Newton's method and periodic solutions of nonlinear wave equations

We prove the existence of periodic solutions of the nonlinear wave equation \documentclass{article}\pagestyle{empty}\begin{document}$$ \mathop \partial \nolimits_t^2 u = \mathop \partial \nolimits_x^2 u - g(x,u), $$\end{document} satisfying either Dirichlet or periodic boundary conditions on the int...

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Bibliographic Details
Published in:Communications on pure and applied mathematics 1993-12, Vol.46 (11), p.1409-1498
Main Authors: Craig, Walter, Wayne, C. Eugene
Format: Article
Language:English
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Summary:We prove the existence of periodic solutions of the nonlinear wave equation \documentclass{article}\pagestyle{empty}\begin{document}$$ \mathop \partial \nolimits_t^2 u = \mathop \partial \nolimits_x^2 u - g(x,u), $$\end{document} satisfying either Dirichlet or periodic boundary conditions on the interval [O, π]. 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. © 1993 John Wiley & Sons, Inc.
ISSN:0010-3640
1097-0312
DOI:10.1002/cpa.3160461102