A Local Sine-gordon Hierarchy and Its Algebro-geometric Solutions

We derive a new zero-curvature formalism for the sine-Gordon (sG) equation which permits the introduction of a local sine-Gordon hierarchy (in contrast to the traditionally accepted nonlocal higher-order sG equations). In complete analogy to other completely integrable hierarchies of soli-ton equations, such as the KdV, AKNS, and Toda hierarchies, our local sG hierarchy is recursively constructed by means of a fundamental polynomial formalism involving a spectral parameter. We further illustrate our approach by developing the basic algebro-geometric setting for the sG hierarchy, including Baker–Akhiezer functions, trace formulas, Dubrovin-type equations, and theta function representations for its algebro-geometric solutions. Although we mainly focus on sG-type hierarchies, our formalism also yields (local) hierarchies for the sinh-Gordon, elliptic sine-Gordon, elliptic sinh-Gordon, and Liouville-type equations.