On quasi-periodic solutions for a generalized Boussinesq equation

Shi, Yuguang; Xu, Ji‐Qing; Xu, Xindong

doi:10.1016/j.na.2014.04.007

Cited by 14 publications

(5 citation statements)

References 29 publications

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“…To complete one KAM step, we need to prove the new perturbation P + still has the special form defined in (A5), i.e., P + ∈ A. Recall its definition (24) and it could be rewritten as…”

Section: Verification Of Condition (A5) After Transformationmentioning

confidence: 99%

“…The infinite dimensional KAM theory is the extension of classical KAM theory, its advantage is the construction of a local normal form in a neighborhood of the obtained solutions in addition to the existence of quasiperiodic solutions, the normal form method is used to understand the dynamics of the corresponding quasi-periodic solutions. Both the CWB method and the infinite dimensional KAM theory have been well developed for one dimensional Hamiltonian PDEs, see [1,13,14,19,20,22,24,25,26,28,29,30] and the references therein.…”

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confidence: 99%

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Quasi-periodic solutions for a class of beam equation system

Shi¹,

Xu²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper, we establish an abstract infinite dimensional KAM theorem. As an application, we use the theorem to study the higher dimensional beam equation system    u 1tt + ∆ 2 u 1 + σu 1 + u 1 u 2 2 = 0 u 2tt + ∆ 2 u 2 + µu 2 + u 2 1 u 2 = 0 under periodic boundary conditions, where 0 < σ ∈ [σ 1 , σ 2 ], 0 < µ ∈ [µ 1 , µ 2 ] are real parameters. By establishing a block-diagonal normal form, we obtain the existence of a Whitney smooth family of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamic system.

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“…To complete one KAM step, we need to prove the new perturbation P + still has the special form defined in (A5), i.e., P + ∈ A. Recall its definition (24) and it could be rewritten as…”

Section: Verification Of Condition (A5) After Transformationmentioning

confidence: 99%

mentioning

confidence: 99%

Quasi-periodic solutions for a class of beam equation system

Shi¹,

Xu²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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“…This important information reminds us of KAM theory for infinite dimensional Hamiltonian system. Under hinged boundary conditions, the authors [24,25] obtained the existence of small amplitude quasi-periodic solutions for f (u) = u 3 and f (u) = u 5 .…”

Section: Yanling Shi Junxiang Xu and Xindong Xumentioning

confidence: 99%

“…For the details, see the appendix in [24]. Thus, the corresponding symplectic structure is the standard one i j≥1 dq −j ∧ dq j .…”

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confidence: 99%

Quasi-periodic solutions of generalized Boussinesq equation with quasi-periodic forcing

Shi¹,

Xu²,

Xu³

2017

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper, one-dimensional quasi-periodically forced generalized Boussinesq equation utt − uxx + uxxxx + εφ(t)(u + u 3)xx = 0 with hinged boundary conditions is considered, where ε is a small positive parameter, φ(t) is a real analytic quasi-periodic function in t with frequency vector ω = (ω 1 , ω 2 , • • • , ωm). It is proved that, under a suitable hypothesis on φ(t), there are many quasi-periodic solutions for the above equation via KAM theory.

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