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Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. (English) Zbl 0708.35087

The paper studies the non-linear wave equation \[ u_{tt}- u_{xx}+v(x)u(x,t)+\epsilon u^ 3(x,t)=0. \] It is shown that for a large class of potentials, v(x), one can use Kolmogorov, Arnold, Moser (KAM) methods to construct periodic and quasi-periodic solutions (in time) for this equation. The author seeks to extend the KAM ideas to a suitable infinite dimensional setting, namely, perturbation of completely integrable partial differential equations, by drawing heavily on the Hamiltonian nature of the problem. The method is based on a perturbation of the known solutions of \(\epsilon =0\) of the above equation and attempts to prove that these solutions persist when \(\epsilon\neq 0\). It is shown that the above equation is equivalent to an infinite system of coupled ordinary differential equations, which are the equations of motion for a Hamiltonian system. The discussion also points out that if one had chosen, more complicated polynomial nonlinearity rather than cubic, the Hamiltonian form persists with more complicated polynomial interactions. The Hamiltonian is made tractable with proper transformation of the variables. Finally, some estimates of the denominators are derived.
Reviewer: N.D.Sengupta

MSC:

35Q58 Other completely integrable PDE (MSC2000)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
35L70 Second-order nonlinear hyperbolic equations
35B25 Singular perturbations in context of PDEs
35B10 Periodic solutions to PDEs
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