On a Second Discretization of the ZS-AKNS Spectral Problem: Revisit

TL;DR

This paper revisits a 1989 discrete spectral problem related to the ZS-AKNS system, revealing its connection to bidirectional discretizations and deriving semi-discrete hierarchies through algebraic and continuum limit analyses.

Contribution

It introduces a new perspective on the second discretization of the ZS-AKNS spectral problem, linking it to higher-dimensional systems and constructing semi-discrete AKNS hierarchies.

Findings

The spectral problem corresponds to a bidirectional discretization of derivatives.

A connection with the differential-difference KP hierarchy is established.

Three semi-discrete AKNS hierarchies are constructed from algebraic structures.

Abstract

In this paper we revisit a discrete spectral problem which was proposed by Ragnisco and Tu in 1989, as a second discretization of the ZS-AKNS spectral problem. We show that the spectral problem corresponds to a bidirectional discretization of the derivative of two wave functions $\phi_{1,x}$ and $\phi_{2,x}$. As a connection with higher dimensional systems, the spectral problem and a related hierarchy can be derived from Lax triads of the differential-difference KP hierarchy via a symmetry constraint. Isospectral and nonisospectral flows derived from the spectral problem compose a Lie algebra. By considering its infinite dimensional subalgebras and continuum limit of recursion operator, three semi-discrete AKNS hierarchies are constructed.