Abstract
We consider the steady Swift–Hohenberg partial differential equation, a one-parameter family of PDEs on the plane that models, for example, Rayleigh–Bénard convection. For values of the parameter near its critical value, we look for small solutions, quasiperiodic in all directions of the plane, and which are invariant under rotations of angle \({\pi/q, q \geqq 4}\). We solve an unusual small divisor problem and prove the existence of solutions for small parameter values, then address their stability with respect to quasi-periodic perturbations.
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Communicated by P. Rabinowitz
This work was partially supported by ANR grant ANR-10-BLAN0102.
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Braaksma, B., Iooss, G. & Stolovitch, L. Existence of Quasipattern Solutions of the Swift–Hohenberg Equation. Arch Rational Mech Anal 209, 255–285 (2013). https://doi.org/10.1007/s00205-013-0627-7
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DOI: https://doi.org/10.1007/s00205-013-0627-7