Non-holonomic and Quasi-integrable deformations of the AB Equations

TL;DR

This paper introduces non-holonomic and quasi-integrable deformations of the AB system, expanding its modeling capabilities in physics and exploring their unique solutions and conserved quantities.

Contribution

It is the first to derive both non-holonomic and quasi-integrable deformations of the AB system, highlighting their properties and solutions.

Findings

Non-holonomic deformation preserves integrability with differential constraints.

Quasi-AB system exhibits asymptotic conserved quantities.

Localized solutions display physically relevant excitations.

Abstract

For the first time, both non-holonomic and quasi-integrable deformations are obtained for the AB system of coupled equations. The AB system models geophysical and atmospheric fluid motion along with ultra-short pulse propagation in nonlinear optics and serves as a generalization of the well-known sine-Gordon equation. The non-holonomic deformation retains integrability subjected to higher-order differential constraints whereas the quasi-AB system, which is partially deviated from integrability, is characterized by an infinite subset of quantities (charges) that are conserved only asymptotically given the solution possesses definite space-time parity properties. Particular localized solutions to both these deformations of the AB system are obtained, some of which are qualitatively unique to the corresponding deformation, displaying similarities with physically observed excitations.