Completion of the integrable coupling systems

TL;DR

This paper introduces a method to complete integrable systems by adding perturbations, enabling the construction of various continuous, discrete, and super integrable couplings, demonstrated on multiple known systems.

Contribution

It proposes a novel procedure for completing integrable systems through perturbations, applicable to a wide range of integrable couplings.

Findings

Constructed completions for several known integrable couplings.

Provided a unified approach for continuous, discrete, and super integrable couplings.

Demonstrated the method on systems like KN, WKI, vAKNS, Volterra, Dirac, and NLS-mKdV.

Abstract

In this paper, we proposed an procedure to construct the completion of the integrable system by adding a perturbation to the generalized matrix problem, which can be used to continuous integrable couplings, discrete integrable couplings and super integrable couplings. As example, we construct the completion of the Kaup-Newell (KN) integrable coupling, the Wadati-Konno-Ichikawa (WKI) integrable couplingsis, vector Ablowitz-Kaup-Newell-Segur (vAKNS) integrable couplings, the Volterra integrable couplings, Dirac type integrable couplings and NLS-mKdV type integrable couplings.