Spots in the Swift--Hohenberg Equation

McCalla, Scott G.; Sandstede, Björn

doi:10.1137/120882111

Cited by 30 publications

(101 citation statements)

References 20 publications

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“…When searching for radial solutions the problem is reduced to the study of some specific solutions of the related nonautonomous second order ODE with "time" r. For some types of nonlinearities the existence of infinitely many (sign-changing) radial solutions was proved in [16]. Those results were extended to more complicated higher order equations [22,26].…”

Section: Introductionmentioning

confidence: 99%

Abundance of entire solutions to nonlinear elliptic equations by the variational method

2020

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We study entire bounded solutions to the equation ∆u − u + u 3 = 0 in R 2 . Our approach is purely variational and is based on concentration arguments and symmetry considerations. This method allows us to construct in a unified way several types of solutions with various symmetries (radial, breather type, rectangular, triangular, hexagonal, etc.), both positive and sign-changing. The method is also applicable for more general equations in any dimension.

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Section: Introductionmentioning

confidence: 99%

Abundance of entire solutions to nonlinear elliptic equations by the variational method

2020

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“…Indeed, in many experimental studies [29,51,83], localized regions of cortical activity closely resemble a spot A solution with the shape of a J 0 (r)-Bessel function at the core with no ring structure. For the localized ring and spot B proofs one requires an additional hypothesis that the stable and unstable manifolds of the ring solution in the far-field normal form transversely intersect; see [82]. Spot B and localized rings have not been observed experimentally and we shall not discuss these further.…”

Section: Introductionmentioning

confidence: 99%

Localized radial bumps of a neural field equation on the Euclidean plane and the Poincaré disc

2013

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We analyze radially symmetric localized bump solutions of an integro-differential neural field equation posed in Euclidean and hyperbolic geometry. The connectivity function and the nonlinear firing rate function are chosen such that radial spatial dynamics can be considered. Using integral transforms, we derive a PDE for the neural field equation in both geometries and then prove the existence of small amplitude radially symmetric spots bifurcating from the trivial state. Numerical continuation is then used to path-follow the spots and their bifurcations away from onset in parameter space. It is found that the radial bumps in Euclidean geometry are linearly stable in a larger parameter region than bumps in the hyperbolic geometry. We also find and path-follow localized structures that bifurcate from branches of radially symmetric solutions with D6-symmetry and D8-symmetry in the Euclidean and hyperbolic cases, respectively. Finally, we discuss the applications of our results in the context of neural field models of short term memory and edges and textures selectivity in a hypercolumn of the visual cortex.

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“…In spite of its robustness, dynamical systems techniques do not seem to be broad enough to capture unsurmountable difficulties in the study of multidimensional patterns. Recently, different research avenues have been exploited: several studies have been done using rigorous numerical analysis [MS13], harmonic analysis techniques [JS15, Jar15, BLBL12], variational techniques [Rab94], or more functional-analytic based techniques [MS18]. Still, many classes of problems remain unsolved, as that of asymmetrical grain boundaries, a case that does not seem to be directly amenable to the spatial dynamics techniques as presented in [HS12,SW14]; in this scenario the far/near decompositions we presented might be relevant for analytical results.…”

Section: Invasion Fronts and The Role Of χ(·)mentioning

confidence: 99%

The Swift–Hohenberg Equation under Directional-Quenching: Finding Heteroclinic Connections Using Spatial and Spectral Decompositions

Monteiro¹,

Yoshinaga

2019

Arch Rational Mech Anal

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We study the existence of patterns (nontrivial, stationary solutions) in the one-dimensional Swift-Hohenberg Equation in a directional quenching scenario, that is, on x ≤ 0 the energy potential associated to the equation is bistable, whereas on x ≥ 0 it is monostable. This heterogeneity in the medium induces a symmetry break that makes the existence of heteroclinic orbits of the type point-to-periodic not only plausible but, as we prove here, true. In this search, we use an interesting result of [FLTW17] in order to understand the multiscale structure of the problem, namely, how multiple scales -fast/slow-interact with each other. In passing, we advocate for a new approach in finding connecting orbits, using what we call "far/near decompositions", relying both on information about the spatial behavior of the solutions and on Fourier analysis. Our method is functional analytic and PDE based, relying minimally on dynamical system techniques and making no use of comparison principles whatsoever.

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Spots in the Swift--Hohenberg Equation

Cited by 30 publications

References 20 publications

Abundance of entire solutions to nonlinear elliptic equations by the variational method

Abundance of entire solutions to nonlinear elliptic equations by the variational method

Localized radial bumps of a neural field equation on the Euclidean plane and the Poincaré disc

The Swift–Hohenberg Equation under Directional-Quenching: Finding Heteroclinic Connections Using Spatial and Spectral Decompositions

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