For the cubic conformal wave equation on the Einstein cylinder, time-periodic solutions form complex trunk-branch bifurcation networks, and a resummed Poincaré-Lindstedt series is claimed to encode these structures.
Trees, trunks, and branches -- bifurcation structure of time-periodic solutions to $u_{tt}-u_{xx}\pm u^{3}=0$
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We propose a systematic approach to analysing the complex structure of time-periodic solutions to the cubic wave equation on an interval with Dirichlet boundary conditions first reported in arXiv:2407.16507. The analysis we present is based on a detailed study of sparse mode interactions suggested by the previous numerical work. Our results complement prior rigorous existence proofs and suggest that solutions exist for any frequency, however, they may be arbitrarily large.
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Time-periodic solutions of the conformally invariant wave equation on the Einstein cylinder
For the cubic conformal wave equation on the Einstein cylinder, time-periodic solutions form complex trunk-branch bifurcation networks, and a resummed Poincaré-Lindstedt series is claimed to encode these structures.