Geometry of KdV. IV: Abel sums, Jacobi variety, and theta function in the scattering case (Q750771)

[For part II see the second author, J. Stat. Phys. 46, No.5-6, 1115-1143 (1987; Zbl 0689.35076); part III is to appear.] The authors study the geometry of Korteweg-de Vries equations with Abel sums, Jacobi variety, and theta function in the scattering case. They identify the ``curve'', ``theta function'', and ``theta divisor'', and confirm that they share much of the familiar beautiful geometry. A cursory view of the full phase geometry is presented. But the principal object of study is a single invariant manifold and its complexification. [For part V see the second author, Commun. Pure Appl. Math. 42, No.5, 687-701 (1989; Zbl 0699.58062).]