Attractors of non-linear Hamiltonian one-dimensional wave equations

A theory is constructed for attractors of all finite-energy solutions of conservative one-dimensional wave equations on the whole real line. The attractor of a non-degenerate (that is, generic) equation is the set of all stationary solutions. Each finite-energy solution converges as $t\to\pm\infty$ to this attractor in the Frechet topology determined by local energy seminorms. The attraction is caused by energy dissipation at infinity. Our results provide a mathematical model of Bohr transitions (“quantum jumps”) between stationary states in quantum systems.