二维分数阶离散非线性Schrodinger方程
The Two-Dimensional Fractional Discrete Nonlinear Schrodinger Equation
摘要：
研究了二维离散非线性薛定谔（DNLS）方程的分数阶形式，其中通常的离散拉普拉斯算子被其分数阶形式所取代，该分数阶形式依赖于一个分数阶指数$s$，该分数阶指数介于恒等算子（$s=0$）和通常的离散二维拉普拉斯算子（$s=1$）之间.这种替换导致在$s$的低值处的位点之间的长程耦合，降低带宽并导致准简并态。初始本地化的激发的均方位移被证明是弹道在任何时候与“速度”，单调增加的分数指数$s$。我们还计算了体模和表面模的非线性模及其稳定性。的调制稳定性被认为是随着分数指数的增加而增加。初始局域激发的捕获显示了作为非线性强度的函数的自捕获转变，其阈值随着$s$的值而增加。在线性限值中，在小的$s$值处存在线性捕获。这种行为与带宽的减小及其相关的准简并度的增加有关。
英文摘要：
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 ($s=1$). 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.
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