This article is cited in
3 papers
Tangential Polynomials and Matrix KdV Elliptic Solitons
A. Treibichab a Université d'Artois, France
b Universidad de la República, Uruguaj
Abstract:
Let
$(X,q)$ be an elliptic curve marked at the origin. Starting from any cover
$\pi\colon\Gamma\to X$ of an elliptic curve
$X$ marked at
$d$ points
$\{\pi_i\}$ of the fiber
$\pi^{-1}(q)$ and satisfying a particular criterion, Krichever constructed a family of
$d\times d$ matrix KP solitons, that is, matrix solutions, doubly periodic in
$x$, of the KP equation. Moreover, if
$\Gamma$ has a meromorphic function
$f\colon\Gamma\to\mathbb{P}^1$ with a double pole at each
$p_i$, then these solutions are doubly periodic solutions of the matrix KdV equation
$U_t=\frac14(3UU_x+3U_xU+U_{xxx})$. 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.
$\Gamma$ is hyperelliptic and each
$p_i$ is a Weierstrass
point of
$\Gamma$. 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.
Keywords:
KP equation, KdV equation, compact Riemann surface, vector Baker–Akhiezer function, ruled surface.
UDC:
517.9
Received: 10.10.2015
DOI:
10.4213/faa3251
Fulltext:
PDF file (234 kB)