On the continuous resonant equation for NLS: I. Deterministic analysis

Germain, Pierre; Hani, Zaher; Thomann, Laurent

doi:10.48550/arxiv.1501.03760

Cited by 20 publications

(93 citation statements)

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“…We note that the Hamiltonian ( 17) could be studied in its own right as a quartic dynamical system. Classical analogs of this quantum resonant system, with various different choices of the interaction coefficients C (including the concrete problem we are considering here as a special case), have often surfaced in recent literature: resonant systems of gravitational AdS perturbations [8][9][10][11][12][13] formulated in relation to the conjectured AdS instability [14,15], resonant systems of nonlinear wave equations in AdS [16][17][18][19] and the related Gross-Pitaevskii equation for Bose-Einstein condensates [20][21][22][23], a solvable model of turbulence in the form of a specific resonant system called the cubic Szegő equation [24], as well as studies of a large class of partially solvable resonant systems [25]. Quantum resonant systems, on the other hand, have been introduced and studied in [7] from a perspective geared toward quantum chaos theory.…”

Section: Quantum Fields In Adsmentioning

confidence: 99%

Spectroscopy instead of scattering: particle experimentation in AdS spacetime

Evnin

2018

Preprint

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Particle experiments are difficult at weak coupling because interactions are rare and a huge number of collision attempts are needed to attain significant precision. One often hears that 'one Higgs boson is produced in a billion of collisions at LHC.' In this essay, we fantasize about possible advantages afforded in this regard by performing experiments in anti-de Sitter (AdS) spacetime instead of a usual collider in a nearly-flat spacetime. Being a perfectly resonant cavity, the AdS spacetime enhances all nonlinear interactions, which therefore produce effects of order one no matter how small the couplings are, provided that one waits long enough. These effects are encoded in spectroscopic data, namely, the fine splittings of the energy levels which would have been highly degenerate in AdS if no interactions were present. Over long times, such energy shifts let different components of wavefunctions drift completely out-of-phase, producing large effects for arbitrarily small interactions.

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Section: Quantum Fields In Adsmentioning

confidence: 99%

Spectroscopy instead of scattering: particle experimentation in AdS spacetime

Evnin

2018

Preprint

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“…(10) The classical version of this resonant Hamiltonian is frequently encountered in studies of weakly nonlinear long-term dynamics of resonant PDEs, see, for instance, [18] for applications to the nonlinear Schrödiger equation in a harmonic potential, the classical version of (1).…”

Section: A General Energy Levelsmentioning

confidence: 99%

“…The classical dynamics corresponding to (10) can be consistently truncated to the set of modes with the maximal amount of rotation for a given energy, namely, the modes in (3) satisfying n = m. Such a truncation has appeared in the literature under the name of the Lowest Landau Level (LLL) equation [18][19][20]. One does not in general expect that consistent classical truncations have direct implications in the quantum theory, since quantum variables cannot be simply set to zero.…”

Section: B the Lll Truncationmentioning

confidence: 99%

“…Indeed, the splitting of the energy levels at linear order in the coupling param-eter is governed by the resonant LLL Hamiltonian [12]. The classical dynamics of this Hamiltonian has been investigated in a few works [18][19][20], driven by rather different motivations. Explicit classical solutions are known for the LLL Hamiltonian, and these known solutions have mode amplitude spectra exactly periodic in time [19].…”

Section: Introductionmentioning

confidence: 99%

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Time-periodic quantum states of weakly interacting bosons in a harmonic trap

De Clerck,

Evnin

2020

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We consider identical quantum bosons with weak contact interactions in a two-dimensional isotropic harmonic trap, and focus on states at the Lowest Landau Level (LLL). At linear order in the coupling parameter g, we exploit the rich algebraic structure of the problem to give an explicit construction of a large family of quantum states with energies of the form E0 + gE1/4 + O(g 2 ), where E0 and E1 are integers. As a result, any superposition of these states evolves periodically with a period of at most 8π/g until, at much longer time scales of order 1/g 2 , corrections to the energies of order g 2 become important and may upset this perfectly periodic behavior. We further construct coherent-like combinations of these states that naturally connect to classical dynamics in an appropriate regime, and explain how our findings relate to the known time-periodic features of the corresponding weakly nonlinear classical theory. We briefly comment on possible generalizations of our analysis to other numbers of spatial dimensions and other analogous physical systems.

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“…Thus, it was shown that the cubic Szegő equation accurately describes the weakly nonlinear long-time dynamics of some sectors of the half-wave equation [15,16] and the wave guide Schrödinger equation [21]. In closer contact with physics applications, resonant systems arise as approximations for the dynamics of Bose-Einstein condensates [22][23][24][25][26] and in Anti-de Sitter (AdS) spacetimes [27][28][29][30][31][32][33], the latter topic extensively studied in relation to AdS instability [13,14]. Resonant systems constructed in this way often show powerful analytic structures and admit special solutions [34,35], even in the absence of Lax-integrability that characterizes the cubic Szegő equation and other systems that we focus on here.…”

Section: Introductionmentioning

confidence: 99%

Turbulent cascades in a truncation of the cubic Szego equation and related systems

Biasi,

Evnin

2020

Preprint

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The cubic Szegő equation has been studied as an integrable model for deterministic turbulence, starting with the foundational work of Gérard and Grellier. We introduce a truncated version of this equation, wherein a majority of the Fourier mode couplings are eliminated while the signature features of the model are preserved, namely, a Lax-pair structure and a nested hierarchy of finite-dimensional dynamically invariant manifolds. Despite the impoverished structure of the interactions, the turbulent behaviors of our new equation are stronger in an appropriate sense than for the original cubic Szegő equation. We construct explicit analytic solutions displaying exponential growth of Sobolev norms. We furthermore introduce a family of models that interpolate between our truncated system and the original cubic Szegő equation, along with a few other related deformations. All of these models possess Lax pairs, invariant manifolds, and display a variety of turbulent cascades. We additionally mention numerical evidence that shows an even stronger type of turbulence in the form of a finite-time blow-up in some different, closely related dynamical systems.

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On the continuous resonant equation for NLS: I. Deterministic analysis

Cited by 20 publications

References 0 publications

Spectroscopy instead of scattering: particle experimentation in AdS spacetime

Spectroscopy instead of scattering: particle experimentation in AdS spacetime

Time-periodic quantum states of weakly interacting bosons in a harmonic trap

Turbulent cascades in a truncation of the cubic Szego equation and related systems

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