Resonant Hamiltonian systems and weakly nonlinear dynamics in AdS spacetimes

TL;DR

This paper reviews weakly nonlinear dynamics in AdS spacetimes, focusing on resonant systems, turbulence, and connections to quantum chaos, to understand the AdS instability conjecture and long-term evolution of small perturbations.

Contribution

It provides a comprehensive overview of resonant Hamiltonian systems in AdS and their relation to quantum chaos, highlighting new insights into nonlinear dynamics and stability.

Findings

Resonant approximation captures long-term evolution of small initial data.

Connections established between AdS dynamics and nonlinear Schrödinger equations.

Insights into quantum chaos influence on classical AdS dynamics.

Abstract

Weakly nonlinear dynamics in anti-de Sitter (AdS) spacetimes is reviewed, keeping an eye on the AdS instability conjecture and focusing on the resonant approximation that accurately captures in a simplified form the long-term evolution of small initial data. Topics covered include turbulent and regular motion, dynamical recurrences analogous to the Fermi-Pasta-Ulam phenomena in oscillator chains, and relations between AdS dynamics and nonrelativistic nonlinear Schrodinger equations in harmonic potentials. Special mention is given to the way the classical dynamics of weakly nonlinear strongly resonant systems is illuminated by perturbative considerations within the corresponding quantum theories, in particular, in relation to quantum chaos theory.