Green's function for the Laplace–Beltrami operator on a toroidal surface

Green, Christopher C.; Marshall, J. S.

doi:10.1098/rspa.2012.0479

Cited by 17 publications

(19 citation statements)

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“…The Neumann Green's function for the Laplace-Beltrami operator will be central to this extension, and in general must be computed numerically. However, recently, explicit formulae for it have been obtained for the torus [33] and for certain cylinders of revolution [29].…”

Section: A)mentioning

confidence: 99%

“…In this sub-section we highlight some results for the linear stability of steady-state periodic spot patterns for the Brusselator (1.2) when the spots are centered in the limit ǫ → 0 at the lattice points of a general oblique Bravais lattice Λ with fixed area |Ω| of the primitive cell of the lattice. To leading order in ν = −1/ log ǫ, the linearization of the steady-state periodic spot pattern has a zero eigenvalue when D = D 0c /ν, where 33) and where w(ρ) is the ground-state satisfying (A.1 c) . This leading-order threshold depends only on the area |Ω| of the primitive cell, and is independent of the specific lattice Λ. Analogous to that analyzed in §3.1 for spot patterns on a finite domain, this zero eigenvalue corresponds to a competition instability of the spot amplitudes.…”

Section: Periodic Spot Patternsmentioning

confidence: 99%

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Spots, traps, and patches: asymptotic analysis of localized solutions to some linear and nonlinear diffusive systems

Ward

2018

Nonlinearity

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Localized spatial patterns commonly occur for various classes of linear and nonlinear diffusive processes. In particular, localized spot patterns, where the solution concentrates at discrete points in the domain, occur in the nonlinear reactiondiffusion (RD) modeling of diverse phenomena such as chemical patterns, biological morphogenesis, and the spatial distribution of urban crime. In a 2-D spatial domain we survey some recent and new results for the existence, linear stability, and slow dynamics of localized spot patterns by using the Brusselator RD model as the prototypical example. In the context of linear diffusive systems with localized solution behavior, we will discuss some previous results for the determination of the mean first capture time for a Brownian particle in a 2-D domain with localized traps, and the determination of the persistence threshold of a species in a 2-D landscape with patchy food resources. Common features in the analysis of all of these spatially localized patterns are emphasized, including the key role of certain matrices involving various Green's functions, and the derivation and study of new classes of interacting particle systems and discrete variational problems arising from various asymptotic reductions. The mathematical tools include matched asymptotic analysis based on strong localized perturbation theory, spectral analysis, the analysis of nonlocal eigenvalue problems, and bifurcation theory. Some specific open problems are highlighted and, more broadly, we will discuss a few new research frontiers for the analysis of localized patterns in multi-dimensional domains.

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Section: A)mentioning

confidence: 99%

Section: Periodic Spot Patternsmentioning

confidence: 99%

Spots, traps, and patches: asymptotic analysis of localized solutions to some linear and nonlinear diffusive systems

Ward

2018

Nonlinearity

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“…The purpose of the present study is constructing Stuart vortex on the surface of a torus, a compact surface having different geometric features from the plane and the sphere. Since vortex dynamics on the toroidal surface has recently been formulated with using the analytic representation of Green's function on the surface [9], a few steady flows with vortex structures are known up to now: it has been shown in [10] that point vortices located at the antipodal positions, and a polygonal ring configuration of identical N point vortices along the line of a latitude become point vortex equilibria.…”

Section: Introductionmentioning

confidence: 99%

Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus

Sakajo

2019

Proc. R. Soc. A.

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A steady solution of the incompressible Euler equation on a toroidal surface T R , r of major radius R and minor radius r is provided. Its streamfunction is represented by an exact solution to the modified Liouville equation, ∇ T R , r 2 ψ = c e d ψ + ( 8 / d ) κ , where ∇ T R , r 2 and κ denote the Laplace–Beltrami operator and the Gauss curvature of the toroidal surface respectively, and c , d are real parameters with cd < 0. This is a generalization of the flows with smooth vorticity distributions owing to Stuart (Stuart 1967 J. Fluid Mech. 29 , 417–440. ( doi:10.1017/S0022112067000941 )) in the plane and Crowdy (Crowdy 2004 J. Fluid Mech. 498 , 381–402. ( doi:10.1017/S0022112003007043 )) on the spherical surface. The flow consists of two point vortices at the innermost and the outermost points of the toroidal surface on the same line of a longitude, and a smooth vorticity distribution centred at their antipodal position. Since the surface of a torus has non-constant curvature and a handle structure that are different geometric features from the plane and the spherical surface, we focus on how these geometric properties of the torus affect the topological flow structures along with the change of the aspect ratio α = R / r . A comparison with the Stuart vortex on the flat torus is also made.

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“…The Green function for Laplace-Beltrami operator was calculated recently [24] and several other studies were performed in order to solve the equation for a particle propagating along a toroidal surface [25][26][27][28]. However, up to now, there is no solution [29].…”

Section: Introductionmentioning

confidence: 99%

Exact Solutions to Schrödinger Equation for a Charged Particle on a Torus in Uniform Electric and Magnetic Fields

Schmidt

2020

Braz J Phys

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Green's function for the Laplace–Beltrami operator on a toroidal surface

Cited by 17 publications

References 28 publications

Spots, traps, and patches: asymptotic analysis of localized solutions to some linear and nonlinear diffusive systems

Spots, traps, and patches: asymptotic analysis of localized solutions to some linear and nonlinear diffusive systems

Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus

Exact Solutions to Schrödinger Equation for a Charged Particle on a Torus in Uniform Electric and Magnetic Fields

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