论文标题
二维分数离散非线性schrodinger方程
The Two-Dimensional Fractional Discrete Nonlinear Schrodinger Equation
论文作者
论文摘要
We study a fractional version of the two-dimensional discrete nonlinear Schrödinger (DNLS) equation, where the usual discrete Laplacian is replaced by its fractional form that depends on a fractional exponent $s$ that interpolates between the case of an identity operator ($s=0$) and that of the usual discrete 2D Laplacian ($s=1$).这种替换导致站点之间的远距离耦合，在低值$ s $的情况下，带宽降低并导致准排分状态。最初定位的激发的均方根位移始终以“速度”为弹道，该速度与分数指数$ s $单调增加。我们还计算了非线性模式及其在体积和表面模式的稳定性。观察到的稳定性随着分数指数的增加而增加。最初局部兴奋的捕获表明，自动捕获过渡是非线性强度的函数，其阈值随着$ s $的值而增加。在线性限制中，在小$ s $值中持续存在线性捕获。这种行为与带宽的减少及其相关的准排行纪相关。
We study a fractional version of the two-dimensional discrete nonlinear Schrödinger (DNLS) equation, where the usual discrete Laplacian is replaced by its fractional form that depends on a fractional exponent $s$ that interpolates between the case of an identity operator ($s=0$) and that of the usual discrete 2D Laplacian ($s=1$). This replacement leads to a long-range coupling among sites that, at low values of $s$, decreases the bandwidth and leads to quasi-degenerate states. The mean square displacement of an initially-localized excitation is shown to be ballistic at all times with a `speed' that increases monotonically with the fractional exponent $s$. We also compute the nonlinear modes and their stability for both, bulk and surface modes. The modulational stability is seen to increase with an increase in the fractional exponent. The trapping of an initially localized excitation shows a selftrapping transition as a function of nonlinearity strength, whose threshold increases with the value of $s$. In the linear limit, there persists a linear trapping at small $s$ values. This behavior is connected with the decrease of the bandwidth and its associated increase in quasi-degeneracy.
