An algorithm to generate integrable systems is extended to the super case. Some new examples of superextensions of integrable systems are illustrated. We also generalize the trace identity due to Tu to the super case and use it to establish Hamiltonian structures of superextensions of integrable systems under consideration.

https://iopscience.iop.org/article/10.1088/0305-4470/30/2/023/pdf

An approach to generate superextensions of integrable systems

A supersymmetric generalization of the Krichever map is used to construct algebro-geometric solutions to the various super Kadomtsev-Petviashvili (SKP) hierarchies. The geometric data required consist of a suitable algebraic supercurve of genusg (generallynot a super Riemann surface) with a distinguished point and local coordinates (z, θ) there, and a generic line bundle of degreeg−1 with a local trivialization near the point. The resulting solutions to the Manin-Radul SKP system describe coupled deformations of the line bundle and the supercurve itself, in contrast to the ordinary KP system which deforms line bundles but not curves. Two new SKP systems are introduced: an integrable “Jacobian” system whose solutions describe genuine Jacobian flows, deforming the bundle but not the curve; and a nonintegrable “maximal” system describing independent deformations of bundle and curve. The Kac-van de Leur SKP system describes the same deformations as the maximal system, but in a different parametrization.

/pdf/the-geometry-of-the-super-kp-flows-uv7hn3gwrp.pdf

The geometry of the super KP flows

We study the geometry and cohomology of algebraic super curves, using a new contour integral for holomorphic differentials. For a class of super curves (``generic SKP curves'') we define a period matrix. We show that the odd part of the period matrix controls the cohomology of the dual curve. The Jacobian of a generic SKP curve is a smooth supermanifold; it is principally polarized, hence projective, if the even part of the period matrix is symmetric. In general symmetry is not guaranteed by the Riemann bilinear equations for our contour integration, so it remains open whether Jacobians are always projective or carry theta functions. 
These results on generic SKP curves are applied to the study of algebro-geometric solutions of the super KP hierarchy. The tau function is shown to be, essentially, a meromorphic section of a line bundle with trivial Chern class on the Jacobian, rationally expressible in terms of super theta functions when these exist. Also we relate the tau function and the Baker function for this hierarchy, using a generalization of Cramer's rule to the supercase.

Supercurves, their Jacobians, and super KP equations

Darboux transformation for the Manin-Radul supersymmetric KdV equation

Wide classes of explicit solutions of the Manin‐Radul and Jacobian supersymmetric KP hierarchies are constructed by using line bundles over complex supercurves based on the Riemann sphere. Their construction extends several ideas of the standard KP theory, such as wave functions, ∂‐equations and τ‐functions. Thus, supersymmetric generalizations of N‐soliton solutions, including a new purely odd ‘‘solitino’’ solution, as well as rational solutions, are found and characterized.

/pdf/explicit-solutions-of-supersymmetric-kp-hierarchies-248x2nvpj9.pdf

Explicit solutions of supersymmetric KP hierarchies:Supersolitons and solitinos

Bilinear residue formulas are established for the super-KP hierarchy and the ortho-symplectic super-KP hierarchy. Furthermore, superframes corresponding to the ortho-symplectic super-KP hierarchy are completely characterized. Soliton solutions to the super-KP hierarchy are given.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1104179075

Kimio Uneo

Papers

Algebraic study on the super-KP hierarchy and the ortho-symplectic super-KP hierarchy