Abstract: In this article we study periodic solutions of the nonlinear wave equation (NLW)
Our main result is a characterization of two fundamental properties of periodic solutions: their frequencies and their spatial localization. We consequently find the following relationship between the spatial and temporal properties of periodic solutions:
Theorem 1: Suppose f is sufficiently smooth and let u be a nontrivial 2*pi/omega - periodic solution of NLW. Then (omega)^2 is less than or equal to f '(0).
Theorem 2: Suppose that f '(0) neq m^2*(omega)^2 for
any m in Z.
Let u be a 2*pi/omega - periodic solution to NLW. Then
exp(alpha*|x|)u is in L^2(S_omega X Rn)
for all alpha satisfying
(alpha)^2 < f'(0) - [sqrt(f '(0)/(omega)^2)]^2*(omega)^2,
where [a] denotes the integer part of a.
Here, S_omega is the circle of circumference 2*pi/omega,
and L^2(S_omega X Rn) denotes the space of 2*pi/omega
-periodic functions (in time) and L^2 in space.
Theorem 1 states that the allowable frequencies for periodic solutions is bounded above by an easily computable number: sqrt(f '(0)). Thus, if f '(0) is less than or equal to 0, then NLW has no (finite-energy) periodic solutions. Theorem 2 states that, in an average sense, periodic solutions are exponentially localized in space, and the degree of localization is given explicitly. Thus, periodic solutions are 'particle-like' entities.
To prove these results we introduce a method which is related to one that has been developed, to a great success, in the scattering theory of Schroedinger operators: that of positive commutators. Our result extends the celebrated work of J-M Coron (Periode Minimale pour une Corde Vibrante de Longeur Infinie, C. R. Acad. Sci. Paris Ser. I Math. 294, 127-129 (1982)).