Modified AKNS model, Riccati-type pseudo-potential approach and infinite towers of quasi-conservation laws

TL;DR

This paper introduces a Riccati-type pseudo-potential approach to a modified AKNS system, revealing infinite towers of anomalous and non-local conservation laws, with implications for various nonlinear physical systems.

Contribution

It develops a unified method to derive infinite quasi-conservation and conservation laws for the modified AKNS and related models, expanding understanding of their integrability properties.

Findings

Uncovered infinite towers of anomalous conservation laws.

Derived exact non-local conservation laws for the system.

Linked the modified AKNS model to other nonlinear systems like NLS and MNLS.

Abstract

A dual Riccati-type pseudo-potential formulation is introduced for a modified AKNS system (MAKNS) and infinite towers of novel anomalous conservation laws are uncovered. In addition, infinite towers of exact non-local conservation laws are uncovered in a linear formulation of the system. It is shown that certain modifications of the non-linear Schr\"odinger model (MNLS) can be obtained through a reduction process starting from the MAKNS model. So, the novel infinite sets of quasi-conservation laws and related anomalous charges are constructed by an unified and rigorous approach based on the Riccati-type pseudo-potential method, for the standard NLS and modified MNLS cases, respectively. The non-local properties, the complete list of towers of infinite number of anomalous charges and the (non-local) exact conservation laws of the quasi-integrable systems, such as the deformed Bullough-Dodd, Toda, KdV and SUSY sine-Gordon systems can be studied in the framework presented in this paper. Our results may find many applications since the AKNS-type system arises in several branches of non-linear physics, such as Bose-Einstein condensation, superconductivity and soliton turbulence.