Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0609183

Open Boundaries for the Nonlinear Schrodinger Equation

A. Soffer·C. Stucchio
View original source ↗Read source document ↗Export metadata ↓

Abstract

We present a new algorithm, the Time Dependent Phase Space Filter (TDPSF) which is used to solve time dependent Nonlinear Schrodinger Equations (NLS). The algorithm consists of solving the NLS on a box with periodic boundary conditions (by any algorithm). Periodically in time we decompose the solution into a family of coherent states. Coherent states which are outgoing are deleted, while those which are not are kept, reducing the problem of reflected (wrapped) waves. Numerical results are given, and rigorous error estimates are described. The TDPSF is compatible with spectral methods for solving the interior problem. The TDPSF also fails gracefully, in the sense that the algorithm notifies the user when the result is incorrect. We are aware of no other method with this capability.

Explore related subjects

math.NAcs.NAmath-phmath.APmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

A. Soffer, C. Stucchio. 2006-12-14. Open Boundaries for the Nonlinear Schrodinger Equation. https://doi.org/10.1016/j.jcp.2007.01.020

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A continuous scale space of diffeomorphisms

In this paper, we define and study a nested family of reproducing kernel Hilbert spaces of vector fields that is indexed by a range of scales, from which we construct a reproducing kernel Hilbert space of scale-dependent vector fields. We provide a characterization of the reproducing kernel of that space, with numerical approximations ensuring quick evaluations when this kernel does not have a closed form. We then introduce a multiscale version of the large deformation diffeomorphic metric mapping (LDDMM) problem and prove the existence of solutions. Finally, we provide numerical experiments performing landmark matching using multiscale LDDMM.

math.NA↗
arXiv2026

Hierarchical Matrices with Tucker Blocks: Construction and Forward Evaluation

In this paper, we introduce hierarchical matrices with Tucker blocks (HMTB) for data-sparse approximation of matrices arising from nonoscillatory and mildly oscillatory kernels. Building on the hierarchical matrix framework, HMTB matrices replace conventional low-rank blocks with Tucker representations. Using high-dimensional interpolation and tensor compression and contractions, we develop algorithms for constructing HMTB matrices and performing matrix-vector multiplication. For an $N \times N$ matrix, these algorithms have linear and quasi-linear complexities in $N$, respectively. Numerical comparisons with an interpolation-based $\mathcal{H}$-matrix implementation demonstrate storage savings and, in several cases, reduced construction times, while maintaining competitive forward-evaluation costs. The framework is also applicable to quasi-uniform grids.

math.NA↗
arXiv2026

Data-driven model order reduction for wave propagation in materials with damage and nonlinearities

In this work, we consider wave propagation in materials characterized by nonlinear properties or damage. To accelerate the simulations of the resulting high-dimensional problems, we apply model order reduction methods. Depending on the knowledge of the underlying equations and the availability of their discrete operators, intrusive methods (here projection-based approaches based on proper orthogonal decomposition (POD)) or non-instrusive methods (here data-driven approaches including dynamic mode decomposition (DMD) and operator inference (OpInf)) can be used. We recall the theoretical foundations of the methods and apply them to the problem of wave propagation. In three different numerical examples, we evaluate the performance of the reduction techniques.

math.NA↗