论文标题
连续限量2D分数非线性schrödinger方程
Continuum Limit of 2D Fractional Nonlinear Schrödinger Equation
论文作者
论文摘要
We prove that the solutions to the discrete Nonlinear Schrödinger Equation (DNLSE) with non-local algebraically-decaying coupling converge strongly in $L^2(\mathbb{R}^2)$ to those of the continuum fractional Nonlinear Schrödinger Equation (FNLSE), as the discretization parameter tends to zero.证明依赖于尖锐的分散估计值，这些估计产生了离散参数中均匀的Strichartz估计值。振荡性积分渐近学的主要项的明确计算用于表明，作为非局部性参数$α\ in（1,2）$接近边界，分散估计的最佳常数爆炸。
We prove that the solutions to the discrete Nonlinear Schrödinger Equation (DNLSE) with non-local algebraically-decaying coupling converge strongly in $L^2(\mathbb{R}^2)$ to those of the continuum fractional Nonlinear Schrödinger Equation (FNLSE), as the discretization parameter tends to zero. The proof relies on sharp dispersive estimates that yield the Strichartz estimates that are uniform in the discretization parameter. An explicit computation of the leading term of the oscillatory integral asymptotics is used to show that the best constants of a family of dispersive estimates blow up as the non-locality parameter $α\in (1,2)$ approaches the boundaries.
