Tag Archives: Exactly Solvable and Integrable Systems

Black Hole Greybody Factors from Korteweg-de Vries Integrals: Computation [CL]

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http://arxiv.org/abs/2301.01096


It has recently been shown that the dynamics of perturbed non-rotating black holes (BHs) admits an infinite number of symmetries that are generated by the flow of the Korteweg-de Vries (KdV) equation. These symmetries lead to an infinite number of conserved quantities that can be obtained as integrals of differential polynomials in the potential appearing in the gauge-invariant master equations describing the BH perturbations, the KdV integrals. These conserved quantities are the same for all the possible potentials, which means that they are invariant under Darboux transformations, and they fully determine the BHs transmission amplitudes, or greybody factors, via a moment problem. In this paper we introduce a new semi-analytical method to obtain the greybody factors associated with BH scattering processes by solving the moment problem using only the KdV integrals. The method is based on the use of Pad\’e approximants and we check it first by comparing with results from the case of a P\”oschl-Teller potential, for which we have analytical expressions for the greybody factors. Then, we apply it to the case of a Schwarzschild BH and compare with results from computations based on the Wentzel-Kramers-Brillouin (WKB) approximation. It turns out that the new method provides accurate results for the BH greybody factors for all frequencies. The method is also computationally very efficient.

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M. Lenzi and C. Sopuerta
Wed, 4 Jan 23
3/43

Comments: 21 pages, 13 figures, 26 plots

Collinear and triangular solutions to the three-body problem in the parameterized post-Newtonian formalism [CL]

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http://arxiv.org/abs/2212.00198


This paper investigates the three-body problem in the parameterized post-Newtonian (PPN) formalism, for which we focus on a coplanar case in a class of fully conservative theories characterized by the Eddington-Robertson parameters $\beta$ and $\gamma$. It is shown that there can still exist a collinear equilibrium configuration and a triangular one, each of which is a generalization of the post-Newtonian equilibrium configuration in general relativity. The collinear configuration can exist for arbitrary mass ratio, $\beta$, and $\gamma$. On the other hand, the PPN triangular configuration depends on the nonlinearity parameter $\beta$ but not on $\gamma$. For any value of $\beta$, the equilateral configuration is possible, if and only if three finite masses are equal or two test masses orbit around one finite mass. For general mass cases, the PPN triangle is not equilateral as in the post-Newtonian case. It is shown also that the PPN displacements from the standard Lagrange points $L_1$, $L_2$ and $L_3$ depend on $\beta$ and $\gamma$, whereas those to $L_4$ and $L_5$ rely only on $\beta$.

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Y. Nakamura and H. Asada
Fri, 2 Dec 22
79/81

Comments: 8 pages, 2 figures

Extreme mass ratio inspirals into black holes surrounded by matter [CL]

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http://arxiv.org/abs/2205.08516


Inspirals of stellar-mass compact objects into massive black holes, known as extreme mass ratio inspirals (EMRIs), are one of the key targets for upcoming space-based gravitational-wave detectors. In this paper we take the first steps needed to systematically incorporate the effect of external gravitating matter on EMRIs. We model the inspiral as taking place in the field of a Schwarzschild black hole perturbed by the gravitational field of a far axisymmetric distribution of mass enclosing the system. We take into account the redshift, frame-dragging, and quadrupolar tide caused by the enclosing matter, thus incorporating all effects to inverse third order of the characteristic distance of the enclosing mass. Then, we use canonical perturbation theory to obtain the action-angle coordinates and Hamiltonian for mildly eccentric precessing test-particle orbits in this background. Finally, we use this to efficiently compute mildly eccentric inspirals in this field and document their properties. This work shows the advantages of canonical perturbation theory for the modeling EMRIs, especially in the cases when the background deviates from the standard black hole fields.

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L. Polcar, G. Lukes-Gerakopoulos and V. Witzany
Thu, 19 May 22
51/61

Comments: N/A

AI Poincaré 2.0: Machine Learning Conservation Laws from Differential Equations [CL]

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http://arxiv.org/abs/2203.12610


We present a machine learning algorithm that discovers conservation laws from differential equations, both numerically (parametrized as neural networks) and symbolically, ensuring their functional independence (a non-linear generalization of linear independence). Our independence module can be viewed as a nonlinear generalization of singular value decomposition. Our method can readily handle inductive biases for conservation laws. We validate it with examples including the 3-body problem, the KdV equation and nonlinear Schr\”odinger equation.

