Series expansions of symmetric elliptic integrals

Fukushima, Toshio

doi:10.1090/s0025-5718-2011-02531-7

Cited by 9 publications

(6 citation statements)

References 39 publications

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“…There are various asymptotic formulae for elliptic integrals. [31] The result of Karp and Sitnik was found to be more accurate than the prior result of Carlson and Gustafson, in the sense of average absolute and relative errors, over a wider range of parameters. [32,33] As → 0, the Karp and Sitnik representation of the divergent term in Eq.…”

Section: The Biot-savart Integral and Local Induction Modelsmentioning

confidence: 63%

Non-Hamiltonian Kelvin wave generation on vortices in Bose-Einstein condensates

Strong¹,

Carr²

2018

Preprint

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Ultra-cold quantum turbulence is expected to decay through a cascade of Kelvin waves. These helical excitations couple vorticity to the quantum fluid causing long wavelength phonon fluctuations in a Bose-Einstein condensate. This interaction is hypothesized to be the route to relaxation for turbulent tangles in quantum hydrodynamics. The local induction approximation is the lowest order approximation to the Biot-Savart velocity field induced by a vortex line and, because of its integrability, is thought to prohibit energy transfer by Kelvin waves. Using the Biot-Savart description, we derive a generalization to the local induction approximation which predicts that regions of large curvature can reconfigure themselves as Kelvin wave packets. While this generalization preserves the arclength metric, a quantity conserved under the Eulerian flow of vortex lines, it also introduces a non-Hamiltonian structure on the geometric properties of the vortex line. It is this non-Hamiltonian evolution of curvature and torsion which provides a resolution to the missing Kelvin wave motion. In this work, we derive corrections to the local induction approximation in powers of curvature and state them for utilization in vortex filament methods. Using the Hasimoto transformation, we arrive at a nonlinear integro-differential equation which reduces to a modified nonlinear Schrödinger type evolution of the curvature and torsion on the vortex line. We show that this modification seeks to disperse localized curvature profiles. At the same time, the non-Hamiltonian break in integrability bolsters the deforming curvature profile and simulations show that this dynamic results in Kelvin wave propagation along the dispersive vortex medium.

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Section: The Biot-savart Integral and Local Induction Modelsmentioning

confidence: 63%

Non-Hamiltonian Kelvin wave generation on vortices in Bose-Einstein condensates

Strong¹,

Carr²

2018

Preprint

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“…This approximation was later improved by Carlson and Gustafson in [14]. On the other hand, new convergent expansions of R D (x, y, z) have been obtained in [17,21] and [22].…”

Section: (): V-volmentioning

confidence: 95%

An Analytic Representation of the Second Symmetric Standard Elliptic Integral in Terms of Elementary Functions

et al. 2022

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We derive new convergent expansions of the symmetric standard elliptic integral $$R_D(x,y,z)$$ R D ( x , y , z ) , for $$x, y,z\in {\mathbb {C}}{\setminus }(-\infty ,0]$$ x , y , z ∈ C \ ( - ∞ , 0 ] , in terms of elementary functions. The expansions hold uniformly for large and small values of one of the three variables x, y or z (with the other two fixed). We proceed by considering a more general parametric integral from which $$R_D(x,y,z)$$ R D ( x , y , z ) is a particular case. It turns out that this parametric integral is an integral representation of the Appell function $$F_1(a;b,c;a+1;x,y)$$ F 1 ( a ; b , c ; a + 1 ; x , y ) . Therefore, as a byproduct, we deduce convergent expansions of $$F_1(a;b,c;a+1;x,y)$$ F 1 ( a ; b , c ; a + 1 ; x , y ) . We also compute error bounds at any order of the approximation. Some numerical examples show the accuracy of the expansions and their uniform features.

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“…[9]). More precise behavior can be found in [3] and references therein. We give a short proof of (A.6).…”

Section: 4mentioning

confidence: 98%

Long range scattering for the complex-valued Klein-Gordon equation with quadratic nonlinearity in two dimensions

Masaki

Segata

Uriya

2018

Preprint

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In this paper, we study large time behavior of complexvalued solutions to nonlinear Klein-Gordon equation with a gauge invariant quadratic nonlinearity in two spatial dimensions. To find a possible asymptotic behavior, we consider the final value problem. It turns out that one possible behavior is a linear solution with a logarithmic phase correction as in the real-valued case. However, the shape of the logarithmic correction term has one more parameter which is also given by the final data. In the real case the parameter is constant so one cannot see its effect. However, in the complex case it varies in general. The one dimensional case is also discussed.

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Series expansions of symmetric elliptic integrals

Cited by 9 publications

References 39 publications

Non-Hamiltonian Kelvin wave generation on vortices in Bose-Einstein condensates

Non-Hamiltonian Kelvin wave generation on vortices in Bose-Einstein condensates

An Analytic Representation of the Second Symmetric Standard Elliptic Integral in Terms of Elementary Functions

Long range scattering for the complex-valued Klein-Gordon equation with quadratic nonlinearity in two dimensions

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