Quasi-periodic solutions of nonlinear wave equations with quasi-periodic forcing. (English) Zbl 1183.37122
Summary: This work focuses on one-dimensional (1D) quasi-periodically forced nonlinear wave equations. This means studying
\[ u_{tt}-u_{xx}+\mu u+\varepsilon\phi(t)h(u)=0,\quad \mu>0, \]
with Dirichlet boundary conditions, where \(\varepsilon\) is a small positive parameter, \(\phi(t)\) is a real analytic quasi-periodic function in \(t\) with frequency vector \(\omega=(\omega_1,\omega_2,\dots,\omega_m)\) and the nonlinearity \(h\) is a real analytic odd function of the form
\[ h(u)=\eta_1u+\eta_{2\overline r+1}u^{2\overline r+1}+\sum_{k\geq\overline r+1}\eta_{2k+1}u^{2k+1},\quad \eta_1,\eta_{2\overline r+1}\neq 0,\;\overline r\in\mathbb N. \]
It is shown that, under a suitable hypothesis on \(\phi(t)\) and \(h\), there are many quasi-periodic solutions for the above equation via KAM theory.
\[ u_{tt}-u_{xx}+\mu u+\varepsilon\phi(t)h(u)=0,\quad \mu>0, \]
with Dirichlet boundary conditions, where \(\varepsilon\) is a small positive parameter, \(\phi(t)\) is a real analytic quasi-periodic function in \(t\) with frequency vector \(\omega=(\omega_1,\omega_2,\dots,\omega_m)\) and the nonlinearity \(h\) is a real analytic odd function of the form
\[ h(u)=\eta_1u+\eta_{2\overline r+1}u^{2\overline r+1}+\sum_{k\geq\overline r+1}\eta_{2k+1}u^{2k+1},\quad \eta_1,\eta_{2\overline r+1}\neq 0,\;\overline r\in\mathbb N. \]
It is shown that, under a suitable hypothesis on \(\phi(t)\) and \(h\), there are many quasi-periodic solutions for the above equation via KAM theory.
MSC:
| 37K55 | Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems |
| 35L70 | Second-order nonlinear hyperbolic equations |
| 35L05 | Wave equation |
Keywords:
infinite dimensional Hamiltonian systems; Kolmogorov-Arnold-Moser (KAM) theory; quasi-periodically forced nonlinear wave equations; quasi-periodic solutions; invariant torusReferences:
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