Tangential Polynomials and Matrix KdV Elliptic Solitons

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.