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We implement the new transform method for solving boundary value problems developed by Fokas for periodic boundary conditions. The approach presented here is not a replacement for classical methods nor is it necessarily an improvement. However, in addition to establishing that periodic problems can indeed be solve by the new transform method (which enhances… (More)

The classical methods for solving initial-boundary-value problems for linear partial differential equations with constant coefficients rely on separation of variables, and specific integral transforms. As such, they are limited to specific equations, with special boundary conditions. Here we review a method introduced by Fokas, which contains the classical… (More)

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… (More)

We solve the focusing and defocusing nonlinear Schrödinger (NLS) equations numerically by implementing the inverse scattering transform. The computation of the scattering data and of the NLS solution are both spectrally con-vergent. Initial conditions in a suitable space are treated. Using the approach of Biondini and Bui [3] we numerically solve… (More)

- Percy A Deift, Govind Menon, Sheehan Olver, Thomas Trogdon
- Proceedings of the National Academy of Sciences…
- 2014

The authors present evidence for universality in numerical computations with random data. Given a (possibly stochastic) numerical algorithm with random input data, the time (or number of iterations) to convergence (within a given tolerance) is a random variable, called the halting time. Two-component universality is observed for the fluctuations of the… (More)

The authors present empirical universal distributions for the halting time (measured by the number of iterations to reach a given accuracy) of optimization algorithms applied to two random systems: spin glasses and deep learning. Given an algorithm, which we take to be both the optimization routine and the form of the random landscape, the fluctuations of… (More)

In honor of Peter Lax on his 90th birthday. Abstract We consider the Laguerre Unitary Ensemble (aka, Wishart Ensemble) of sample covariance matrices A = XX * , where X is an N × n matrix with iid standard complex normal entries. Under the scaling n = N + √ 4cN , c > 0 and N → ∞, we show that the rescaled fluctuations of the smallest eigenvalue, largest… (More)

In this letter we describe how to compute the finite-genus solutions of the Korteweg-de Vries equation using a Riemann-Hilbert problem that is satisfied by the Baker-Akhiezer function corresponding to a Schrödinger operator with finite-gap spectrum. The recovery of the corresponding finite-genus solution is performed using the asymptotics of the… (More)

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