# American Institute of Mathematical Sciences

January  2020, 40(1): 1-32. doi: 10.3934/dcds.2020001

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

 1 Institute of Physics, Jagiellonian University, Kraków, Poland 2 Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1, Canada

Received  August 2018 Published  October 2019

Fund Project: This research was supported by the Polish National Science Centre grant no. 2017/26/A/ST2/00530.

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 bifurcating stationary states: one is a constrained maximizer of energy and the other one is a local constrained minimizer of the energy, where the constraints are due to other conserved quantities of the resonant system. For the second eigenmode, we obtain two local constrained minimizers of the energy, which are also orbitally stable in the time evolution. All other bifurcating states are saddle points of energy under these constraints and their stability in the time evolution is unknown.

Citation: Piotr Bizoń, Dominika Hunik-Kostyra, Dmitry Pelinovsky. Stationary states of the cubic conformal flow on $\mathbb{S}^3$. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 1-32. doi: 10.3934/dcds.2020001
##### References:
 [1] A. Biasi, P. Bizoń, B. Craps and O. Evnin, Exact lowest-Landau-level solutions for vortex precession in Bose-Einstein condensates, Phys. Rev. A, 96 (2017), 053615. [2] A. Biasi, P. Bizoń and O. Evnin, Solvable cubic resonant systems, Commun. Math. Phys., 369 (2019), 433-456.  doi: 10.1007/s00220-019-03365-z. [3] P. Bizoń, B. Craps, O. Evnin, D. Hunik, V. Luyten and M. Maliborski, Conformal flow on $\mathbb{S}^3$ and weak field integrability in $AdS_4$, Commun. Math. Phys., 353 (2017), 1179-1199.  doi: 10.1007/s00220-017-2896-8. [4] P. Bizoń, D. Hunik–Kostyra and D. Pelinovsky, Ground state of the conformal flow on $\mathbb{S}^3$, Commun. Pure Appl. Math., 72 (2019), 1123-1151.  doi: 10.1002/cpa.21815. [5] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Undergraduate Texts in Mathematics, 251, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4613-8159-4. [6] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2. [7] P. Gérard, P. Germain and L. Thomann, On the cubic lowest Landau level equation, Arch. Rat. Mech. Appl., 231 (2019), 1073-1128.  doi: 10.1007/s00205-018-1295-4. [8] P. Gérard and S. Grellier, The cubic Szegő equation, Ann. Scient. Éc. Norm. Sup., 43 (2010), 761-810.  doi: 10.24033/asens.2133. [9] P. Gérard and S. Grellier, Invariant tori for the cubic Szegő equation, Invent. Math., 187 (2012), 707-754.  doi: 10.1007/s00222-011-0342-7. [10] P. Gérard and E. Lenzmann, A Lax pair structure for the half-wave maps equation, Lett. Math. Phys., 108 (2018), 1635-1648.  doi: 10.1007/s11005-017-1044-x. [11] P. Germain, Z. Hani and L. Thomann, On the continuous resonant equation for NLS: I. Deterministic analysis, J. Math. Pur. App., 105 (2016), 131-163.  doi: 10.1016/j.matpur.2015.10.002. [12] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. Ⅱ, J. Funct. Anal., 94 (1990), 308-348.  doi: 10.1016/0022-1236(90)90016-E. [13] E. Lenzmann and A. Schikorra, On energy-critical half-wave maps to $\mathcal{S}^2$, Invent. Math., 213 (2018), 1-82.  doi: 10.1007/s00222-018-0785-1. [14] D. E. Pelinovsky, Localization in Periodic Potentials: From Schrödinger Operators to the Gross-Pitaevskii Equation, LMS Lecture Note Series, 390, Cambridge University Press, Cambridge, 2011.  doi: 10.1017/CBO9780511997754. [15] O. Pocovnicu, Traveling waves for the cubic Szegő equation on the real line, Anal. PDE, 4 (2011), 379-404.  doi: 10.2140/apde.2011.4.379. [16] J. Thirouin, Classification of traveling waves for a quadratic Szegő equation, Discr. Cont. Dynam. Syst., 39 (2019), 3099-3122.  doi: 10.3934/dcds.2019128.

