Spectral Theory of $\mathrm {sl}(3,\mathbb {C})$ Auxiliary Linear Problem with $\mathbb {Z}_2\times \mathbb {Z}_2\times \mathbb {Z}_2$ Reduction of Mikhailov Type

Abstract

We consider an auxiliary system $L_{S_{\pm 1}}$, see (1) below, used recently as L operator in a Lax pair for some soliton equations. $L_{S_{\pm 1}}$ could be regarded as a generalization of a pole gauge Generalized Zakharov-Shabat system on $\mathrm {sl}(3,\mathbb {C})$ on the whole real axis involving rational dependence on the spectral parameter. We consider the system on the condition that its ‘potentials’ u(x) and v(x) tend sufficiently fast to constant values $u_0,v_0$ when $x\rightarrow \pm \infty$ in the general situation when both $u_0,v_0\ne 0$. We show that in this case the spectral theory for $L_{S_{\pm 1}}$ should be considered on a suitable Riemann surface and discuss the symmetry properties of the fundamental analytic solutions to $L_{S_{\pm 1}}\psi =0$.