2012
DOI: 10.1007/s00220-012-1621-x
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On the Continuum Limit for Discrete NLS with Long-Range Lattice Interactions

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 (−∆) 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 −∆ describes the dispersion … Show more

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Cited by 203 publications
(123 citation statements)
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“…For given constant ∈ (0, 1], the fractional Sobolev norm ‖⋅‖ H 2 and semi-norm |⋅| H 2 over • V h are defined as [19,22,47]…”
Section: Preliminarymentioning
confidence: 99%
“…For given constant ∈ (0, 1], the fractional Sobolev norm ‖⋅‖ H 2 and semi-norm |⋅| H 2 over • V h are defined as [19,22,47]…”
Section: Preliminarymentioning
confidence: 99%
“…In , Ervin and Roop established the equivalence of fractional derivatives spaces and fractional Sobolev spaces under homogeneous Dirichelet boundary conditions and listed some properties of fractional differential operators, which are useful for the analysis of the stability and convergence in the context of finite element and spectral method. Recently, the discrete counterpart, that is, fractional Sobolev norms and embedding inequalities, are introduced by Kirkpatrick et al in . The discrete norms and embedding inequality play a key role in the analysis of the finite difference scheme.…”
Section: Analysis Of the Difference Schemementioning
confidence: 99%
“…Following the similar technique in , we can derive the following two lemmas. Lemma (Discrete Sobolev inequality).…”
Section: Analysis Of the Difference Schemementioning
confidence: 99%
“…Ervin discussed a finite element approximation to nonlinear diffusion equation, which contain fractional diffusion operator D2αfalse(1false/2<α1false). Guo et al and Kirkpatrick et al obtained a global unique solution of nonlinear Schrödinger equation on double-struckR with fractional operator ( − Δ) α . Wei et al presented a local discontinuous Galerkin finite element method to approximate fractional Schrödinger equation.…”
Section: Introductionmentioning
confidence: 99%
“…Guo et al 17 and Kirkpatrick et al 18 obtained a global unique solution of nonlinear Schrödinger equation on R with fractional operator ( − Δ) . Wei et al 19,20 presented a local discontinuous Galerkin finite element method to approximate fractional Schrödinger equation.…”
Section: Introductionmentioning
confidence: 99%