标题： 关于中心二次假设：可积模型与Painlevé约化 显示英文标题

标题： On the central quadric ansatz: integrable models and Painleve reductions

摘要： 托德（Tod）以及后来的敦亚斯基（Dunajski）和托德观察到，博伊尔-芬利（Boyter-Finley, BF）方程和色散less Kadomtsev-Petviashvili（dKP）方程具有这样的解：这些解的水平面在独立变量空间中为中心二次曲面（即所谓的中心二次曲面假设）。研究显示，这类解的通用形式分别由Painlevé方程PIII和PII描述。 本文的目的有三点： -- 基于流体力学约化方法，我们分类了拥有中心二次曲面假设的可积模型。这导致了五种典型形式（包括BF和dKP）。 -- 将中心二次曲面假设应用于这五种典型形式中的每一个，我们得到了所有Painlevé方程PI - PVI，其中PVI对应于我们的分类中的通用情况。 -- 我们认为，来自中心二次曲面假设的解构成了由流体力学约化方法提供的双相解的一个子类。 显示英文摘要

摘要： It was observed by Tod and later by Dunajski and Tod that the Boyer-Finley (BF) and the dispersionless Kadomtsev-Petviashvili (dKP) equations possess solutions whose level surfaces are central quadrics in the space of independent variables (the so-called central quadric ansatz). It was demonstrated that generic solutions of this type are described by Painleve equations PIII and PII, respectively. The aim of our paper is threefold: -- Based on the method of hydrodynamic reductions, we classify integrable models possessing the central quadric ansatz. This leads to the five canonical forms (including BF and dKP). -- Applying the central quadric ansatz to each of the five canonical forms, we obtain all Painleve equations PI - PVI, with PVI corresponding to the generic case of our classification. -- We argue that solutions coming from the central quadric ansatz constitute a subclass of two-phase solutions provided by the method of hydrodynamic reductions.