Method of Squared Eigenfunction Potentials in Integrable Hierarchies of KP Type
Method of Squared Eigenfunction Potentials in Integrable Hierarchies of KP Type
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DOI：
10.1007/s002200050338
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发表时间：
1997-01
影响因子：
2.4
通讯作者：
H. Aratyn;E. Nissimov;S. Pacheva
中科院分区：
文献类型：
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作者：
H. Aratyn;E. Nissimov;S. Pacheva
本文系统地发展了平方本征函数势（SEP）方法，用以描述和获得Kadomtsev-Petviashvili（KP）族及其约化的新信息。详细讨论了与τ函数法的相互关系。的主要结果，它形成了我们的SEP方法的基础上，是证明一般KP系列的任何本征函数可以表示为一个频谱积分的Baker-Akhiezer（BA）波函数的频谱密度表示SEP。事实上，（伴随）BA函数的频谱表示，反过来，可以被认为是KP系列的定义方程。SEP方法随后被用来显示如何减少的完整的KP系列的约束KP（cKPrm）系列可以完全在相关的τ-函数的线性约束方程。SEP的概念在用通用的Sato Grassmannian语言描述cKPrmhierarchies和找到作用于底层τ函数的非等谱Virasoro对称生成元方面是至关重要的。本文用SEP方法写出了约束KP族的广义BinaryDarboux-Bäcklund变换，证明了其轨道对应于一个新的四方格上的户田模型.结果，我们得到了τ-函数的一系列新的行列式解，推广了已知的Wronskian（多孤子）解.最后简要讨论了随机矩阵模型在凝聚态物理中的应用。
The method of squared eigenfunction potentials (SEP) is developed systematically to describe and gain new information about the Kadomtsev–Petviashvili (KP) hierarchy and its reductions. Interrelation to the τ-function method is discussed in detail. The principal result, which forms the basis of our SEP method, is the proof that any eigenfunction of the general KP hierarchy can be represented as a spectral integral over the Baker–Akhiezer (BA) wave function with a spectral density expressed in terms of SEP. In fact, the spectral representations of the (adjoint) BA functions can, in turn, be considered as defining equations for the KP hierarchy. The SEP method is subsequently used to show how the reduction of the full KP hierarchy to the constrained KP (cKPrm) hierarchies can be given entirely in terms of linear constraint equations on the pertinent τ-functions. The concept of SEP turns out to be crucial in providing a description of cKPrmhierarchies in the language of the universal Sato Grassmannian and finding the non-isospectral Virasoro symmetry generators acting on the underlying τ-functions. The SEP method is used to write downgeneralized binaryDarboux-Bäcklund transformations for constrained KP hierarchies whose orbits are shown to correspond to a new Toda model on asquarelattice. As a result, we obtain a series of new determinant solutions for the τ-functions generalizing the known Wronskian (multi-soliton) solutions. Finally, applications to random matrix models in condensed matter physics are briefly discussed.