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

Abstract

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.

Quasi-periodic solutions for a class of beam equation system

“…Now we solve equations (31)-(33) one by one: Solving (31): by the expansion (25), (31) is turned into i( k, ω + k ,ω )F k100,k000 = P k100,k000 , |k| + |k| = 0, i( k, ω + k ,ω )F k000,k100 = P k000,k100 , |k| + |k| = 0.…”

“…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.…”

Novel dynamics of the Fokas-Lenells model in Birefringent fibers applying different integration algorithms

“…Optical soliton disturbance is one of the most discernible bounds of exploration in the telecommunication industry [1][2][3][4][5][6]. The realm of studying soliton transmission in optical fibers has flourished in the last few decades due to its rich prominence of occurrences explaining these events [7][8][9][10][11].…”

The existence of full dimensional invariant tori for an almost-periodically forced nonlinear beam equation

“…By the similar method in [22], Rui and Si [15] turned the inhomogeneous Schrödinger equation √ −1u t − u xx + mu + φ(t)|u| 2 u = εg(t) into a complex ordinary differential equation uniformly in space variables and a PDE with zero equilibrium point, then constructed the invariant tori or quasi-periodic solutions of the PDE by KAM method. For the existence of quasi-periodic solutions for non-autonomous PDEs, there are a couple of other references, see [11,18,20,24]. Up to now, there are only a few results about the existence of almost-periodic solutions for Hamiltonian partial differential equations (HPDEs) with almost-periodic forcing.…”