Geometry of Spectral Curves and All Order Dispersive Integrable System

Understand

We propose a definition for a Tau function and a spinor kernel (closely related to Baker-Akhiezer functions), where times parametrize slow (of order 1/N) deformations of an algebraic plane curve.

• This definition consists of a formal asymptotic series in powers of 1/N, where the coefficients involve theta functions whose phase is linear in N and therefore features generically fast oscillations when N is large.
• The large N limit of this construction coincides with the algebro-geometric solutions of the multi-KP equation, but where the underlying algebraic curve evolves according to Whitham equations.
• We check that our conjectural Tau function satisfies Hirota equations to the first two orders, and we conjecture that they hold to all orders.