A Nonlocal Kac-van Moerbeke Equation Admitting N-Soliton Solutions

Ruijsenaars, S. N. M.

doi:10.2991/jnmp.2002.9.s1.16

Cited by 3 publications

(2 citation statements)

References 19 publications

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“…also [9,10]). Their eigenfunctions are of the form W(a, b, µ; x, p) = e ixp 1 − Here we have a, b ∈ C N , whereas the generalized norming 'constants', µ 1 (x), .…”

Section: Introductionmentioning

confidence: 97%

Isometric Reflectionless Eigenfunction Transforms for Higher-order AOs

Ruijsenaars¹

2005

JNMP

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In a previous paper (Regular and Chaotic Dynamics 7 (2002), 351-391, Ref. [1]), we obtained various results concerning reflectionless Hilbert space transforms arising from a general Cauchy system. Here we extend these results, proving in particular an isometry property conjectured in Ref. [1]. Crucial input for the proof comes from previous work on a special class of relativistic Calogero-Moser systems. Specifically, we exploit results on action-angle maps for the pertinent systems and their relation to the 2D Toda soliton tau-functions. The reflectionless transforms may be viewed as eigenfunction transforms for an algebra of higher-order analytic difference operators.

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“…also [9,10]). Their eigenfunctions are of the form W(a, b, µ; x, p) = e ixp 1 − Here we have a, b ∈ C N , whereas the generalized norming 'constants', µ 1 (x), .…”

Section: Introductionmentioning

confidence: 97%

Isometric Reflectionless Eigenfunction Transforms for Higher-order AOs

Ruijsenaars¹

2005

JNMP

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“…(We mention in passing that for reflectionless second order A Os this doubling phenomenon admits a quite general formulation, cf. [9,10]. )…”

Section: Introductionmentioning

confidence: 96%

Quadratic transformations for a function that generalizes 2F1 and the Askey-Wilson polynomials

Ruijsenaars¹

2007

Ramanujan J

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In previous papers we introduced and studied a 'relativistic' hypergeometric function R(a + , a − , c; v,v) that satisfies four hyperbolic difference equations of AskeyWilson type. Specializing the family of couplings c ∈ C 4 to suitable two-dimensional subfamilies, we obtain doubling identities that may be viewed as generalized quadratic transformations. Specifically, they give rise to a quadratic transformation for 2 F 1 in the 'nonrelativistic' limit, and they yield quadratic transformations for the AskeyWilson polynomials when the variables v orv are suitably discretized. For the general coupling case, we also study the bearing of several previous results on the AskeyWilson polynomials.

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A note on N-soliton solutions for the viscid incompressible Navier–Stokes differential equation

Meulens¹

2022

AIP Advances

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Repetitive curling of the incompressible viscid Navier–Stokes differential equation leads to a higher-order diffusion equation. Substituting this equation into the Navier–Stokes differential equation transposes the latter into the Korteweg–De Vries–Burgers-equation with the Weierstrass p-function as the soliton solution. However, a higher-order derivative of the studied variable produces the so-called N-soliton solution, which is comparable with the N-soliton solution of the Kadomtsev–Petviashvili equation. Experiments have made it clear that the system behaves like a coupled (an)harmonic oscillator on a discrete collapsed-state level. The streamlines obtained are derivatives of the error function as a function of the obtained Lax functional of the particle filaments dynamics induced by the (hypothetical) Calogero–Moser many-body system with elliptical potential and are the so-called Hermite functions. Hermite tried to introduce doubly periodic Hermite functions (the so-called Hermite problem) using coefficients related to the Weierstrass p-function. A solution-sensitive analysis of the incompressible viscid Navier–Stokes equation is performed using the Lamb vector. Cases with a meaningful potential-energy contribution require a particle interaction model with an N-soliton solution using a hierarchy-like solution of the Kadomtsev–Petviashvili equation. A three-soliton solution is emulated for the cylinder-wake problem. Our analytical results are put in perspective by comparison with two well-studied benchmark cases of fluid dynamics: the cylinder-wake problem and the driven-lid problem. The time-average velocity distribution (limit of streamline patterns) is consistent with published results and is enclosed in an asymmetrical lemniscate.

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A Nonlocal Kac-van Moerbeke Equation Admitting N-Soliton Solutions

Cited by 3 publications

References 19 publications

Isometric Reflectionless Eigenfunction Transforms for Higher-order AOs

Isometric Reflectionless Eigenfunction Transforms for Higher-order AOs

Quadratic transformations for a function that generalizes 2F1 and the Askey-Wilson polynomials

A note on N-soliton solutions for the viscid incompressible Navier–Stokes differential equation

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