Rational solutions of nonlinear evolution equations, vertex operators, and bispectrality (Q1188228)

By definition, a differential operator \(L(x,\partial_ x)\) has the bispectral property if there exists a differential operator \(B(k,\partial_ k)\) and a nonconstant function \(\theta(x)\) such that the equations: \(L\varphi=k\varphi\), \(B\varphi=\theta(x)\varphi\) have common nonzero solutions \(\varphi(x,k)\). \textit{J. J. Duistermaat} and \textit{F. A. Grünbaum} [Commun. Math. Phys. 103, 177-240 (1986; Zbl 0625.34007)] showed that the Schrödinger operator \(L(x,\partial_ x)=-\partial^ 2_ x+v\) has the bispectral property only if the potential \(v\) is a rational solution of the KdV equation. Here the author investigates the analogous question about the AKNS/ZS operator \(L=\left[{\partial_ x\atop r} {-q\atop-\partial_ x}\right]\), where \(q=q(x,t_ 2,\dots,t_ m)\), \(r=r(x,t_ 2,\dots,t_ m)\).