A reducibility result for a class of linear wave equations on\n mathbbT^d

2017/02/22 by Riccardo Montalto, Montalto, Riccardo · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #35L10 #37K55 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods for differential equations #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1702.06880

openalex publication_date 2017/02/22 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

We prove a reducibility result for a class of quasi-periodically forced\nlinear wave equations on the d-dimensional torus mathbbT^d of the form\n
partial_tt v -
Delta v +
varepsilon
cal P(
omega t)[v] = 0 where\nthe perturbation cal P(\ω t) is a second order operator of the form\n cal P(\ω t) = - a(\ω t) \Δ - cal R(\ω t), the frequency\n\ω \∈ cal R^\ν is in some Borel set of large Lebesgue measure, the\nfunction a : mathbbT^\ν \→ cal R (independent of the space variable)\nis sufficiently smooth and cal R(\ω t) is a time-dependent finite rank\noperator. This is the first reducibility result for linear wave equations with\nunbounded perturbations on the higher dimensional torus mathbbT^d. As a\ncorollary, we get that the linearized Kirchhoff equation at a smooth and\nsufficiently small quasi-periodic function is reducible.\n

Citations

• Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory

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• Reducibility, Lyapunov exponent, pure point spectra property for quasi-periodic wave operator

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