Breathing Modes, Quartic Nonlinearities and Effective Resonant Systems

Abstract

A breathing mode in a Hamiltonian system is a function on the phase space whose evolution is exactly periodic for all solutions of the equations of motion. Such observables are familiar from nonlinear dynamics in harmonic traps or Anti-de Sitter spacetimes, with applications to the physics of cold atomic gases, general relativity and high-energy physics. We discuss the implications of breathing modes in weakly nonlinear regimes, assuming that both the Hamiltonian and the breathing mode are linear functions of the coupling parameter, taken to be small. For a linear system, breathing modes dictate resonant relations between the normal frequencies. These resonant relations imply that arbitrarily small nonlinearities may produce large effects over long times. The leading effects of the nonlinearities in this regime are captured by the corresponding effective resonant system. The breathing mode of the original system translates into an exactly conserved quantity of this effective resonant system under simple assumptions that we explicitly specify. If the nonlinearity in the Hamiltonian is quartic in the canonical variables, as is common in many physically motivated cases, further consequences result from the presence of the breathing modes, and some nontrivial explicit solutions of the effective resonant system can be constructed. This structure explains in a uniform fashion a series of results in the recent literature where this type of dynamics is realized in specific Hamiltonian systems, and predicts other situations of interest where it should emerge.

Time-periodicities in holographic CFTs

“…In practice, we shall focus on a non-gravitating φ 4 scalar within the maximally rotating sector (states that have the maximal possible amount of angular momentum for a given energy). We shall explain, following [31], that similar patterns should be expected in maximally rotating sectors of gravitating systems, but recovering them explicitly would require substantial technical work beyond the scope of our treatment. As outlined already in [41], the problem of finding these energy shifts can be reduced to diagonalizing a specific quantum resonant system [42], whose Hamiltonian is a quartic combination of creation-annihilation operators.…”

“…One may also take a nonrelativistic limit of gravitating systems in AdS, obtaining a Hartree equation in a harmonic potential, which also displays some perfect periodic returns [29]. The underlying mathematical structure responsible for these weakly nonlinear periodic behaviors, common for AdS fields and their nonrelativistic harmonically trapped limits, has been made manifest in [30,31].…”

“…Conservation of Z can be traced back [31] to the fact that the center-of-mass of any system in AdS performs simple perfectly periodic motions irrespectively of the complexity of the dynamics of other degrees of freedom (just as the center-of-mass in Minkowski space moves with a constant velocity). Conservation of Z hints at solvable features in (2.36) that we shall explore below, while the fact that it comes from something as generic as the center-ofmass motion in AdS makes one expect that similar features would be seen in other, more complicated systems, and not just for the self-interacting probe scalar (2.18).…”

“…Conservation of Z hints at solvable features in (2.36) that we shall explore below, while the fact that it comes from something as generic as the center-ofmass motion in AdS makes one expect that similar features would be seen in other, more complicated systems, and not just for the self-interacting probe scalar (2.18). While the center-of-mass motion in AdS may seem a triviality, it imposes powerful relations between the mode couplings in the normal mode basis [31] and leads to solvable features in the corresponding resonant systems.…”

“…One might wonder how the systems in the class constructed in [30], with their timeperiodic behaviors, could arise as resonant approximations to Hamiltonian PDEs of physical interest. This question has been addressed in [31]. The crucial ingredient for the construction of the time-periodic solutions of the resonant system [30] is conservation of the quantity Z defined by (2.39).…”