## WAVE COMPUTATION ON THE HYPERBOLIC DOUBLE DOUGHNUT (902)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Bachelot-motet902wavecomputation,

author = {Agnès Bachelot-motet},

title = {WAVE COMPUTATION ON THE HYPERBOLIC DOUBLE DOUGHNUT},

year = {902}

}

### OpenURL

### Abstract

Abstract. We compute the waves propagating on the compact surface of constant negative curvature and genus 2. We adopt a variational approach using finite elements. We have to implement the action of the fuchsian group by suitable boundary conditions of periodic type. A spectral analysis of the wave allows to compute the spectrum and the eigenfunctions of the Laplace-Beltrami operator. We test the exponential decay due to a localized dumping and the ergodicity of the geodesic flow.

### Citations

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(Show Context)
Citation Context ...domain F ⊂ D by a suitable relation of equivalence ∼ : K = D/Γ = F/ ∼ . This beautiful object has many fascinating properties as regards the classical and quantum chaos (classical references are [1], =-=[5]-=-). Several important computational investigations of the spectrum were performed by using a stationnary method by R. Aurich and F. Steiner [4]. Moreover there has been much recent interest for the cos... |

14 |
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Citation Context ...lassical and quantum chaos (classical references are [1], [5]). Several important computational investigations of the spectrum were performed by using a stationnary method by R. Aurich and F. Steiner =-=[4]-=-. Moreover there has been much recent interest for the cosmological models with non trivial topology (a seminal work is the famous “Cosmic Topology” by M. Lachièze-Rey and J-P. Luminet [7]). In this c... |

13 |
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(Show Context)
Citation Context ...utions of the damped wave equation ∂ 2 t ψ − ∆Kψ + a∂tψ = 0. When 0 ≤ a ∈ L ∞ (K) and a > 0 on ∂F, the ergodicity of the geodesic flow assures that the geometric control condition of Rauch and Taylor =-=[9]-=- is satisfied. Our numerical experiments agree with their theoretical results, stating that the energy decays exponentially. II. The Hyperbolic Double Doughnut. In this part we describe the constructi... |

12 |
Ergodic Theory and Topological
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(Show Context)
Citation Context ...g0g −1 −1 1 g2g3 )(g−1 −1 0 g1g2 g3) = Id. The Hyperbolic Double Doughnut is the quotient manifold (II.12) K := D/Γ, endowed with the hyperbolic metric ds2 K induced by ds2 D . We know, see e.g. [1], =-=[2]-=-, [5], that K is a two dimensional C∞ compact manifold, without boundary, its sectional curvature is constant, equal to −1, and its genus, that is the number of “holes”, is 2. The geodesic flow is ver... |

10 | Eigenfrequencies and expansions for damped wave equations
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(Show Context)
Citation Context ... 1 2 √ π . Therefore any finite energy solution ψ(t, x, y) of ∂2 t ψ − ∆Kψ = 0 has an expansion of the form ∑ q eiqt ψq(x, y) (such expansions exist also for the damped wave equation, when a > 0, see =-=[6]-=-). More precisely, if we denote <, > the scalar product in L 2 (K), we write (IV.3) ψ(t, x, y) = 1 4π (< ∂tψ(0, .), 1 > t+ < ψ(0, .), 1 >) + ∑ sin qt < ∂tψ(0, .), ψq > q ψq(x, y)+ < ψ(0, .), ψq > cos ... |

10 |
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Citation Context ...) 2 cos ωT − cos ωT cos qT ] ) T − . 2q IV.7. Damped waves. We test our scheme for the damped wave equation (III.3) when the damping function a ≥ 0 is non zero (for deep theoretical results, see [6], =-=[8]-=-, [9]). We know that the energy of any finite energy solution decays exponentially (uniformly with respect to the initial energy) iff the dumping a satisfies the assumption of geometric control introd... |

5 |
Chaos On The Pseudosphere
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(Show Context)
Citation Context ...ntal domain F ⊂ D by a suitable relation of equivalence ∼ : K = D/Γ = F/ ∼ . This beautiful object has many fascinating properties as regards the classical and quantum chaos (classical references are =-=[1]-=-, [5]). Several important computational investigations of the spectrum were performed by using a stationnary method by R. Aurich and F. Steiner [4]. Moreover there has been much recent interest for th... |

4 | Ringing the eigenmodes from compact manifolds, preprint gr-qc/9802066
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- 1997
(Show Context)
Citation Context ...st for the cosmological models with non trivial topology (a seminal work is the famous “Cosmic Topology” by M. Lachièze-Rey and J-P. Luminet [7]). In this context, K has been studied as a paradigm in =-=[3]-=-, where a schema based on the finite differences on a euclidean grid was used to solve the D’Alembertian. In this paper we compute the solutions of the wave equations in the time domain, by using a va... |