On the existence of branches of time-periodic solutions to the nonlinear vibrating string equation
The author is interested in the construction of time periodic solutions to the nonlinear wave equation \(y_{tt} = (\widehat \sigma (y_x))_x\), \(0 < x < 1\), with boundary conditions \(y(0,t) = y(1,t) = 0\), \(t \in \mathbb{R}\), where \(\widehat \sigma (\cdot)\) is an odd function, in particular \(\widehat \sigma (\gamma) = |\gamma |^{m - 1} \gamma (1 + R (\gamma)) \), \(R\) even. Firstly the variables are scaled to obtain a boundary value problem for a new function \(u(x, \tau)\) in the rectangle \(0 \leq \tau \leq \pi/2\); \(0 \leq x \leq 1/2\). Then an equivalent problem is stated for the Riemann invariants \(\alpha\) and \(\beta\) of the equation for \(u(x, \tau)\). Finally the problem is reduced to the determination of a pair of functions \(\varphi, \psi\) with values in \([0, \pi/2]\) satisfying the equations: \[ (\varphi - \psi)_\tau + c (\varphi, \psi) (\varphi - \psi)_x = 0, \quad (\varphi + \psi)_\tau - c (\varphi, \psi) (\varphi + \psi)_x = 0 \] with boundary conditions: \[ \varphi (0, \tau) = 0, \;\varphi (1/2, \tau) = \pi/2,\;0 \leq \tau \leq \pi/2, \quad \psi (x,0) = 0,\;\psi (x, \pi/2) = \pi/2, \;0 \leq x \leq 1/2, \] where \(c (\varphi, \psi)\) is a given function of \((\varphi, \psi)\). This problem is transformed in a form which can be solved by a fixed point argument.
scientific article; zbMATH DE number 736832