小格点间距区域差分方程的多尺度展开及邻近性和可积性检验。I
Multiscale expansions of difference equations in the small lattice spacing regime, and a vicinity and integrability test. I
摘要：
我们提出了一个算法程序i）研究之间的“距离”的可积偏微分方程和任何离散化，在小晶格间距的制度，并在同一时间，ii）测试（渐近）可积性的离散化。这种方法应该提供，特别是，有用的和具体的信息是如何好的任何数值计划用于集成一个给定的可积偏微分方程。该方法以非线性Schroedinger方程的一个相当一般的10参数离散族为例，由以下三个步骤组成：i）按照Degasperis-Manakov- Santini方法，构造离散系统的所有阶的通解的连续多尺度展开; ii）应用Degasperis-Procesi（DP）可积性检验对这种扩展进行检验，以检验离散系统的渐近可积性及其与连续极限的“距离”; iii）使用DP测试的主要输出来通过新颖且简单的公式构造离散系统的无穷多个近似对称性和运动常数。
英文摘要：
We propose an algorithmic procedure i) to study the ``distance'' between an integrable PDE and any discretization of it, in the small lattice spacing epsilon regime, and, at the same time, ii) to test the (asymptotic) integrability properties of such discretization. This method should provide, in particular, useful and concrete informations on how good is any numerical scheme used to integrate a given integrable PDE. The procedure, illustrated on a fairly general 10-parameter family of discretizations of the nonlinear Schroedinger equation, consists of the following three steps: i) the construction of the continuous multiscale expansion of a generic solution of the discrete system at all orders in epsilon, following the Degasperis - Manakov - Santini procedure; ii) the application, to such expansion, of the Degasperis - Procesi (DP) integrability test, to test the asymptotic integrability properties of the discrete system and its ``distance'' from its continuous limit; iii) the use of the main output of the DP test to construct infinitely many approximate symmetries and constants of motion of the discrete system, through novel and simple formulas.
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