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Newton’s method and periodic solutions of nonlinear wave equations. (English) Zbl 0794.35104
We prove the existence of periodic solutions of the nonlinear wave equation \(\partial^ 2_ tu=\partial^ 2_ xu-g(x,u)\), satisfying either Dirichlet or periodic boundary conditions on the interval \([0,\pi]\). The coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation. Using a version of Newton’s method, we show that this equation has solutions provided the nonlinearity \(g(x,u)\) satisfies certain generic conditions of nonresonance and genuine nonlinearity.

MSC:
35L70 Second-order nonlinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B10 Periodic solutions to PDEs
35B32 Bifurcations in context of PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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