Discretisations of Constrained KP Hierarchies
Discretisations of Constrained KP Hierarchies
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发表时间：
2014-06
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通讯作者：
R. Willox;Madoka Hattori
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作者：
R. Willox;Madoka Hattori
我们提出了一个离散的模拟所谓的对称减少或“约束”KP层次。作为结果，我们得到了一些著名的连续可积系统，如非线性薛定谔方程，Broer-Kaup方程和Yajima-Oikawa系统，以及它们的Lax对在空间和时间上的可积离散。它将表明，这些离散化也引起了一个离散的描述相关联的可积系统的整个层次。Broer-Kaup方程和Yajima-Oikawa系统的离散化被认为是新的。自非线性Schrodinger（NLS）方程（1）或Korteweg-de弗里斯（KdV）方程（17）的可积差分格式首次被发现至今，已有35年多的历史，可积离散系统的研究已经取得了显著的进展。然而，尽管在20世纪80年代取得了巨大的概念上和技术上的进步（例如，见一系列引人注目的论文，其中（6，7）构成了中间部分），但在那个十年的后半期，对可积离散化的兴趣暂时减弱了，因为大多数研究活动集中在更适合于连续系统的可积性的纯分析方法上。然而，在接下来的十年里，所有这一切都发生了巨大的变化，这主要是由于一系列同时发生但起初看似不相关的发现。首先是在二维引力场理论模型的背景下（重新）发现了Painleve I方程的离散形式（3），这导致了离散Painleve方程结果的爆炸，并发展了离散映射的可积性测试（12）（参见[1]）。(13)对于该领域的历史和数学细节的回顾）。其次，发现了第一个孤立子元胞自动机（46），随后发现该系统通过一个特殊的极限过程（超离散极限）与离散KdV方程密切相关（47）。这些结果和随后的实现，即通过“ultradesisation”获得的系统实际上与某些完全可解的晶格模型（在其晶体极限）有关，在量子可积和经典可积细胞自动机中产生了大量的研究活动。然而，多年来，一个奇怪的情况已经发展。尽管在可积自动机或热带可积系统领域已经取得了相当大的进展，尽管与各种各样的对称代数相关联的这种系统的许多例子已经被构造出来，但1+1维经典离散可积系统应该
We present a discrete analogue of the so-called symmetry reduced or 'constrained' KP hierarchy. As a result we obtain integrable discretisations, in both space and time, of some well-known continuous integrable systems such as the nonlinear Schrodinger equation, the Broer-Kaup equation and the Yajima-Oikawa system, together with their Lax pairs. It will be shown that these discretisations also give rise to a discrete description of the entire hierarchy of associated integrable systems. The discretisations of the Broer- Kaup equation and of the Yajima-Oikawa system are thought to be new. There has been remarkable progress in the study of integrable discrete sys- tems, since the initial discovery of integrable difference schemes for the nonlinear Schrodinger (NLS) equation (1) or the Korteweg-de Vries (KdV) equation (17), now more than 35 years ago. However, although tremendous conceptual as well as tech- nical advances were made during the 1980's (see e.g. the remarkable series of pa- pers of which (6, 7) constitute the middle part), interest in integrable discretisations temporarily waned during the latter half of that decade, as most research activity focused on purely analytical approaches to integrability, better suited to continuous systems. All this changed dramatically however during the following decade, and this mainly due to a series of simultaneous but at first seemingly unrelated discov- eries. First there was the (re-)discovery of a discrete form of the Painleve I equation in the context of a field-theoretical model of 2-dimensional gravity (3), which led to an explosion of results on discrete Painleve equations, and to the development of integrability tests for discrete mappings (12) (cf. (13) for a review of the history and the mathematical particulars of the field). Secondly, there was the discovery of the first solitonic cellular automaton (46), followed by the discovery that this system is intimately related to the discrete KdV equation through a special limiting proce- dure, the ultradiscrete limit (47). These results and the subsequent realization that systems obtained through 'ultradiscretisation' are in fact related to certain exactly solvable lattice models (at their crystal limits), spawned an enormous amount of research activity in both quantum integrable as well as classically integrable cellular automata. However, over the years, a strange situation has developed. Whereas considerable advances have been made in the field of integrable automata, or trop- ical integrable systems as they are also known, and whereas numerous examples of such systems associated to a large variety of symmetry algebras have been con- structed, the 1+1 dimensional, classical, discrete integrable systems that ought to