Searcharxiv⌕ Search

arXiv subjects

Tim Van Hoose

Publications and source records attributed to Tim Van Hoose.

7 recordsLinked to original sources
arXiv2026

On Analyticity of Solitons for the Periodic Dispersion-Managed Nonlinear Schr\"odinger Equation

We explore the regularity of energy maximizers for the Lagrangian of a periodic dispersion managed fiber optic at fixed intensity, on a torus of length $L$ and with vanishing average dispersion. We show that the Fourier coefficients decay at a polynomial rate, and then upgrade this to exponential decay, so that the maximizers are analytic in space for large enough $L$. In addition, by an asymptotic comparison to the optimizers on the real line, we prove that the solutions are non-trivial. We also consider a conjecture that the maximizer necessarily has an underlying symmetry inherent to both the energy functional and the resulting Euler-Lagrange equation. All of our results are supported with illustrative numerical experiments.

math.AP↗
arXiv2025

Modified Scattering for Nonlocal Nonlinear Schr\"odinger Equations

We prove a modified scattering and sharp $L^\infty$ decay result for both the Hartree and Schr\"odinger-Bopp-Podolsky equations in dimensions $2$ and $3$ using the testing by wavepackets approach due to Ifrim and Tataru. We show that modified scattering and sharp pointwise decay occur for these equations at a regularity much lower than previous results due to Hayashi-Naumkin and Kato-Pusateri, and as a corollary also show that the results on power-type scattering-critical NLS due to Hayashi-Naumkin can be proven under minimal regularity assumptions.

math.AP↗
arXiv2024

Modified Scattering for the Hartree Nonlinear Schr\"odinger Equation

We prove sharp $L^\infty$ decay and modified scattering for the Hartree nonlinear Schr\"odinger equation in dimensions $2$ and $3$ using the testing by wavepackets method of Ifrim and Tataru. We show that the scattering behavior happens at a regularity well below that of earlier results of Hayashi-Naumkin and Kato-Pusateri.

math.AP↗
arXiv2022

Averaging for the dispersion-managed NLS

We establish global-in-time averaging for the $L^2$-critical dispersion-managed nonlinear Schr\"odinger equation in the fast dispersion management regime. In particular, in the case of nonzero average dispersion, we establish averaging with any subcritical data, while in the case of a strictly positive dispersion map, we obtain averaging for data in $L^2$.

math.AP↗
arXiv2021

Well-posedness and blowup for the dispersion-managed nonlinear Schr\"odinger equation

We consider the nonlinear Schr\"odinger equation with periodic dispersion management. We first establish global-in-time Strichartz estimates for the underlying linear equation with suitable dispersion maps. As an application, we establish a small-data scattering result for the $3d$ cubic equation. Finally, we use a virial argument to demonstrate the existence of blowup solutions for the $3d$ cubic equation with piecewise constant dispersion map.

math.AP↗
arXiv2021

Modified scattering for a dispersion-managed nonlinear Schrödinger equation

We prove sharp $L^\infty$ decay and modified scattering for a one-dimensional dispersion-managed cubic nonlinear Schrödinger equation with small initial data chosen from a weighted Sobolev space. Specifically, we work with an averaged version of the dispersion-managed NLS in the strong dispersion management regime. The proof adapts techniques from Hayashi-Naumkin and Kato-Pusateri, which established small-data modified scattering for the standard $1d$ cubic NLS.

math.AP↗