Resonant Hamiltonian systems and weakly nonlinear dynamics in AdS spacetimes

Evnin, Oleg

doi:10.48550/arxiv.2104.09797

Cited by 3 publications

(6 citation statements)

References 127 publications

(438 reference statements)

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“…These are precisely the classical equations of motion corresponding to the resonant Hamiltonian (5.1). Similar derivations hold for a variety of other physical equations of motion in different numbers of dimensions taken as the starting point [68]. The key ingredients are weak quartic nonlinearities and a spatial domain with a discrete highly resonant spectrum of normal mode frequencies.…”

Section: Resonant Systems As Weakly Nonlinear Approximationsmentioning

confidence: 73%

“…Quantum resonant systems (and their classical limits) arise as consistent weakly nonlinear approximations to a variety of physical problems featuring weak nonlinearities in strongly resonant domains. A recent review can be found in [68]. While their Hilbert spaces are infinite-dimensional, the Hamiltonians are block-diagonal in the Fock basis, with all blocks of finite sizes (although there is no bound on the size).…”

Section: Quantum Resonant Systemsmentioning

confidence: 99%

“…Hamiltonians of the form (5.1) naturally emerge in the description of weakly nonlinear dynamics with weak quartic interactions in strongly resonant bounded domains (a review can be found in [68]). We first start with the classical version of the story, and as a very simple particular case consider the nonlinear Schrödinger equation in one dimension with a harmonic potential:…”

Section: Resonant Systems As Weakly Nonlinear Approximationsmentioning

confidence: 99%

“…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,[64][65][66], as well as generic chaotic dynamics expected from a nonlinear system with an infinite number of degrees of freedom when the mode couplings C nmkl are chosen randomly. A recent review can be found in [68]. In contrast to the rich array of classical dynamical behaviors, the corresponding quantum theory is very economical in its structure and can be explored via an operation as simple as diagonalizing finite-sized numerical matrices [37].…”

Section: Introductionmentioning

confidence: 99%

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Bounds on quantum evolution complexity via lattice cryptography

Craps¹,

Clerck²,

Evnin³

et al. 2022

Preprint

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We address the difference between integrable and chaotic motion in quantum theory as manifested by the complexity of the corresponding evolution operators. Complexity is understood here as the shortest geodesic distance between the time-dependent evolution operator and the origin within the group of unitaries. (An appropriate 'complexity metric' must be used that takes into account the relative difficulty of performing 'nonlocal' operations that act on many degrees of freedom at once.) While simply formulated and geometrically attractive, this notion of complexity is numerically intractable save for toy models with Hilbert spaces of very low dimensions. To bypass this difficulty, we trade the exact definition in terms of geodesics for an upper bound on complexity, obtained by minimizing the distance over an explicitly prescribed infinite set of curves, rather than over all possible curves. Identifying this upper bound turns out equivalent to the closest vector problem (CVP) previously studied in integer optimization theory, in particular, in relation to lattice-based cryptography. Effective approximate algorithms are hence provided by the existing mathematical considerations, and they can be utilized in our analysis of the upper bounds on quantum evolution complexity. The resulting algorithmically implemented complexity bound systematically assigns lower values to integrable than to chaotic systems, as we demonstrate by explicit numerical work for Hilbert spaces of dimensions up to ∼ 10 4 .

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Section: Resonant Systems As Weakly Nonlinear Approximationsmentioning

confidence: 73%

Section: Quantum Resonant Systemsmentioning

confidence: 99%

Section: Resonant Systems As Weakly Nonlinear Approximationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Bounds on quantum evolution complexity via lattice cryptography

Craps¹,

Clerck²,

Evnin³

et al. 2022

Preprint

Self Cite

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show abstract

“…When the spherical symmetry assumption is removed [3,22], we use an ansatz based on Hopf coordinates 2 (η, ξ 1 , ξ 2 ) ∈ 0, π 2 × [0, 2π) × [0, 2π) rather than the standard spherical coordinates. The Laplace-Beltrami operator on S 3 in these coordinates reads as…”

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confidence: 99%

Non-linear periodic waves on the Einstein cylinder

Chatzikaleas¹,

Smulevici²

2022

Preprint

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Motivated by the study of small amplitudes non-linear waves in the Anti-de-Sitter spacetime and in particular the conjectured existence of periodic in time solutions to the Einstein equations, we construct families of arbitrary small time-periodic solutions to the conformal cubic wave equation and the spherically-symmetric Yang-Mills equations on the Einstein cylinder R × S 3 . For the conformal cubic wave equation, we consider both spherically-symmetric solutions and complexed-valued aspherical solutions with an ansatz relying on the Hopf fibration of the 3-sphere. In all three cases, the equations reduce to 1+1 semi-linear wave equations. Our proof relies on a theorem of Bambusi-Paleari for which the main assumption is the existence of a seed solution, given by a non-degenerate zero of a non-linear operator associated with the resonant system. For the problems that we consider, such seed solutions are simply given by the mode solutions of the linearized equations. Provided that the Fourier coefficients of the systems can be computed, the non-degeneracy conditions then amount to solving infinite dimensional linear systems. Since the eigenfunctions for all three cases studied are given by Jacobi polynomials, we derive the different Fourier and resonant systems using linearization and connection formulas as well as integral transformation of Jacobi polynomials. In the Yang-Mills case, the original version of the theorem of Bambusi-Paleari is not applicable because the non-linearity of smallest degree is nonresonant. The resonant terms are then provided by the next order non-linear terms with an extra correction due to backreaction terms of the smallest degree non-linearity and we prove an analogous theorem in this setting. Finally, we emphasize that, in view of the KAM type method used, the time periodic solutions that we construct exist only when the small perturbative parameter belongs to a Cantor-like set.

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Fermi-Pasta-Ulam phenomena and persistent breathers in the harmonic trap

Biasi,

Evnin,

Malomed

2021

Preprint

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We consider the long-term weakly nonlinear evolution governed by the two-dimensional nonlinear Schrödinger (NLS) equation with an isotropic harmonic oscillator potential. The dynamics in this regime is dominated by resonant interactions between quartets of linear normal modes, accurately captured by the corresponding resonant Hamiltonian system. In the framework of this system, we identify Fermi-Pasta-Ulam-like recurrence phenomena, whereby the normal-mode spectrum passes in close proximity of the initial configuration, and two-mode states with time-independent mode amplitude spectra that translate into long-lived breathers of the original NLS equation. We comment on possible implications of these findings for nonlinear optics and matter-wave dynamics in Bose-Einstein condensates.

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Resonant Hamiltonian systems and weakly nonlinear dynamics in AdS spacetimes

Cited by 3 publications

References 127 publications

Bounds on quantum evolution complexity via lattice cryptography

Bounds on quantum evolution complexity via lattice cryptography

Non-linear periodic waves on the Einstein cylinder

Fermi-Pasta-Ulam phenomena and persistent breathers in the harmonic trap

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