STRONG CONVERGENCE FOR DISCRETE NONLINEAR SCHRODINGER EQUATIONS IN THE CONTINUUM LIMIT

초록

We consider discrete nonlinear Schrodinger 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 [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 T. Numer. Anal., 47 (2009), pp. 1366-1390, L. I. Ignat and E. Zuazua, T. 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 Schrodinger 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.

키워드