On the Continuum Limit for Discrete NLS with Long-Range Lattice Interactions
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Staffilani_on the continuum.pdf
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Author(s)
Kirkpatrick, Kay
Lenzmann, Enno
Staffilani, Gigliola
Date Issued
November 2012
Journal
Communications in Mathematical Physics
Publisher
Springer-Verlag
Citation
Kirkpatrick, Kay, Enno Lenzmann, and Gigliola Staffilani. “On the Continuum Limit for Discrete NLS with Long-Range Lattice Interactions.” Communications in Mathematical Physics 317, no. 3 (February 17, 2013): 563-591.
Version
Original manuscript
Abstract
We consider a general class of discrete nonlinear Schrödinger equations (DNLS) on the lattice hZ with mesh size h > 0. In the continuum limit when h → 0, we prove that the limiting dynamics are given by a nonlinear Schrödinger equation (NLS) on R with the fractional Laplacian (−Δ)[superscript α] as dispersive symbol. In particular, we obtain that fractional powers 1/2 < α < 1 arise from long-range lattice interactions when passing to the continuum limit, whereas the NLS with the usual Laplacian −Δ describes the dispersion in the continuum limit for short-range or quick-decaying interactions (e. g., nearest-neighbor interactions).
Our results rigorously justify certain NLS model equations with fractional Laplacians proposed in the physics literature. Moreover, the arguments given in our paper can be also applied to discuss the continuum limit for other lattice systems with long-range interactions.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1007/s00220-012-1621-x