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##### References:
 [1] A. Biasi, P. Bizoń, B. Craps and O. Evnin, Exact lowest-Landau-level solutions for vortex precession in Bose-Einstein condensates, Phys. Rev. A, 96 (2017), 053615. [2] A. Biasi, P. Bizoń and O. Evnin, Solvable cubic resonant systems, Commun. Math. Phys., 369 (2019), 433-456.  doi: 10.1007/s00220-019-03365-z. [3] P. Bizoń, B. Craps, O. Evnin, D. Hunik, V. Luyten and M. Maliborski, Conformal flow on $\mathbb{S}^3$ and weak field integrability in $AdS_4$, Commun. Math. Phys., 353 (2017), 1179-1199.  doi: 10.1007/s00220-017-2896-8. [4] P. Bizoń, D. Hunik–Kostyra and D. Pelinovsky, Ground state of the conformal flow on $\mathbb{S}^3$, Commun. Pure Appl. Math., 72 (2019), 1123-1151.  doi: 10.1002/cpa.21815. [5] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Undergraduate Texts in Mathematics, 251, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4613-8159-4. [6] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2. [7] P. Gérard, P. Germain and L. Thomann, On the cubic lowest Landau level equation, Arch. Rat. Mech. Appl., 231 (2019), 1073-1128.  doi: 10.1007/s00205-018-1295-4. [8] P. Gérard and S. Grellier, The cubic Szegő equation, Ann. Scient. Éc. Norm. Sup., 43 (2010), 761-810.  doi: 10.24033/asens.2133. [9] P. Gérard and S. Grellier, Invariant tori for the cubic Szegő equation, Invent. Math., 187 (2012), 707-754.  doi: 10.1007/s00222-011-0342-7. [10] P. Gérard and E. Lenzmann, A Lax pair structure for the half-wave maps equation, Lett. Math. Phys., 108 (2018), 1635-1648.  doi: 10.1007/s11005-017-1044-x. [11] P. Germain, Z. Hani and L. Thomann, On the continuous resonant equation for NLS: I. Deterministic analysis, J. Math. Pur. App., 105 (2016), 131-163.  doi: 10.1016/j.matpur.2015.10.002. [12] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. Ⅱ, J. Funct. Anal., 94 (1990), 308-348.  doi: 10.1016/0022-1236(90)90016-E. [13] E. Lenzmann and A. Schikorra, On energy-critical half-wave maps to $\mathcal{S}^2$, Invent. Math., 213 (2018), 1-82.  doi: 10.1007/s00222-018-0785-1. [14] D. E. Pelinovsky, Localization in Periodic Potentials: From Schrödinger Operators to the Gross-Pitaevskii Equation, LMS Lecture Note Series, 390, Cambridge University Press, Cambridge, 2011.  doi: 10.1017/CBO9780511997754. [15] O. Pocovnicu, Traveling waves for the cubic Szegő equation on the real line, Anal. PDE, 4 (2011), 379-404.  doi: 10.2140/apde.2011.4.379. [16] J. Thirouin, Classification of traveling waves for a quadratic Szegő equation, Discr. Cont. Dynam. Syst., 39 (2019), 3099-3122.  doi: 10.3934/dcds.2019128.
The smallest eigenvalues of $L_+$ (left) and $L_-$ (right) for the stationary state (1.4) for the upper sign normalized by $\lambda = 1$
The smallest eigenvalues of $L_+$ (left) and $L_-$ (right) for the stationary state (1.4) for the lower sign normalized by $\lambda-\omega = 1$
The smallest eigenvalues of $L_+$ (left) and $L_-$ (right) for the branch bifurcating from the second eigenmode at $\omega_6 = 3/35$ with normalization $\lambda-\omega = 1$
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