The τ-Functions of the gAKNS Equations

Abstract

The AKNS equations (see [1])

$$\left. {\matrix{ {i{q_t} = - {1 \over 2}{q_{xx}} + {q^2}r} \cr {i{r_t} = {1 \over 2}{r_{xx}} - q{r^2}} \cr } } \right\}$$

are generally agreed to be one of the most basic examples of an integrable evolutionary system. They may be regarded as a complexified version of the physically important nonlinear Schrödinger equation(s)

$${i{q_t} = - {1 \over 2}{q_{xx}} \pm {q^2}\bar q}$$

to which they reduce if we impose one of the conditions q = ±̄r. The equations (1.1) are the simplest (in some sense) integrable system associated with the Lie algebra sl(2, ℂ); as such they have natural generalizations in which sl(2, ℂ) is replaced by some other simple Lie algebra g. I shall refer to these as the gAKNS equations. In the case when g is sl(n, ℂ) the gAKNS equations are exactly the equations referred to as the AKNS-D equations in [3]; for a general simple Lie algebra g it seems that they were first introduced in [9]. The equations (1.2), in which x and t must be considered to be real, are associated with the real subalgebras su(2) and su(1,1) of sl(2,ℂ), and will not be discussed explicitly here.