alexa KP solitons and total positivity for the Grassmannian


Journal of Generalized Lie Theory and Applications

Author(s): Yuji Kodama, Lauren Williams

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Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev and Petviashvili proposed a two-dimensional nonlinear dispersive wave equation now known as the KP equation. It is well-known that one can use the Wronskian method to construct a soliton solution to the KP equation from each point of the real Grassmannian Grk,n. More recently, several authors (Biondini and Chakravarty, J Math Phys 47:033514, 2006; Biondini and Kodama, J. Phys A Math Gen 36:10519–10536, 2003; Chakravarty and Kodama, J Phys A Math Theor 41:275209, 2008; Chakravarty and Kodama, Stud Appl Math 123:83–151, 2009; Kodama, J Phys A Math Gen 37:11169–11190, 2004) have studied the regular solutions that one obtains in this way: these come from points of the totally non-negative part of the Grassmannian (Grk,n)≥0. In this paper we exhibit a surprising connection between the theory of total positivity for the Grassmannian, and the structure of regular soliton solutions to the KP equation.

This article was published in Inventiones mathematicae and referenced in Journal of Generalized Lie Theory and Applications

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