Strong Convergence for Discrete Nonlinear Schrödinger equations in the Continuum Limit

Abstract

We consider discrete nonlinear Schrödinger equations (DNLS) on the lattice hZ d whose linear part is determined by the discrete Laplacian which accounts only for nearest neighbor interactions, or by its fractional power. We show that in the continuum limit h → 0, solutions to DNLS converge strongly in L 2 to those to the corresponding continuum equations, but a precise rate of convergence is also calculated. In particular cases, this result improves weak convergence in Kirkpatrick, Lenzmann and Staffilani [17]. Our proof is based on a suitable adjustment of dispersive PDE techniques to a discrete setting. Notably, we employ uniform-in-h Strichartz estimates for discrete linear Schrödinger equations in [10], which quantitatively measure dispersive phenomena on the lattice. Our approach could be adapted to a more general setting like [17] as long as the desired Strichartz estimates are obtained.

Invariant Measures for Integrable Spin Chains and an Integrable Discrete Nonlinear Schrödinger Equation

“…We begin by constructing global solutions to (11) for almost every initial data chosen according to the measure dµ β wn . We will do so by proving that increasingly large finite-volume solutions to the Ablowitz-Ladik system (21) converge to a solution to (11), uniformly on compact regions of spacetime.…”

“…. , K} × R → C denote the unique global solution to (21) with initial data α K (0) = {α n (0)} |n|≤K constructed in Proposition 2.9.…”

“…Then Ω = Ω Tn is a set of full measure. Moreover, for an initial data α(0) = {α n (0)} n∈Z ∈ Ω, the unique global solutions α K : Z × R → C to (21) with truncated initial data α K (0) = {α n (0)} |n|≤K satisfy 4≤K∈2 Z sup |t|≤T,n∈Z e −2 n |α 2K n (t) − α K n (t)| < ∞ for any T > 0, which shows that α K converge uniformly on compact regions of spacetime.…”

“…Fix T > 0 and 4 ≤ K ∈ 2 Z . By invariance of the measure for the finitedimensional system (21), we obtain…”

“…In that paper, Vaninsky considers the defocusing problem on the circle and constructs an invariant measure associated to the conservation law at one degree of regularity higher than the Hamiltonian. For convergence in the deterministic setting, see [21], which works in the energy space, and references therein.…”