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A RIEMANNHILBERT PROBLEM IN A RIEMANN SURFACE ∗
"... Abstract One of the inspirations behind Peter Lax’s interest in dispersive integrable systems, as the small dispersion parameter goes to zero, comes from systems of ODEs discretizing 1dimensional compressible gas dynamics [17]. For example, an understanding of the asymptotic behavior of the Toda la ..."
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the method used was the method of Lax and Levermore [16], reducing the asymptotic problem to the solution of a minimization problem with constraints (an “equilibrium measure ” problem). Later, it was found that the asymptotic method of Deift and Zhou (analysis of the associated RiemannHilbert factorization
A steepest descent method for oscillatory Riemann–Hilbert problems: asymptotics for the MKdV equation
 Ann. of Math
, 1993
"... In this announcement we present a general and new approach to analyzing the asymptotics of oscillatory RiemannHilbert problems. Such problems arise, in particular, in evaluating the longtime behavior of nonlinear wave equations solvable by the inverse scattering method. We will restrict ourselves ..."
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Cited by 303 (27 self)
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for the MKdV equation leads to a RiemannHilbert factorization problem for a 2 × 2 matrix valued function m = m(·; x, t) analytic in C\R, (1) where m+(z) = m−(z)vx,t, z ∈ R, m(z) → I as z → ∞, m±(z) = lim ε↓0 m(z ± iε; x, t), vx,t(z) ≡ e −i(4tz3 +xz)σ3 v(z)e i(4tz 3 +xz)σ3, σ3 =
From Stationary Phase to Steepest Descent
"... The socalled nonlinear stationaryphasesteepestdescent method for the asymptotic analysis of RiemannHilbert factorization problems has been very successful in providing (i) rigorous results on long time, long range and semiclassical asymptotics for ..."
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Cited by 3 (0 self)
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The socalled nonlinear stationaryphasesteepestdescent method for the asymptotic analysis of RiemannHilbert factorization problems has been very successful in providing (i) rigorous results on long time, long range and semiclassical asymptotics for
Asymptotics via Steepest Descent for an Operator RiemannHilbert Problem
, 1999
"... In this paper, we take the first step towards an extension of the nonlinear steepest descent method of Deift, Its and Zhou to the case of operator RiemannHilbert problems. In particular, we provide long range asymptotics for a Fredholm determinant arising in the computation of the probability of fi ..."
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of finding a string of n adjacent parallel spins up in the antiferromagnetic ground state of the spin 1/2 XXX Heisenberg Chain. Such a determinant can be expressed in terms of the solution of an operator RiemannHilbert factorization problem. Typeset by A M ST E X 2 SPYRIDON KAMVISSIS 1. INTRODUCTION
On the Semiclassical Limit of the Focusing Nonlinear Schrdinger Equation, Phys
 Lett. A
, 1998
"... We study the semiclassical behavior of the focusing nonlinear Schrödinger equation in 1+1 dimensions under discontinuous ”barrier ” initial data and we describe the violent oscillations arising in terms of theta functions. The construction of proofs relies on the analysis of the associated Riemann ..."
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Cited by 25 (2 self)
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RiemannHilbert factorization problem. Typeset by AMSTEX 1
Stability of the periodic Toda lattice: Higher order asymptotics
, 2008
"... In a recent paper we have considered the longtime asymptotics of the periodic Toda lattice under a short range perturbation and we have proved that the perturbed lattice asymptotically approaches a modulated lattice. In the present paper we capture the higher order asymptotics, at least away from ..."
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Cited by 7 (7 self)
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asymptotics arise from ”local” Riemann–Hilbert factorizationproblems on small crosses centered on the stationary phase points. We discover that the analysis of such a local problem can be done in a chart around each stationary phase point and reduces to a Riemann–Hilbert factorization problem on the complex
SEMICLASSICAL FOCUSING NLS WITH STEPLIKE INITIAL DATA
, 2009
"... We study the semiclassical behavior of the focusing nonlinear Schrödinger equation in 1+1 dimensions under discontinuous ”barrier ” initial data and we describe the violent oscillations arising in terms of theta functions. The construction of proofs relies on (i) the analysis of the associated Riema ..."
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RiemannHilbert factorization problem (ii) the analysis of the resulting maximin variational problem for a Green’s potential with external field and in particular the proof of existence of a regular solution, which enables the construction of the socalled gfunction transformation and hence
ON THE METHOD OF STEEPEST DESCENT
, 2005
"... ABSTRACT: We review the history of the nonlinear steepest descent method for the asymptotic evaluation of the solutions of RiemannHilbert factorization problems. We stress some recent results on the ”nonselfadjoint ” extension of the theory. In particular we consider the case of the semiclassical ..."
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ABSTRACT: We review the history of the nonlinear steepest descent method for the asymptotic evaluation of the solutions of RiemannHilbert factorization problems. We stress some recent results on the ”nonselfadjoint ” extension of the theory. In particular we consider the case
Solutions of the Painlevé VI Equation from Reduction of Integrable Hierarchy in a Grassmannian Approach
, 804
"... We present an explicit method to perform similarity reduction of a RiemannHilbert factorization problem for a homogeneous GL(N, C) loop group and use our results to find solutions to the Painlevé VI equation for N = 3. The tau function of the reduced hierarchy is shown to satisfy the σform of the ..."
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We present an explicit method to perform similarity reduction of a RiemannHilbert factorization problem for a homogeneous GL(N, C) loop group and use our results to find solutions to the Painlevé VI equation for N = 3. The tau function of the reduced hierarchy is shown to satisfy the σ
Results 1  10
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