Nonexistence of spatially localized free vibrations for a class of nonlinear wave equations

We prove the nonexistence of free vibrations of arbitrary period with polynomially decreasing profiles for a large class of nonlinear wave equations in one space dimension Our class of admissible models includes examples of non integrable wave equations with certain polynomial nonlinearities, as well as examples of completely integrable ones with exponential nonlinearities related to Mikhailov's equations. Our result thus proves a particular case of a conjecture first formulated by Eleonskii, Kulagin, Novozhilova and Silin, and dispels some confusion regarding the relationship between the existence of so-called breather-solutions and the complete integrability of the wave equation. Our class of admissible nonlinearities also contains a particular instance of the nonlinear scalar Higgs' equation, but does not contain the Sine-Gordon equation which is known to possess a 2π-periodic solution in time with exponential fall-off in the spatial direction. Our results may be considered as complementary to recent results by Coron and Weinstein. Our arguments are entirely global, and rest upon methods from the calculus of variations. Work supported in part by the Los Alamos National Laboratory under contract COL-2335, by a University of Texas summer grant and by the ETH-Forschungsinstitut für Mathematik.