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J. A. Murdock, Perturbations: Theory and Methods
1987
Earlier work this paper cites.
V. E. Zakharov, V. S. L’vov and G. Falkovich, Kolmogorov spectra of turbulence I: wave turbulence,
1992
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M. Lakshmanan and R. Sahadevan, Painlevé analysis, Lie symmetries and integrability of coupled nonlinear oscillators of polynomial type, Phys. Rep. 224
1993
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A. J. Majda, D. W. McLaughlin, and E. G. Tabak, A one-dimensional model for dispersive wave turbulence , J. Nonlin. Sci. 7
1997
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M. Bruschi and F. Calogero, Integrable systems of quartic oscillators, Phys. Lett. A 273
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J. B. Garnett, Bounded analytic functions,
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J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation
Original
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P. Gérard and S. Grellier, The cubic Szegő equation,
Original
2010
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S. Nazarenko, Wave turbulence,
2011
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P. Bizoń and A. Rostworowski, On weakly turbulent instability of anti-de Sitter space,
Original
2011
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P. Gérard and S. Grellier, Effective integrable dynamics for a certain nonlinear wave equation,
Original
2012
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2016

Conformal flow on $S^3$ and weak field integrability in AdS$_4$

Bizoń, Piotr, Craps, Ben, Evnin, Oleg et al.

Understand

We consider the conformally invariant cubic wave equation on the Einstein cylinder $\mathbb{R} \times \mathbb{S}^3$ for small rotationally symmetric initial data.

This simple equation captures many key challenges of nonlinear wave dynamics in confining geometries, while a conformal transformation relates it to a self-interacting conformally coupled scalar in four-dimensional anti-de Sitter spacetime (AdS$_4$) and connects it to various questions of AdS stability.
We construct an effective infinite-dimensional time-averaged dynamical system accurately approximating the original equation in the weak field regime.
It turns out that this effective system, which we call the conformal flow, exhibits some remarkable features, such as low-dimensional invariant subspaces, a wealth of stationary states (for which energy does not flow between the modes), as well as solutions with nontrivial exactly periodic energy flows.

Built on

J. A. Murdock, Perturbations: Theory and Methods
1987
Earlier work this paper cites.
V. E. Zakharov, V. S. L’vov and G. Falkovich, Kolmogorov spectra of turbulence I: wave turbulence,
1992
Earlier work this paper cites.
M. Lakshmanan and R. Sahadevan, Painlevé analysis, Lie symmetries and integrability of coupled nonlinear oscillators of polynomial type, Phys. Rep. 224
1993
Earlier work this paper cites.
A. J. Majda, D. W. McLaughlin, and E. G. Tabak, A one-dimensional model for dispersive wave turbulence , J. Nonlin. Sci. 7
1997
Earlier work this paper cites.
M. Bruschi and F. Calogero, Integrable systems of quartic oscillators, Phys. Lett. A 273
2004
Earlier work this paper cites.
J. B. Garnett, Bounded analytic functions,
2007
Earlier work this paper cites.
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation
Original
2010
Earlier work this paper cites.
P. Gérard and S. Grellier, The cubic Szegő equation,
Original
2010
Earlier work this paper cites.
S. Nazarenko, Wave turbulence,
2011
Earlier work this paper cites.
P. Bizoń and A. Rostworowski, On weakly turbulent instability of anti-de Sitter space,
Original
2011
Earlier work this paper cites.
P. Gérard and S. Grellier, Effective integrable dynamics for a certain nonlinear wave equation,
Original
2012
Earlier work this paper cites.