The geometry of real sine-Gordon wavetrains

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Abstract

The characterization ofreal, N phase, quasiperiodic solutions of the sine-Gordon equation has been an open problem. In this paper we achieve this result, employing techniques of classical algebraic geometry which have not previously been exploited in the soliton literature. A significant by-product of this approach is a naturalalgebraic representation of the full complex isospectral manifolds, and an understanding of how the real isospectral manifolds are embedded. By placing the problem in this general context, these methods apply directly to all soliton equations whose multiphase solutions are related to hyperelliptic functions.

Supported in part by NSF Grant No. MCS-8202288
Supported in part by NSF Grant No. MCS-8002969
Communicated by S.-T. Yau