Classification of Soliton Graphs On Totally positive Grassmannian
Classification of Soliton Graphs On Totally positive Grassmannian
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全正格拉斯曼孤子图的分类
DOI：
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发表时间：
2015
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影响因子：
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通讯作者：
Ji
中科院分区：
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作者：
Ji
已知一类非线性波动方程存在稳定的孤立波解，这些孤立波解是规则的、非衰减的，并且在xy平面上沿着不同的直线上局部化。这种方程最著名的例子是KP方程，它为浅水波提供了一个很好的模型。这些解被称为线孤子解，它们形成类似于网状结构的复杂的线孤子相互作用模式。在这里，由这些孤子解产生的图案将被称为孤子图。在这篇论文中，我们主要考虑KP方程的孤子解。KP方程的孤子解uA（x，y，t）可以由真实的Grassmannian Gr（N，M）的一个点A构造，并证明了孤子解uA（x，y，t）的正则性等价于A的全非负性，即A是全非负Grassmannian的一个元素，记为Gr（N，M）≥0.本文的主要目的是利用几何组合学对孤立子图进行分类。为此，我们考虑KP族的孤子解，它由KP方程的对称性组成，由相容时间变量序列t =（t3，t4，· · ·）参数化。每个孤子图可以表示为一个点配置A，其中A的每个元素表示一组孤子参数，如波数和传播方向。然后我们考虑了点组态A的一种细分，我们称之为孤子细分，并将孤子细分与多时间t中的多面体扇结构联系起来。
It has been known that certain class of nonlinear wave equations admits stable solitary wave solutions that are regular, non decaying and localized along distinct lines in the xy-plane. The most well-known example of such equations is the KP equation, and it provides an excellent model for shallow water waves. These solutions are known as the line-soliton solutions, and they form complex interaction patterns of line-solitons resembling web-like structures. Here the patterns generated by those soliton solutions will be called the soliton graphs. In this thesis, we consider mainly the soliton solutions of the KP equation. It is well known that a soliton solution uA(x, y, t) of the KP equation can be constructed from a point A of the real Grassmannian Gr(N,M), and has been proven that the regularity of soliton solution uA(x, y, t) is equivalent to the total non-negativity of A, that is, A is an element of totally nonnegative Grassmannian, denoted by Gr(N,M)≥0. The main purpose of the thesis is to classify the soliton graphs using geometric combinatorics. For this purpose, we consider the soliton solutions of the KP hierarchy, which consists of the symmetries of the KP equation parameterized by a sequence of compatible time variables t = (t3, t4, · · · ). Each soliton graph can be expressed as a point configuration A, where each element of A represents a set of soliton parameters such as the wavenumber and the propagation direction. Then we consider a subdivision of the point configuration A, which we call soliton subdivisions, and we relate the soliton subdivisions with polyhedral fan structure in the multi-time t