Integrable Systems and Spacetime Dynamics.

Marcela Cárdenas,Miguel Pino,Francisco Correa,Kristiansen Lara

doi:10.1103/physrevlett.127.161601

Marcela Cárdenas, Miguel Pino + Show 2 more

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https://doi.org/10.1103/physrevlett.127.161601

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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,R) conjugacy classes. Conical singularities and black hole solutions are included in the boundary conditions.

Highlights

Introduction.—Over the years, the significance of integrable systems has been successfully demonstrated and tested in almost all areas of physics
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

Summary

Integrable Systems and Spacetime Dynamics

Marcela Cárdenas,1,* Francisco Correa ,2,† Kristiansen Lara ,1,‡ and Miguel Pino 1,3,§. 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. The Ward conjecture explores the possibility that all integrable systems find a common origin as reductions of self-dual Yang Mills equations [9,10] and has been checked for well-known examples such as Korteweg-de Vries, sine-Gordon, nonlinear Schrödinger, and the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy [11]. First we demonstrate how the nonlinear integrable AKNS hierarchy arises as the field equations of general relativity in three dimensions with negative cosmological constant.

The auxiliary functions ΩÆ and ωÆ are defined as ΩÆ

Pn δHn δp