Functional Analysis and Its Applications

, Volume 50, Issue 4, pp 308–318 | Cite as

Tangential polynomials and matrix KdV elliptic solitons

  • A. TreibichEmail author


Let (X, q) be an elliptic curve marked at the origin. Starting from any cover π: Γ → X of an elliptic curve X marked at d points {π i } of the fiber π −1(q) and satisfying a particular criterion, Krichever constructed a family of d × d matrix KP solitons, that is, matrix solutions, doubly periodic in x, of the KP equation. Moreover, if Γ has a meromorphic function f: Γ → P1 with a double pole at each p i , then these solutions are doubly periodic solutions of the matrix KdV equation U t = 1/4(3UU x + 3U x U + U xxx ). In this article, we restrict ourselves to the case in which there exists a meromorphic function with a unique double pole at each of the d points {p i }; i.e. Γ is hyperelliptic and each pi is a Weierstrass point of Γ. More precisely, our purpose is threefold: (1) present simple polynomial equations defining spectral curves of matrix KP elliptic solitons; (2) construct the corresponding polynomials via the vector Baker–Akhiezer function of X; (3) find arbitrarily high genus spectral curves of matrix KdV elliptic solitons.

Key words

KP equation KdV equation compact Riemann surface vector Baker–Akhiezer function ruled surface 


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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Université d’ArtoisArrasFrance
  2. 2.Universidad de la República, RNMontevideoUruguay

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