The EllipticGL(n)Dynamical Quantum Group as an𝔥-Hopf Algebroid

Using the language of𝔥-Hopf algebroids which was introduced by Etingof and Varchenko, we construct a dynamical quantum group,ℱell(GL(n)), from the elliptic solution of the quantum dynamical Yang-Baxter equation with spectral parameter associated to the Lie algebra𝔰𝔩n. We apply the generalized FRST construction and obtain an𝔥-bialgebroidℱell(M(n)). Natural analogs of the exterior algebra and their matrix elements, elliptic minors, are defined and studied. We show how to use the cobraiding to prove that the elliptic determinant is central. Localizing at this determinant and constructing an antipode we obtain the𝔥-Hopf algebroidℱell(GL(n)).

Numerical study of a multiscale expansion of the Korteweg–de Vries equation and Painlevé-II equation

The Cauchy problem for the Korteweg–de Vries (KdV) equation with small dispersion of order ϵ 2 , ϵ ≪1, is characterized by the appearance of a zone of rapid modulated oscillations. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wavenumber and frequency are not constant but evolve according to the Whitham equations. Whereas the difference between the KdV and the asymptotic solution decreases as ϵ in the interior of the Whitham oscillatory zone, it is known to be only of order ϵ 1/3 near the leading edge of this zone. To obtain a more accurate description near the leading edge of the oscillatory zone, we present a multiscale expansion of the solution of KdV in terms of the Hastings–McLeod solution of the Painlevé-II equation. We show numerically that the resulting multiscale solution approximates the KdV solution, in the small dispersion limit, to the order ϵ 2/3 .

Exponentially Convergent Approximation to the Elliptic Solution Operator

We develop an accurate approximation of the normalized hyperbolic operator sine family generated by a strongly positive operator A in a Banach space X which represents the solution operator for the elliptic boundary value problem. The solution of the corresponding inhomogeneous boundary value problem is found through the solution operator and the Green function. Starting with the Dunford — Cauchy representation for the normalized hyperbolic operator sine family and for the Green function, we then discretize the integrals involved by the exponentially convergent Sinc quadratures involving a short sum of resolvents of A. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [0, 1].