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Multicritical unitary random matrix ensembles and the general Painlevé II equation
, 2008
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The RiemannHilbert approach to double scaling limit of random matrix eigenvalues near the ”birth of a cut” transition
, 2007
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Numerical study of a multiscale expansion of the Korteweg de Vries equation and . . .
 PROC. ROYAL. SOC. A
, 2008
"... The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order ǫ 2, ǫ≪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 freq ..."
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Cited by 7 (6 self)
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The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order ǫ 2, ǫ≪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 ǫ in the interior of the Whitham oscillatory zone, it is known to be only of order ǫ 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 HastingsMcLeod solution of the PainlevéII equation. We show numerically that the resulting multiscale solution approximates the KdV solution, in the small dispersion limit, to the order ǫ 2/3.
First colonization of a hard edge in random matrix theory
, 804
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Mesoscopic colonization of a spectral band
, 904
"... We consider the unitary matrix model in the limit where the size of the matrices become infinite and in the critical situation when a new spectral band is about to emerge. In previous works the number of expected eigenvalues in a neighborhood of the band was fixed and finite, a situation that was te ..."
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We consider the unitary matrix model in the limit where the size of the matrices become infinite and in the critical situation when a new spectral band is about to emerge. In previous works the number of expected eigenvalues in a neighborhood of the band was fixed and finite, a situation that was termed “birth of a cut ” or “first colonization”. We now consider the transitional regime where this microscopic population in the new band grows without bounds but at a slower rate than the size of the matrix. The local population in the new band organizes in a “mesoscopic ” regime, in between the macroscopic behavior of the full system and the previously studied microscopic one. The mesoscopic colony may form a finite number of new bands, with a maximum number dictated by the degree of criticality of the original potential. We describe the delicate scaling limit that realizes/controls the mesoscopic colony. The method we use is the steepest descent analysis of the RiemannHilbert problem that is satisfied by the associated orthogonal polynomials.
4 GLOBAL ASYMPTOTICS FOR THE CHRISTOFFELDARBOUX KERNEL OF RANDOM MATRIX THEORY
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A numerical study of the small dispersion limit of the Korteweg–de Vries equation and asymptotic solutions, Physica D 241
, 2012
"... Abstract. We study numerically the small dispersion limit for the Kortewegde Vries (KdV) equation ut + 6uux + 2uxxx = 0 for 1 and give a quantitative comparison of the numerical solution with various asymptotic formulae for small in the whole (x, t)plane. The matching of the asymptotic soluti ..."
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Abstract. We study numerically the small dispersion limit for the Kortewegde Vries (KdV) equation ut + 6uux + 2uxxx = 0 for 1 and give a quantitative comparison of the numerical solution with various asymptotic formulae for small in the whole (x, t)plane. The matching of the asymptotic solutions is studied numerically. 1.