Craig W., Wayne , C.E.
Newton's Method and Periodic Solutions of Nonlinear
Wave Equations
(238K, TeX)
ABSTRACT. We prove the existence of periodic solutions
of the nonlinear wave equation
$$
\partial_t^2 u = \partial_x^2 u - g(x,u)~,
$$
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.