arXiv subjects

Justin T. Brereton

Publications and source records attributed to Justin T. Brereton.

3 recordsLinked to original sources

arXiv2018

Invariant measure construction at a fixed mass

In this paper we analyze the derivative nonlinear Schrödinger equation on $\mathbb{T}$ with randomized initial data in $\cap_{s < \frac{1}{2}} H^{s}(\mathbb{T})$ according to a Wiener measure. We construct an invariant measure at each sufficiently small, fixed mass $m$ through an argument that emulates the divergence theorem in infinitely many dimensions. We also prove that the density function needed to construct the Wiener measure is in $L^p$, even after scaling of the Fourier coefficients of the intial data.

math.AP↗

arXiv2016

Almost sure local well-posedness for the supercritical quintic NLS

This paper studies the quintic nonlinear Schrödinger equation on $\mathbb{R}^d$ with randomized initial data below the critical regularity $H^{\frac{d-1}{2}}$. The main result is a proof of almost sure local well-posedness given a Wiener Randomization of the data in $H^s$ for $s \in (\frac{d-2}{2}, \frac{d-1}{2})$. The argument further develops the techniques introduced in the work of Á. Bényi, T. Oh and O. Pocovnicu on the cubic problem. The paper concludes with a condition for almost sure global well-posedness.

math.AP↗

arXiv2012

Some principles for mountain pass algorithms, and the parallel distance

The problem of computing saddle points is important in certain problems in numerical partial differential equations and computational chemistry, and is often solved numerically by a minimization problem over a set of mountain passes. We point out that a good global mountain pass algorithm should have good local and global properties. Next, we define the parallel distance, and show that the square of the parallel distance has a quadratic property. We show how to design algorithms for the mountain pass problem based on perturbing parameters of the parallel distance, and that methods based on the parallel distance have midrange local and global properties.

math.NA↗