紧支集下散焦Davey-Stewartson II方程的数值散射
Numerical scattering for the defocusing Davey-Stewartson II equation for initial data with compact support
摘要：
在这项工作中，我们提出了散焦Davey-Stewartson（DS）II方程的数值散射的谱算法，其初始数据在圆盘上具有紧支撑，即，用于解决d杆问题。我们的算法使用极坐标和实现的径向依赖和傅立叶谱方法的方位角依赖的切比雪夫谱方案。重点放在建设复杂的几何光学（CGO）的解决方案，这是需要在散射方法DS。我们讨论了两种不同的方法：第一种方法构造了d杆系统的基本解，并将CGO条件应用于后者。这对于谱参数$k$的模的小值特别有效。第二种方法使用一个不动点迭代的重新制定的d-杆系统包含的谱参数明确，付出的代价有更简单的渐近。该方法被示出为磁盘的特征函数的例子，并示出表现出频谱收敛，即，数值误差随配置点数目的指数衰减。给出了大$|k|$反射系数的渐近公式。
英文摘要：
In this work we present spectral algorithms for the numerical scattering for the defocusing Davey-Stewartson (DS) II equation with initial data having compact support on a disk, i.e., for the solution of d-bar problems. Our algorithms use polar coordinates and implement a Chebychev spectral scheme for the radial dependence and a Fourier spectral method for the azimuthal dependence. The focus is placed on the construction of complex geometric optics (CGO) solutions which are needed in the scattering approach for DS. We discuss two different approaches: The first constructs a fundamental solution to the d-bar system and applies the CGO conditions on the latter. This is especially efficient for small values of the modulus of the spectral parameter $k$. The second approach uses a fixed point iteration on a reformulated d-bar system containing the spectral parameter explicitly, a price paid to have simpler asymptotics. The approaches are illustrated for the example of the characteristic function of the disk and are shown to exhibit spectral convergence, i.e., an exponential decay of the numerical error with the number of collocation points. An asymptotic formula for large $|k|$ is given for the reflection coefficient.
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