Tangential polynomials and matrix KdV elliptic solitons

Abstract

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_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. 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.