A Birkhoff-Lewis type theorem for the nonlinear wave equation (Q982284)

The authors look for periodic (in time) solutions of the one-dimensional equation \(u_{tt}-u_{xx}+\mu u+f(u)=0\) satisfying boundary conditions \(u(t,0)=u(t,\pi )=0,\) where \(\mu >0\), \(f\) is odd real analytic function, \(f'(0)=0\) and \(f'''(0)\neq 0.\) They prove that there exists a Cantor like set \({\mathcal C}\) such that for all \(\varepsilon \in{\mathcal C}\) there exists a \(2\pi /\varepsilon \)-periodic analytic solution of the above problem, which is even in \(t\) and sine-series in \(x\). Other properties of the solution are established.