Geodesic
Attractors of non-linear Hamiltonian one-dimensional wave equations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 55 (2000) no. 1, pp. 43-92

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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. 
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     author = {A. I. Komech},
     title = {Attractors of non-linear {Hamiltonian} one-dimensional wave equations},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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A. I. Komech. Attractors of non-linear Hamiltonian one-dimensional wave equations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 55 (2000) no. 1, pp. 43-92. http://geodesic.mathdoc.fr/item/RM_2000_55_1_a1/