Titlebook: KP Solitons and the Grassmannians; Combinatorics and Ge Yuji Kodama Book 2017 The Author(s) 2017 KP equation.soliton solutions in two-dimen

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Introduction to the Real Grassmannian,am, which gives a graphical interpretation of the permutation [103]. The pipedream will be useful to describe the spatial structure of the KP soliton as we will see in the later chapters. (See, for example, [15, 44, 45, 49] for the general information on the Grassmannian, the Young diagram and the symmetric group of permutations.)

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Book 2017borators, the book presents new developments focused on an interplay between the theory of solitons and the combinatorics of finite-dimensional Grassmannians, in particular, the totally nonnegative (TNN) parts of the Grassmannians..The book begins with a brief introduction to the theory of the Kadom

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The Deodhar Decomposition for the Grassmannian and the Positivity,the flag variety due to Marsh and Rietsch [83]. We conclude this section to give an algorithm to compute an explicit form of the matrix . and to discuss the positivity of .. Most of the materials presented here can be also found in [72].

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Two-Dimensional Solitons, soliton solutions in the determinant form and show that their wave parameters for these solutions are chosen from conic curves, that is, the KP soliton from the parabola, the two-dimensional Toda soliton from the hyperbola, and the Davey-Stewartson soliton from the circle.

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KP Solitons on ,,nd are generated from the points in an irreducible component of the . dimension, ., of .. We also discuss some combinatorial properties of those solutions. For the simplest cases of non-resonant interactions, the total number of such N-soliton solutions is given by a Catalan number ..

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Soliton Graphs,s the soliton graphs for the matrix . and give coordinates for all of the trivalent vertices, which then allows one to completely describe the soliton graph. Most of this chapter will be devoted to the case when ., with the final section explaining how the same ideas can be applied to the case when ..