Results 1  10
of
34
Multicritical unitary random matrix ensembles and the general Painlevé II equation
, 2008
"... ..."
Universality of the breakup profile for the KdV equation in the small dispersion limit using the RiemannHilbert approach
, 2008
"... ..."
On universality of critical behaviour in Hamiltonian PDEs, in Geometry
 Topology, and Mathematical Physics, Amer. Math. Soc. Transl. Ser
"... on the occasion of his 70th birthday. Abstract. Our main goal is the comparative study of singularities of solutions to the systems of first order quasilinear PDEs and their perturbations containing higher derivatives. The study is focused on the subclass of Hamiltonian PDEs with one spatial dimensi ..."
Abstract

Cited by 21 (4 self)
 Add to MetaCart
(Show Context)
on the occasion of his 70th birthday. Abstract. Our main goal is the comparative study of singularities of solutions to the systems of first order quasilinear PDEs and their perturbations containing higher derivatives. The study is focused on the subclass of Hamiltonian PDEs with one spatial dimension. For the systems of order one or two we describe the local structure of singularities of a generic solution to the unperturbed system near the point of “gradient catastrophe ” in terms of standard objects of the classical singularity theory; we argue that their perturbed companions must be given by certain special solutions of Painlevé equations and their generalizations. Contents
Critical edge behavior in unitary random matrix ensembles and the thirty fourth Painlevé transcendent
, 2008
"... ..."
Painlevé I asymptotics for orthogonal polynomials with respect to a varying quartic weight
, 2008
"... ..."
The birth of a cut in unitary random matrix ensembles
 Int Math Res Notices, 2008(article ID rnm166):40
"... We study unitary random matrix ensembles in the critical regime where a new cut arises away from the original spectrum. We perform a double scaling limit where the size of the matrices tends to infinity, but in such a way that only a bounded number of eigenvalues is expected in the newborn cut. It t ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
(Show Context)
We study unitary random matrix ensembles in the critical regime where a new cut arises away from the original spectrum. We perform a double scaling limit where the size of the matrices tends to infinity, but in such a way that only a bounded number of eigenvalues is expected in the newborn cut. It turns out that limits of the eigenvalue correlation kernel are given by Hermite kernels corresponding to a finite size Gaussian Unitary Ensemble (GUE). When modifying the double scaling limit slightly, we observe a remarkable transition each time the new cut picks up an additional eigenvalue, leading to a limiting kernel interpolating between GUEkernels for matrices of size k and size k + 1. We prove our results using the RiemannHilbert approach. 1
Numerical study of a multiscale expansion of KdV and CamassaHolm equation arXiv:mathph/0702038
"... Abstract. We study numerically solutions to the Kortewegde Vries and CamassaHolm equation close to the breakup of the corresponding solution to the dispersionless equation. The solutions are compared with the properly rescaled numerical solution to a fourth order ordinary differential equation, th ..."
Abstract

Cited by 12 (8 self)
 Add to MetaCart
(Show Context)
Abstract. We study numerically solutions to the Kortewegde Vries and CamassaHolm equation close to the breakup of the corresponding solution to the dispersionless equation. The solutions are compared with the properly rescaled numerical solution to a fourth order ordinary differential equation, the second member of the Painlevé I hierarchy. It is shown that this solution gives a valid asymptotic description of the solutions close to breakup. We present a detailed analysis of the situation and compare the Kortewegde Vries solution quantitatively with asymptotic solutions obtained via the solution of the Hopf and the Whitham equations. We give a qualitative analysis for the CamassaHolm equation 1.