Fokas A S
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK.
Phys Rev Lett. 2006 May 19;96(19):190201. doi: 10.1103/PhysRevLett.96.190201.
The derivation and solution of integrable nonlinear evolution partial differential equations in three spatial dimensions has been the holy grail in the field of integrability since the late 1970s. The celebrated Korteweg-de Vries and nonlinear Schrödinger equations, as well as the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. Do there exist integrable analogs of these equations in three spatial dimensions? In what follows, I present a positive answer to this question. In particular, I first present integrable generalizations of the KP and DS equations, which are formulated in four spatial dimensions and which have the novelty that they involve complex time. I then impose the requirement of real time, which implies a reduction to three spatial dimensions. I also present a method of solution.
自20世纪70年代末以来，三维空间中可积非线性演化偏微分方程的推导与求解一直是可积性领域的圣杯。著名的科特韦格 - 德弗里斯方程和非线性薛定谔方程，以及卡多姆采夫 - 彼得维谢夫利（KP）方程和戴维 - 斯图尔特森（DS）方程，分别是一维和二维空间中可积演化方程的典型例子。在三维空间中是否存在这些方程的可积类似物呢？在接下来的内容中，我将对这个问题给出肯定的答案。具体而言，我首先给出KP方程和DS方程的可积推广形式，它们是在四个空间维度中表述的，并且具有涉及复时间这一新颖之处。然后我施加实时间的要求，这意味着降维到三个空间维度。我还给出了一种求解方法。
