Department of Mathematics, Ohio State University, Columbus, OH 43210, USA.
Proc Natl Acad Sci U S A. 2011 May 31;108(22):8984-9. doi: 10.1073/pnas.1102627108. Epub 2011 May 11.
Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev and Petviashvili [Kadomtsev BB, Petviashvili VI (1970) Sov Phys Dokl 15:539-541] proposed a two-dimensional nonlinear dispersive wave equation now known as the KP equation. It is well-known that the Wronskian approach to the KP equation provides a method to construct soliton solutions. The regular soliton solutions that one obtains in this way come from points of the totally nonnegative part of the Grassmannian. In this paper we explain how the theory of total positivity and cluster algebras provides a framework for understanding these soliton solutions to the KP equation. We then use this framework to give an explicit construction of certain soliton contour graphs and solve the inverse problem for soliton solutions coming from the totally positive part of the Grassmannian.
自 1970 年 Kadomtsev 和 Petviashvili[Kadomtsev BB, Petviashvili VI (1970) Sov Phys Dokl 15:539-541]提出现在被称为 KP 方程的二维非线性弥散波方程以来，人们一直在研究 KP 方程的孤子解。众所周知，KP 方程的 Wronskian 方法提供了一种构造孤子解的方法。通过这种方法得到的正则孤子解来自 Grassmannian 的全非负部分的点。在本文中，我们解释了全正理论和簇代数如何为理解 KP 方程的这些孤子解提供了一个框架。然后，我们使用这个框架来给出某些孤子轮廓图的显式构造，并解决来自 Grassmannian 的全正部分的孤子解的逆问题。
