F. Gesztesy and H. Holden
A combined sine-Gordon and modified Korteweg--de Vries hierarchy and its
algebro-geometric solutions
(127K, LaTeX)
ABSTRACT. We derive a zero-curvature formalism for a combined sine-Gordon (sG) and
modified Korteweg-de Vries (mKdV) equation which yields a local sGmKdV
hierarchy. In complete analogy to other completely integrable hierarchies
of soliton equations, such as the KdV, AKNS, and Toda hierarchies, the
sGmKdV 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
sGmKdV 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 equations,
our formalism also yields the sinh-Gordon, elliptic sine-Gordon, elliptic
sinh-Gordon, and Liouville-type equations combined with the mKdV hierarchy.