English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We present efficient and accurate numerical methods for computing the groundstate and dynamics of the nonlinear Schr\"odinger equation (NLSE) with nonlocalinteractions based on a fast and accurate evaluation of the long-rangeinteractions via the nonuniform fast Fourier transform (NUFFT). We begin with areview of the fast and accurate NUFFT based method in \cite{JGB} for nonlocalinteractions where the singularity of the Fourier symbol of the interactionkernel at the origin can be canceled by switching to spherical or polarcoordinates. We then extend the method to compute other nonlocal interactionswhose Fourier symbols have stronger singularity at the origin that cannot becanceled by the coordinate transform. Many of these interactions do not decayat infinity in the physical space, which adds another layer of complexity sinceit is more difficult to impose the correct artificial boundary conditions forthe truncated bounded computational domain. The performance of our methodagainst other existing methods is illustrated numerically, with particularattention on the effect of the size of the computational domain in the physicalspace. Finally, to study the ground state and dynamics of the NLSE, we proposeefficient and accurate numerical methods by combining the NUFFT method forpotential evaluation with the normalized gradient flow using backward EulerFourier pseudospectral discretization and time-splitting Fourier pseudospectralmethod, respectively. Extensive numerical comparisons are carried out betweenthese methods and other existing methods for computing the ground state anddynamics of the NLSE with various nonlocal interactions. Numerical results showthat our scheme performs much better than those existing methods in terms ofboth accuracy and efficiency.

Computing the ground state and dynamics of the nonlinear Schr\"odinger equation with nonlocal interactions via the nonuniform FFT