos: An Interdisciplinary Journal of Nonlinear Science, Mar 2018

   Abs DOI Bib

   Many ecosystems show both self-organized spatial patterns and multistability of possible states. The combination of these two phenomena in different forms has a significant impact on the behavior of ecosystems in changing environments. One notable case is connected to tristability of two distinct uniform states together with patterned states, which has recently been found in model studies of dryland ecosystems. Using a simple model, we determine the extent of tristability in parameter space, explore its effects on the system dynamics, and consider its implications for state transitions or regime shifts. We analyze the bifurcation structure of model solutions that describe uniform states, periodic patterns, and hybrid states between the former two. We map out the parameter space where these states exist, and note how the different states interact with each other. We further focus on two special implications with ecological significance, breakdown of the snaking range and complex fronts. We find that the organization of the hybrid states within a homoclinic snaking structure breaks down as it meets a Maxwell point where simple fronts are stationary. We also discover a new series of complex fronts between the uniform states, each with its own velocity. We conclude with a brief discussion of the significance of these findings for the dynamics of regime shifts and their potential control.

   @article{zelnik2018implications,
     author = {Zelnik, Yuval R. and Gandhi, Punit and Knobloch, Edgar and Meron, Ehud},
     title = {Implications of tristability in pattern-forming ecosystems},
     journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science},
     volume = {28},
     number = {3},
     pages = {033609},
     year = {2018},
     month = mar,
     issn = {1054-1500},
     doi = {10.1063/1.5018925},
     url = {https://doi.org/10.1063/1.5018925},
     eprint = {https://pubs.aip.org/aip/cha/article-pdf/doi/10.1063/1.5018925/14617421/033609\_1\_online.pdf}
   }

2017

1. Phys. Rev. Fluids

   Slanted snaking of localized Faraday waves

   Bastián Pradenas, Isidora Araya, Marcel G. Clerc, Claudio Falcón, Punit Gandhi, and Edgar Knobloch

   Physical Review Fluids, Mar 2017

   Abs DOI arXiv Bib

   We report on an experimental, theoretical, and numerical study of slanted snaking of spatially localized parametrically excited waves on the surface of a water-surfactant mixture in a Hele-Shaw cell. We demonstrate experimentally the presence of a hysteretic transition to spatially extended parametrically excited surface waves when the acceleration amplitude is varied, as well as the presence of spatially localized waves exhibiting slanted snaking. The latter extend outside the hysteresis loop. We attribute this behavior to the presence of a conserved quantity, the liquid volume trapped within the meniscus, and introduce a universal model based on symmetry arguments, which couples the wave amplitude with such a conserved quantity. The model captures both the observed slanted snaking and the presence of localized waves outside the hysteresis loop, as demonstrated by numerical integration of the model equations.

   @article{pradenas2017slanted,
     title = {Slanted snaking of localized {F}araday waves},
     author = {Pradenas, Basti\'an and Araya, Isidora and Clerc, Marcel G. and Falc\'on, Claudio and Gandhi, Punit and Knobloch, Edgar},
     journal = {Physical Review Fluids},
     volume = {2},
     issue = {6},
     pages = {064401},
     numpages = {12},
     year = {2017},
     publisher = {American Physical Society},
     doi = {10.1103/PhysRevFluids.2.064401},
     url = {https://link.aps.org/doi/10.1103/PhysRevFluids.2.064401}
   }

2016

1. Springer

   Time-periodic forcing of spatially localized structures

   Punit Gandhi, Cédric Beaume, and Edgar Knobloch

   In Nonlinear Dynamics: Materials, Theory and Experiments, Mar 2016

   Abs DOI Bib

   We study localized statesLocalized statesin the Swift–Hohenberg equation when time-periodic parametric forcing is introduced. The presence of a time-dependent forcing introduces a new characteristic time which creates a series of resonances with the depinning time of the fronts bounding the localized pattern. The organization of these resonances in parameter space can be understood using appropriate asymptotics. A number of distinct canard trajectories involved in the observed transitions is constructed.

   @incollection{gandhi2016time,
     year = {2016},
     booktitle = {Nonlinear Dynamics: Materials, Theory and Experiments},
     volume = {173},
     series = {Springer Proceedings in Physics},
     editor = {Clerc, Marcel G. and Tlidi, Mustafa},
     title = {Time-periodic forcing of spatially localized structures},
     url = {https://link.springer.com/chapter/10.1007/978-3-319-24871-4_23},
     doi = {https://doi.org/10.1007/978-3-319-24871-4_23},
     publisher = {Springer International Publishing},
     author = {Gandhi, Punit and Beaume, C\'edric and Knobloch, Edgar},
     pages = {303--316},
     isbn = {978-3-319-24871-4}
   }

2015

1. SIADS

   A new resonance mechanism in the Swift–Hohenberg equation with time-periodic forcing

   Punit Gandhi, Cédric Beaume, and Edgar Knobloch

   SIAM Journal on Applied Dynamical Systems, Mar 2015

   Abs DOI arXiv Bib

   The generalized Swift–Hohenberg equation with a quadratic-cubic nonlinearity is used to study the persistence and decay of localized patterns in the presence of time-periodic parametric forcing. A novel resonance phenomenon between the forcing period and the time required to nucleate one wavelength of the pattern outside the pinning region is identified. The resonance generates distinct regions in parameter space characterized by the net number of wavelengths gained or lost in one forcing cycle. These regions are well described by an asymptotic theory based on the wavelength nucleation/annihilation time near the boundaries of the pinning region. The resulting theory leads to predictions that are qualitatively correct and, in some cases, provide quantitative agreement with numerical simulations.

   @article{gandhi2015new,
     title = {A new resonance mechanism in the {S}wift--{H}ohenberg equation with time-periodic forcing},
     author = {Gandhi, Punit and Beaume, C\'edric and Knobloch, Edgar},
     journal = {SIAM Journal on Applied Dynamical Systems},
     volume = {14},
     pages = {860--892},
     year = {2015},
     doi = {10.1137/14099468X},
     url = {https://doi.org/10.1137/14099468X},
     eprint = {  https://doi.org/10.1137/14099468X}
   }

2. PRL

   Localized states in periodically forced systems

   Punit Gandhi, Edgar Knobloch, and Cédric Beaume

   Physical Review Letters, Mar 2015

   Abs DOI arXiv Bib

   The theory of stationary spatially localized patterns in dissipative systems driven by time-independent forcing is well developed. With time-periodic forcing, related but time-dependent structures may result. These may consist of breathing localized patterns, or states that grow for part of the cycle via nucleation of new wavelengths of the pattern followed by wavelength annihilation during the remainder of the cycle. These two competing processes lead to a complex phase diagram whose structure is a consequence of a series of resonances between the nucleation time and the forcing period. The resulting diagram is computed for the periodically forced quadratic-cubic Swift–Hohenberg equation, and its details are interpreted in terms of the properties of the depinning transition for the fronts bounding the localized state on either side. The results are expected to shed light on localized states in a large variety of periodically driven systems.

   @article{gandhi2015localized,
     title = {Localized states in periodically forced systems},
     author = {Gandhi, Punit and Knobloch, Edgar and Beaume, C\'edric},
     journal = {Physical Review Letters},
     volume = {114},
     number = {3},
     pages = {034102},
     year = {2015},
     publisher = {APS},
     doi = {10.1103/PhysRevLett.114.034102},
     url = {https://link.aps.org/doi/10.1103/PhysRevLett.114.034102}
   }