Examining Instabilities Due to Driven Scalars in AdS

Abstract

We extend the study of the non-linear perturbative theory of weakly turbulent energy cascades in AdS d+1 to include solutions of driven systems, i.e. those with time-dependent sources on the AdS boundary. This necessitates the activation of non-normalizable modes in the linear solution for the massive bulk scalar field, which couple to the metric and normalizable scalar modes. We determine analytic expressions for secular terms in the renormalization flow equations for any mass, and for various driving functions. Finally, we numerically evaluate these sources for d = 4 and discuss what role these driven solutions play in the perturbative stability of AdS.

Bounds on quantum evolution complexity via lattice cryptography

“…It is known from extensive literature [40][41][42][43][44][45][46][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66] that, depending on the specific values of C, such systems of equations desplay a striking range of behaviors, some of which are at the forefront of contemporary PDE mathematics. Dynamical features that have been specifically studied are turbulent transfers of energy [64][65][66] including finite-time turbulent blow-up [40-42, 44, 46, 66], dynamical recurrence phenomena [40,45,62], integrability [64][65][66], extra conservation laws and solvability within restricted dynamically invariant manifolds [49,51,52,58,59,67], etc.…”

“…The Hamiltonian (1.1) may appear not-too-familiar to many readers, but, as a matter of fact, the corresponding classical Hamiltonian system frequently arises as a controlled approximation to weakly nonlinear partial differential equations (PDEs) in strongly resonant domains, originating from a number of branches of physics and mathematics. Specifically, classical systems corresponding to (1.1), together with some closely related variations, have been studied in the following contexts: gravitational dynamics in anti-de Sitter (AdS) spacetimes [40][41][42][43][44][45][46] (typically, motivated by the AdS instability conjecture [47,48]); related dynamical problems for classical relativistic fields [49][50][51][52][53][54]; nonrelativistic nonlinear Schrödinger equations describing, among other things, the dynamics of Bose-Einstein condensates in harmonic potentials [55][56][57][58][59][60][61][62][63]; and integrable models for turbulence [64][65][66]. These classical systems display, for different choices of C nmkl , a wide range of analytic and dynamical patterns ranging from full solvability [64] to Lax-integrability [64][65][66], partial solvability [49][50][51][52]58,59,67], turbulent cascades [42,…”

Resonant Hamiltonian systems and weakly nonlinear dynamics in AdS spacetimes

“…Studies of AdS dynamics with periodic driving introduced via the boundary conditions at the conformal boundary [75,76]. In [77], this problem was recast in a language closely reminiscent of the resonant approximations for AdS systems with 'reflective' Dirichlet boundary conditions (the 'zero driving' case that we consider here).…”