Rational Degenerations of M-Curves, Totally Positive Grassmannians and KP2-Solitonsстатья

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Дата последнего поиска статьи во внешних источниках: 20 августа 2018 г.

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[1] Abenda S., Grinevich P. G. Rational degenerations of m-curves, totally positive grassmannians and kp2-solitons // Communications in Mathematical Physics. — 2018. — Vol. 361, no. 3. — P. 1029–1081. We establish a new connection between the theory of totally positive Grassmannians and the theory of 𝙼-curves using the finite-gap theory for solitons of the KP equation. Here and in the following KP equation denotes the Kadomtsev–Petviashvili 2 equation [see (1)], which is the first flow from the KP hierarchy. We also assume that all KP times are real. We associate to any point of the real totally positive Grassmannian Gr{TP}(N,M) a reducible curve which is a rational degeneration of an 𝙼-curve of minimal genus g=N(M−N), and we reconstruct the real algebraic-geometric data á la Krichever for the underlying real bounded multiline KP soliton solutions. From this construction, it follows that these multiline solitons can be explicitly obtained by degenerating regular real finite-gap solutions corresponding to smooth 𝙼-curves. In our approach, we rule the addition of each new rational component to the spectral curve via an elementary Darboux transformation which corresponds to a section of a specific projection Gr{TP}(r+1,M−N+r+1) to Gr{TP}(r,M−N+r). [ DOI ]

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