指数配点方法用于立方薛维定方程
Exponential collocation methods for the cubic Schrödinger equation
AI总结：
本文提出并分析了新的指数配点方法，用于高效求解立方薛维定方程的初值问题。方法能精确或近似保持连续能量，通过理论分析和数值实验验证了其高阶精度和稳定性。
摘要：
本文推导并分析了新的指数配点方法，用于高效求解立方薛维定方程在d维环上的初值问题。能量守恒是立方薛维定方程的关键特征。证明了新方法可以达到任意高阶精度，能够精确或近似保持原连续系统的连续能量。详细研究了新方法的存在性、唯一性、正则性、全局收敛性和非线性稳定性。构造了两种实用的指数配点方法，并包含两个数值实验。数值结果展示了新方法在文献中现有数值方法相比的高效性。
英文摘要：
In this paper we derive and analyse new exponential collocation methods to efficiently solve the cubic Schrödinger Cauchy problem on a $d$-dimensional torus. Energy preservation is a key feature of the cubic Schrödinger equation. It is proved that the novel methods can be of arbitrarily high order which exactly or nearly preserve the continuous energy of the original continuous system. The existence and uniqueness, regularity, global convergence, nonlinear stability of the new methods are studied in detail. Two practical exponential collocation methods are constructed and two numerical experiments are included. The numerical results illustrate the efficiency of the new methods in comparison with existing numerical methods in the literature.
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