Tyurin parameters and elliptic analogue of nonlinear Schr\"odinger hierarchy

TL;DR

This paper constructs elliptic analogues of the nonlinear Schrödinger hierarchy incorporating Tyurin parameters, and maps these systems to a Grassmannian framework to analyze their integrable structure.

Contribution

It introduces elliptic analogues of the NLS hierarchy with Tyurin parameters and elucidates their Grassmannian representation, expanding the understanding of elliptic integrable systems.

Findings

Constructed two elliptic analogues of the NLS hierarchy.

Established a Grassmannian mapping for these systems.

Derived solutions forming a Riemann-Hilbert pair.

Abstract

Two "elliptic analogues'' of the nonlinear Schr\"odinger hiererchy are constructed, and their status in the Grassmannian perspective of soliton equations is elucidated. In addition to the usual fields $u,v$, these elliptic analogues have new dynamical variables called ``Tyurin parameters,'' which are connected with a family of vector bundles over the elliptic curve in consideration. The zero-curvature equations of these systems are formulated by a sequence of $2 \times 2$ matrices $A_n(z)$, $n = 1,2,...$, of elliptic functions. In addition to a fixed pole at $z = 0$, these matrices have several extra poles. Tyurin parameters consist of the coordinates of those poles and some additional parameters that describe the structure of $A_n(z)$'s. Two distinct solutions of the auxiliary linear equations are constructed, and shown to form a Riemann-Hilbert pair with degeneration points. The Riemann-Hilbert pair is used to define a mapping to an infinite dimensional Grassmann variety. The elliptic analogues of the nonlinear Schr\"odinger hierarchy are thereby mapped to a simple dynamical system on a special subset of the Grassmann variety.