Strong Convergence for Discrete Nonlinear Schrödinger equations in the Continuum Limit

Abstract

We consider discrete nonlinear Schrödinger equations (DNLS) on the lattice $h\Bbb{Z}^d$ 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 $h\to 0$, 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 [Comm. Math. Phys., 317 (2013), pp. 563--591], but it also does not employ any extra numerical scheme to avoid weak dispersion [L. I. Ignat and E. Zuazua, SIAM J. Numer. Anal., 47 (2009), pp. 1366--1390, L. I. Ignat and E. Zuazua, J. Math. Pures Appl. (9), 98 (2012), pp. 479--517]. 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 [Y. Hong and C. Yang, Discrete Contin. Dyn. Syst., 39 (2019), pp. 3239--3264], which quantitatively measure dispersive phenomena on the lattice. Our approach could be adapted to a more general setting as [K. Kirkpatrick, E. Lenzmann, and G. Staffilani, Comm. Math. Phys., 317 (2013), pp. 563--591] as long as the desired Strichartz estimates are obtained.