Integrability Lost

TL;DR

This paper investigates classical string dynamics in a confining AdS soliton background, demonstrating that unlike in pure AdS_5×S^5, the system is non-integrable and exhibits chaos, confirmed by numerical analysis and Lyapunov exponents.

Contribution

It provides the first detailed analysis showing classical integrability is lost in a more realistic confining background, contrasting with the integrable pure AdS case.

Findings

Numerical evidence of chaotic behavior in string dynamics

Loss of invariant phase space foliation due to chaos

Positive Lyapunov exponent confirming chaos

Abstract

It is known that classical string dynamics in pure AdS_5\times S^5 is integrable and plays an important role in solvability. This is a deep and central issue in holography. Here we investigate similar classical integrability for a more realistic confining background and provide a negative answer. The dynamics of a class of simple string configurations in AdS soliton background can be mapped to the dynamics of a set of non-linearly coupled oscillators. In a suitable limit of small fluctuations we discuss a quasi-periodic analytic solution of the system. However numerics indicates chaotic behavior as the fluctuations are not small. Integrability implies the existence of a regular foliation of the phase space by invariant manifolds. Our numerics shows how this nice foliation structure is eventually lost due to chaotic motion. We also verify a positive Lyapunov index for chaotic orbits. Our dynamics is roughly similar to other known non-integrable coupled oscillators systems like Henon-Heiles equations.