Integrable Systems and Spacetime Dynamics

TL;DR

This paper demonstrates that the AKNS integrable hierarchy can be derived from 3D gravity with negative cosmological constant through novel boundary conditions, linking integrable systems with spacetime dynamics.

Contribution

It introduces a geometrization of the AKNS system via boundary conditions in 3D gravity, revealing a new connection between integrable hierarchies and gravitational configurations.

Findings

AKNS hierarchy arises from 3D gravity equations

Boundary conditions invariant under infinite symmetry group

Includes conical singularities and black hole solutions

Abstract

It is shown that the Ablowitz-Kaup-Newell-Segur (AKNS) integrable hierarchy can be obtained as the dynamical equations of three-dimensional General Relativity with a negative cosmological constant. This geometrization of the AKNS system is possible through the construction of novel boundary conditions for the gravitational field. These are invariant under an asymptotic symmetry group characterized by an infinite set of AKNS commuting conserved charges. Gravitational configurations are studied by means of $SL(2,\mathbb{R})$ conjugacy classes. Conical singularities and black hole solutions are included in the boundary conditions.