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Numerical inverse scattering for the Korteweg–de Vries and modified Korteweg–de Vries equations
 Physica D
"... Recent advances in the numerical solution of Riemann–Hilbert problems allow for the implementation of a Cauchy initial value problem solver for the Korteweg–de Vries equation (KdV) and the defocusing modified Korteweg–de Vries equation (mKdV), without any boundary approximation. Borrowing ideas from ..."
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Recent advances in the numerical solution of Riemann–Hilbert problems allow for the implementation of a Cauchy initial value problem solver for the Korteweg–de Vries equation (KdV) and the defocusing modified Korteweg–de Vries equation (mKdV), without any boundary approximation. Borrowing ideas from the method of nonlinear steepest descent, this method is demonstrated to be asymptotically accurate. The method is straightforward for the case of defocusing mKdV due to the lack of poles in the Riemann–Hilbert problem and the boundedness properties of the reflection coefficient. Solving KdV requires the introduction of poles in the Riemann–Hilbert problem and more complicated deformations. The introduction of a new deformation for KdV allows for the stable asymptotic computation of the solution in the entire (x, t)plane. KdV and mKdV are dispersive equations and this method can fully capture the dispersion with spectral accuracy. Thus, this method can be used as a benchmarking tool for determining the effectiveness of future numerical methods designed to capture dispersion. This method can easily be adapted to other integrable equations with Riemann–Hilbert formulations, such as the nonlinear Schrödinger equation. 1
Nonlinear Steepest Descent and Numerical Solution of Riemann–Hilbert Problems
 Comm. Pure Appl. Math
, 2013
"... The effective and efficient numerical solution of Riemann–Hilbert problems has been demonstrated in recent work. With the aid of ideas from the method of nonlinear steepest descent for Riemann– Hilbert problems, the resulting numerical methods have been shown numerically to retain accuracy as values ..."
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The effective and efficient numerical solution of Riemann–Hilbert problems has been demonstrated in recent work. With the aid of ideas from the method of nonlinear steepest descent for Riemann– Hilbert problems, the resulting numerical methods have been shown numerically to retain accuracy as values of certain parameters become arbitrarily large. Remarkably, this numerical approach does not require knowledge of local parametrices; rather, the deformed contour is scaled near stationary points at a specific rate. The primary aim of this paper is to prove that this observed asymptotic accuracy is indeed achieved. To do so, we first construct a general theoretical framework for the numerical solution of Riemann–Hilbert problems. Second, we demonstrate the precise link between nonlinear steepest descent and the success of numerics in asymptotic regimes. In particular, we prove sufficient conditions for numerical methods to retain accuracy. Finally, we compute solutions to the homogeneous Painleve ́ II equation and the modified Korteweg–de Vries equations to explicitly demonstrate the practical validity of the theory. 1
A numerical methodology for the Painlevé equations
 Journal of Computational Physics
, 2011
"... Abstract. The six Painlevé transcendents PIPV I have both applications and analytic properties that make them stand out from most other classes of special functions. Although they have been the subject of extensive theoretical investigations for about a century, they still have a reputation for bei ..."
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Abstract. The six Painlevé transcendents PIPV I have both applications and analytic properties that make them stand out from most other classes of special functions. Although they have been the subject of extensive theoretical investigations for about a century, they still have a reputation for being numerically challenging. In particular, their extensive pole fields in the complex plane have often been perceived as ‘numerical mine fields’. In the present work, we note that the Painlevé property in fact provides the opportunity for very fast and accurate numerical solutions throughout such fields. When combining a Taylor/Padébased ODE initial value solver for the pole fields with a boundary value solver for smooth regions, numerical solutions become available across the full complex plane. We focus here on the numerical methodology, and illustrate it for the PI equation. In later studies, we will concentrate on mathematical aspects of both the PI and the higher Painlevé transcendents.
Numerical study of higher order analogues of the TracyWidom distribution,” ArXiv eprints
, 2011
"... We study a family of distributions that arise in critical unitary random matrix ensembles. They are expressed as Fredholm determinants and describe the limiting distribution of the largest eigenvalue when the dimension of the random matrices tends to infinity. The family contains the Tracy–Widom dis ..."
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We study a family of distributions that arise in critical unitary random matrix ensembles. They are expressed as Fredholm determinants and describe the limiting distribution of the largest eigenvalue when the dimension of the random matrices tends to infinity. The family contains the Tracy–Widom distribution and higher order analogues of it. We compute the distributions numerically by solving a Riemann–Hilbert problem numerically, plot the distributions, and discuss several properties that they appear to exhibit. 1
A Riemann–Hilbert approach to Jacobi operators and Gaussian quadrature
, 2014
"... The computation of the entries of Jacobi operators associated with orthogonal polynomials has important applications in numerical analysis. From truncating the operator to form a Jacobi matrix, one can apply the Golub–Welsh algorithm to compute the Gaussian quadrature weights and nodes. Furthermore, ..."
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The computation of the entries of Jacobi operators associated with orthogonal polynomials has important applications in numerical analysis. From truncating the operator to form a Jacobi matrix, one can apply the Golub–Welsh algorithm to compute the Gaussian quadrature weights and nodes. Furthermore, the entries of the Jacobi operator are the coefficients in the threeterm recurrence relationship for the polynomials. This provides an efficient method for evaluating the orthogonal polynomials. Here, we present an Ø(N) method to compute the first N rows of Jacobi operators from the associated weight. The method exploits the Riemann–Hilbert representation of the polynomials by solving a deformed Riemann–Hilbert problem numerically. We further adapt this computational approach to certain entire weights that are beyond the reach of current asymptotic Riemann–Hilbert techniques.
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.
Rational approximation, oscillatory Cauchy integrals and Fourier transforms. arXiv Prepr. arXiv1403.2378
, 2014
"... We develop the convergence theory for a wellknown method for the interpolation of functions on the real axis with rational functions. Precise new error estimates for the interpolant are derived using existing theory for trigonometric interpolants. Estimates on the Dirichlet kernel are used to deri ..."
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We develop the convergence theory for a wellknown method for the interpolation of functions on the real axis with rational functions. Precise new error estimates for the interpolant are derived using existing theory for trigonometric interpolants. Estimates on the Dirichlet kernel are used to derive new bounds on the associated interpolation projection operator. Error estimates are desired partially due to a recent formula of the author for the Cauchy integral of a specific class of socalled oscillatory rational functions. Thus, error bounds for the approximation of the Fourier transform and Cauchy integral of oscillatory smooth functions are determined. Finally, the behavior of the differentiation operator is discussed. The analysis here can be seen as an extension of that of Weber (1980) and Weideman (1995) in a modified basis used by Olver (2009) that behaves well with respect to function multiplication and differentiation. 1