Reflective prolate-spheroidal operators and the KP/KdV equations
Reflective prolate-spheroidal operators and the KP/KdV equations
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DOI：
10.1073/pnas.1906098116
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发表时间：
2019-09-10
影响因子：
11.1
通讯作者：
Zurrian, Ignacio
中科院分区：
文献类型：
--
作者：
Casper, W. Riley;Grunbaum, F. Alberto;Zurrian, Ignacio
交换积分和微分算子连接了信号处理、随机矩阵理论和可积系统的主题。以前，这种对的构造是基于直接计算和相关的具体特殊情况，留下了重要的族，例如与Korteweg-de Vries(KdV)方程的有理解相关的算子。我们证明了一个一般的定理，即在Wilson的无穷维广义Grass-mannianGr(Ad)中，与每个波函数相关的积分算子总是映射一个微分算子(在下面定义1的意义下)。这一内在性质源于Kadomtsev-Petviashvili(KP)波函数的Grassmannians对称性，其中直接交换性对于与由威尔逊符号对合固定的波函数相关的算子成立，但一般是违反的。在此基础上，我们证明了第二个主要定理，即计算与所有秩为1的双谱波函数有关的截断广义拉普拉斯变换奇异值的积分算子反映的是一个微分算子。利用90度旋转引理证明了第三个主要定理，即与所有这种KP波函数相关的截断广义傅立叶变换的奇异值计算中的积分算子与一个微分算子可交换。这些方法产生了大量具有长球面性质的积分算子，包括与[Airault，McKean，and Moser，Commun]所考虑的KdV和Kp族的所有有理解相关的积分算子作为特例。纯苹果。数学课。30、95-148(1977)]和[Krichever，Funkcional]。肛门。我普里洛。禅宗。12、76-78(1978)]，分别在1970年代末。文中给出了许多实例。
Commuting integral and differential operators connect the topics of signal processing, random matrix theory, and integrable systems. Previously, the construction of such pairs was based on direct calculation and concerned concrete special cases, leaving behind important families such as the operators associated to the rational solutions of the Korteweg-de Vries (KdV) equation. We prove a general theorem that the integral operator associated to every wave function in the infinite-dimensional adelic Grass-mannian Gr(ad) of Wilson always reflects a differential operator (in the sense of Definition 1 below). This intrinsic property is shown to follow from the symmetries of Grassmannians of Kadomtsev-Petviashvili (KP) wave functions, where the direct commutativity property holds for operators associated to wave functions fixed by Wilson's sign involution but is violated in general. Based on this result, we prove a second main theorem that the integral operators in the computation of the singular values of the truncated generalized Laplace transforms associated to all bispectral wave functions of rank 1 reflect a differential operator. A 90 degrees rotation argument is used to prove a third main theorem that the integral operators in the computation of the singular values of the truncated generalized Fourier transforms associated to all such KP wave functions commute with a differential operator. These methods produce vast collections of integral operators with prolate-spheroidal properties, including as special cases the integral operators associated to all rational solutions of the KdV and KP hierarchies considered by [Airault, McKean, and Moser, Commun. Pure Appl. Math. 30, 95-148 (1977)] and [Krichever, Funkcional. Anal. i Prilo. zen. 12, 76-78 (1978)], respectively, in the late 1970s. Many examples are presented.