Theoretical framework demonstrates linear stability of quasi-periodic solutions in one-dimensional Kirchhoff equations, indicating advances in understanding of eigenvalues.
We prove an abstract infinite dimensional KAM theorem, which could be applied to prove the existence and linear stability of small-amplitude quasi-periodic solutions for one dimensional forced Kirchhoff equations with periodic boundary conditions \[ uₜₜ-(1+∫₀2π |u_x|^2 dx)uₓₓ+ M_ξu+εg(ωt,x) =0, u(t,x+2π)=u(t,x),\] where M_ξ is a real Fourier multiplier, g(ωt,x) is real analytic with forced Diophantine frequencies ω, $ε$ is a small parameter. The paper generalizes the previous results from the simple eigenvalue to the double eigenvalues under the quasi-linear perturbation.
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Chen et al. (2025) studied this question.
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