A Combined Sine-gordon and Modified Korteweg{de Vries Hierarchy and Its Algebro-geometric Solutions

We derive a zero-curvature formalism for a combined sine-Gordon (sG) and modi-ed 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-typeequations, 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. 1. Introduction This paper is concerned with two main issues, a systematic derivation of a local hierarchy of nonlinear evolution equations by embedding the sine-Gordon (sG) equation into the modiied Korteweg{de Vries (mKdV) hierarchy, resulting in what we call the sGmKdV hierarchy, and a simpliied approach to its algebro-geometric solutions. A careful investigation of the (enormous) literature on the sG equation (in light-cone coordinates) u xt = sin(u); (1.1) reveals the fact that relatively little eeort has been spent on deriving solutions which simultaneously satisfy the whole hierarchy of sine-Gordon equations. More signiicantly, the generally accepted hierarchy in the sine-Gordon case, as originally derived by Sasaki and Bullough 68], 69] in 1980, in sharp contrast to other hierarchies of soliton equations such as the KdV, AKNS, Toda, and Gelfand{Dickey hierarchies, appears to be nonlocal in u for all but the rst element (1.1) in the hierarchy, although attempts at deriving a local sine-Gordon hierarchy (which, however, fell short of providing an explicit formalism) were made by S. J. Al'ber and M. S. Al'ber 3] in 1987. In particular, algebro-geometric (or periodic) solutions and their theta function representations are not derived for higher-order sG equations. The current paper focuses on the close connection between the sG equation and the mKdV hierarchy. We ooer an elementary recursive approach to a local hierarchy which combines the sG equation and the mKdV hierarchy in a completely inte-grable manner (and similarly for the sinh-Gordon, elliptic sine-Gordon, elliptic sinh-Gordon, and Liouville-type equations, etc.), in the spirit of previous treatments of the Toda 14], Boussinesq 19], AKNS 46], and KdV 47] hierarchies, respectively. Since, in a sense to be made precise at the end of Section 2, the new hierarchy embeds the sG equation …