Nonlinear oscillations of strings and beams

2025/08/26 by Ficek, Filip, Maliborski, Maciej · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Pattern Formation and Solitons (nlin.PS)

paper · doi:10.48550/arxiv.2508.19091

Abstract

We investigate the time-periodic solutions to the nonlinear wave and beam equations and uncover their intricate, fractal-like structure. In particular, we identify a new class of large-energy solutions with complex mode compositions and propose a systematic framework for their analysis. A Floquet stability study reveals that this class contains solutions that are linearly stable, suggesting that they may play a significant role in the nonlinear dynamics of the systems.

Citations

• New class of time-periodic solutions to the 1D cubic wave equation
• Complex structure of time-periodic solutions decoded in Poincaré-Lindstedt series: the cubic conformal wave equation on \mathbbS³
• Trees, trunks, and branches -- bifurcation structure of time-periodic solutions to u_tt-u_xx± u³=0
• Periodic Solutions for the 1d Cubic Wave Equation with Dirichlet Boundary Conditions
• KAM for the nonlinear beam equation 1: small-amplitude solutions

Cited by

• Time-periodic solutions to the cubic wave equation: an elementary constructive approach

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