• Corpus ID: 119617550

Stationary states of the cubic conformal flow on $\mathbb{S}^3$

@article{Bizo2018StationarySO,
  title={Stationary states of the cubic conformal flow on \$\mathbb\{S\}^3\$},
  author={Piotr Bizoń and Dominika Hunik-Kostyra and Dmitry E. Pelinovsky},
  journal={arXiv: Mathematical Physics},
  year={2018}
}
We consider the resonant system of amplitude equations for the conformally invariant cubic wave equation on the three-sphere. Using the local bifurcation theory, we characterize all stationary states that bifurcate from the first two eigenmodes. Thanks to the variational formulation of the resonant system and energy conservation, we also determine variational characterization and stability of the bifurcating states. For the lowest eigenmode, we obtain two orbitally stable families of the… 
6 Citations

Figures from this paper

Complex plane representations and stationary states in cubic and quintic resonant systems

Weakly nonlinear energy transfer between normal modes of strongly resonant PDEs is captured by the corresponding effective resonant systems. In a previous article, we have constructed a large class

Solvable Cubic Resonant Systems

Weakly nonlinear analysis of resonant PDEs in recent literature has generated a number of resonant systems for slow evolution of the normal mode amplitudes that possess remarkable properties. Despite

Breathing Modes, Quartic Nonlinearities and Effective Resonant Systems

  • O. Evnin
  • Physics
    Symmetry, Integrability and Geometry: Methods and Applications
  • 2020
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 breathing modes are familiar from

Quantum resonant systems, integrable and chaotic

Resonant systems emerge as weakly nonlinear approximations to problems with highly resonant linearized perturbations. Examples include nonlinear Schrödinger equations in harmonic potentials and

Energy returns in global AdS4

Recent studies of the weakly nonlinear dynamics of probe fields in global AdS$_4$ (and of the nonrelativistic limit of AdS resulting in the Gross-Pitaevskii equation) have revealed a number of cases

Melonic Turbulence

We propose a new application of random tensor theory to studies of non-linear random flows in many variables. Our focus is on non-linear resonant systems which often emerge as weakly non-linear

References

SHOWING 1-10 OF 16 REFERENCES

Conformal Flow on S3 and Weak Field Integrability in AdS4

We consider the conformally invariant cubic wave equation on the Einstein cylinder $${\mathbb{R} \times \mathbb{S}^3}$$R×S3 for small rotationally symmetric initial data. This simple equation

Solvable Cubic Resonant Systems

Weakly nonlinear analysis of resonant PDEs in recent literature has generated a number of resonant systems for slow evolution of the normal mode amplitudes that possess remarkable properties. Despite

Invariant tori for the cubic Szegö equation

AbstractWe continue the study of the following Hamiltonian equation on the Hardy space of the circle, $$i\partial_tu=\Pi(|u|^2u),$$ where Π denotes the Szegö projector. This equation can be seen as

On the Cubic Lowest Landau Level Equation

We study dynamical properties of the cubic lowest Landau level equation, which is used in the modeling of fast rotating Bose–Einstein condensates. We obtain bounds on the decay of general stationary

Traveling waves for the cubic Szego equation on the real line

We consider the cubic Szego equation i u_t=Pi(|u|^2u) on the real line, with solutions in the Hardy space on the upper half-plane, where Pi is the Szego projector onto the non-negative frequencies.

Exact lowest-Landau-level solutions for vortex precession in Bose-Einstein condensates

The Lowest Landau Level (LLL) equation emerges as an accurate approximation for a class of dynamical regimes of Bose-Einstein Condensates (BEC) in two-dimensional isotropic harmonic traps in the

Localization in Periodic Potentials: From Schrodinger Operators to the Gross-Pitaevskii Equation

Preface 1. Formalism of the nonlinear Schrodinger equations 2. Justification of the nonlinear Schrodinger equations 3. Existence of localized modes in periodic potentials 4. Stability of localized

On the continuous resonant equation for NLS II: Statistical study

We consider the continuous resonant (CR) system of the 2D cubic nonlinear Schrodinger (NLS) equation. This system arises in numerous instances as an effective equation for the long-time dynamics of

Bifurcation from simple eigenvalues