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Erschienen in:
2017 | OriginalPaper | Buchkapitel
7. KP Solitons on \(\mathrm{Gr}(N, 2N)_{\ge 0}\)
verfasst von : Yuji Kodama
Erschienen in: KP Solitons and the Grassmannians
Verlag: Springer Singapore
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Abstract
In this chapter, we will discuss the reflective symmetry \((x, y, t) \leftrightarrow (-x,-y,-t)\) of the KP equation, and show that this symmetry is a consequence of the duality between the Grassmannian \(\mathrm{Gr}(N, M)\) and \(\mathrm{Gr}(M-N, M)\) in terms of the KP solitons. Using this duality, we construct the KP solitons for \(\mathrm{Gr}(M-N, M)_{\ge 0}\) from those for \(\mathrm{Gr}(N, M)_{\ge 0}\). We then consider a special class of KP solitons for \(\mathrm{Gr}(N, 2N)_{\ge 0}\), which consists of the same set of the asymptotic solitons at both \(y\ll 0\) and \(y\gg 0\), i.e. \(\mathscr {S}_+=\mathscr {S}_-\). The soliton solutions of this type are referred to as N-soliton solutions. The simplest ones of these solutions consist of N line-solitons having non-resonant interactions among those solitons and are generated from the points in an irreducible component of the lowest dimension, N, of \(\text {Gr}(N, 2N)_{\ge 0}\). 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 \(C_N=\frac{1}{N+1}\left( {\begin{array}{c}2N\\ N\end{array}}\right) \).