A particle-system model of the sine-Gordon hierarchy
This is really an extended abstract of author's results, with no proofs of theorems. The author makes the following statement. The first Lax-pair equation to be integrated by constrained harmonic motion (the ''Neumann system'') was KdV. The construction was later: (i) extended to other equations; (ii) viewed as a special case of a class of (constrained) completely integrable systems called rank 2 perturbations; (iii) explained in the more natural context of ''squared eigenfunctions''; (iv) linked with the algebraic geometry that is called for by the angle variables and had been used independently to construct quasiperiodic solutions via Baker-Akhiezer functions. We present an analogous treatment for the sine-Gordon equation; we use a rank-2 perturbation with symmetries to integrate it and spell out the link with algebraic geometry: this involves an unramified double-covering of a hyperelliptic curve, which had appeared in other contexts. The reason for that is that the AKNS eigenfunctions will have essential singularities at two points, 0 and \(\infty\), of the base curve and will be well-defined only when we pass to a double cover. Similar combinations of tangent vectors of the curve at 0 and \(\infty\) may provide a simple way of generating other interesting equations. (AKNS stands for \textit{M. J. Ablowitz}, \textit{D. J. Kaup}, \textit{A. C. Newell}, \textit{H. Segur} [Stud. Appl. Math. 53, 249-315 (1974; Zbl 0408.35068)].) As an application, we give a necessary and sufficient condition for the solution to be real that confirms the preexisting ones. In favorable cases, this can be lifted linearly to the original phase-space \({\mathbb{C}}^{4n}\).
scientific article; zbMATH DE number 3957809