Space curves and solitons of the KP hierarchy | Topics in Integrability

Abstract

It is well known that algebro-geometric solutions of the KdV hierarchy are constructed from the Riemann theta (or Klein sigma) functions associated with hyperelliptic curves, and soliton solutions can be obtained by rational limits of the corresponding curves. In this talk, I will associate a family of singular space curves indexed by the numerical semigroups $\left \langle l, lm+1, \dots, lm+k \right \rangle$ where $m \ge 1$ and $1\le k \le l-1$ with a class of generalized KP solitons. Some of these curves can be deformed into smooth \space curves", and they provide canonical models for the $l$-th generalized KdV hierarchies (KdV hierarchy corresponds to the case $l=2$). If time permits, we will also see how to construct the space curves from a commutative ring of di erential operators in the sense of the well-known Burchnall-Chaundy theory. This talk is based on a joint work with Professor Yuji Kodama.