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The neck instability of bright solitons of the hyperbolic nonlinear Shrödinger equation is investigated. It is shown that this instability originates from a four-wave mixing interaction that links on-axis to off-axis radiation at opposite frequency bands. Our experiment supports this interpretation. Symmetry-breaking instability of solitons has been studied… (More)

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)

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)

- Bernard Deconinck, Thomas Trogdon, Vishal Vasan
- SIAM Review
- 2014

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)

We derive a Riemann–Hilbert problem satisfied by the Baker-Akhiezer function for the finite-gap solutions of the Korteweg-de Vries (KdV) equation. As usual for Riemann-Hilbert problems associated with solutions of integrable equations, this formulation has the benefit that the space and time dependence appears in an explicit, linear and computable way. We… (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)

- Levent Sagun, Thomas Trogdon, Yann LeCun
- ArXiv
- 2015

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)

The purpose of this paper is to establish bounds on the rate of convergence of the conjugate gradient algorithm when the underlying matrix is a random positive definite perturbation of a deterministic positive definite matrix. We estimate all finite moments of a natural halting time when the random perturbation is drawn from the Laguerre unitary ensemble in… (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)