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Z. Liu, V. Madhavan and M. Tegmark
Thu, 24 Mar 22
21/56

Comments: 17 pages, 10 figures

Two-Scale Oscillons [CL]

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http://arxiv.org/abs/1612.07228


Oscillons are spatially stationary, quasi-periodic solutions of nonlinear field theories seen in settings ranging from granular systems, low temperature condensates and early universe cosmology. We describe a new class of oscillon in which the spatial envelope can have “off centre” maxima and pulsate on timescales much longer than the fundamental frequency. These are exact solutions of the 1-D sine-Gordon equation and we demonstrate numerically that similar solutions exist in up to three dimensions for a range of potentials. The dynamics of these solutions match key properties of oscillons that may form after cosmological inflation in string-motivated monodromy scenarios.

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C. Liu and R. Easther
Thu, 22 Dec 16
30/65

Comments: 4 pages, figues; animations and further background at this http URL

Integrable Cosmological Potentials [CL]

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http://arxiv.org/abs/1608.08511


The problem of classification of the Einstein–Friedman cosmological Hamiltonians $H$ with a single scalar inflaton field $\varphi$ that possess an additional integral of motion polynomial in momenta on the shell of the Friedman constraint $H=0$ is considered. Necessary and sufficient conditions for the existence of first, second, and third degree integrals are derived. These conditions have the form of ODEs for the cosmological potential $V(\varphi)$. In the case of linear and quadratic integrals we find general solutions of the ODEs and construct the corresponding integrals explicitly. A new wide class of Hamiltonians that possess a cubic integral is derived. The corresponding potentials are represented in a parametric form in terms of the associated Legendre functions. Six families of special elementary solutions are described and sporadic superintegrable cases are discussed.

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V. Sokolov and A. Sorin
Wed, 31 Aug 16
47/61

Comments: 24 pages, LaTeX, 2 figures

On the astrodynamics applications of Weierstrass elliptic and related functions [EPA]

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http://arxiv.org/abs/1601.04963


Weierstrass elliptic and related functions have been recently shown to enable analytical explicit solutions to classical problems in astrodynamics. These include the constant radial acceleration problem, the Stark problem and the two-fixed center (or Euler’s) problem. In this paper we review the basic technique that allows for these results and we discuss the limits and merits of the approach. Applications to interplanetary trajectory design are then discussed including low-thrust planetary fly-bys and the motion of an artificial satellite under the influence of an oblate primary including $J_2$ and $J_3$ harmonics.

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D. Biscani
Wed, 20 Jan 16
30/58

Comments: Presented at the AAS/AIAA Space Flight Mechanics Meeting, Napa, CA in February 14, 2016

Figures of equilibrium of an inhomogeneous self-gravitating fluid [CL]

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http://arxiv.org/abs/1407.3196


This paper is concerned with the figures of equilibrium of a self-gravitating ideal fluid with density stratification and a steady-state velocity field. As in the classical setting, it is assumed that the figures or their layers uniformly rotate about an axis fixed in space.

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I. Bizyaev, A. Borisov and I. Mamaev
Mon, 14 Jul 14
1/64

Comments: N/A

Integrable cosmological models with non-minimally coupled scalar fields [CL]

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http://arxiv.org/abs/1312.3540


We construct general solutions for flat Friedmann universes filled with a scalar field in induced gravity models and models including the Hilbert-Einstein curvature term plus a scalar field conformally coupled to gravity. The corresponding models are connected with minimally coupled models through the combination of a conformal transformation and a transformation of the scalar field. The explicit forms of the self-interaction potentials for six exactly solvable models are presented here. We argue that although being mathematically in a one-to-one correspondence with the solutions in the minimally coupled models, the solutions in the respective non-minimally coupled models are physically different. This is because the cosmological evolutions seen by an internal observer connected with the cosmic time can be quite different. We give an explicit example of such a difference.

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Tue, 17 Dec 13
55/78