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Classification of Soliton Graphs On Totally positive Grassmannian
Huang, Jihui

2015, Doctor of Philosophy, Ohio State University, Mathematics.
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 pa- rameters 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 space.
Here is a summary of the main results. We develop an explicit algorithm to construct the soliton subdivisions by lifting the configuration A with specific weights for these points in A. We use the Gale transform to identify the polyhedral cones of Gr(1,M)>0 case to the secondary polytopes. Then we extend the Gale transform for the Gr(N,M)>0 case, and give a realizability checking theorem about soliton sub- divisions, which induces a parameterization of a polyhedral cone in multiple time t space by a soliton subdivision. In the end, we present the detailed analysis for the case of Gr(3, 6).
Yuji Kodama (Advisor)
Michael Davis (Committee Member)
David Anderson (Committee Member)
Vladimir Kogan (Committee Member)
92 p.

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Huang, Jihui. "Classification of Soliton Graphs On Totally positive Grassmannian." Electronic Thesis or Dissertation. Ohio State University, 2015. OhioLINK Electronic Theses and Dissertations Center. 25 Sep 2017.

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Huang, Jihui "Classification of Soliton Graphs On Totally positive Grassmannian." Electronic Thesis or Dissertation. Ohio State University, 2015.