A. I. Komech

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We consider the dynamics of a harmonic crystal in d dimensions with n components, d, n ≥ 1 . The initial date is a random function with finite mean density of the energy which also satisfies a Rosenblattor Ibragimov-Linnik-type mixing condition. The random function converges to different space-homogeneous processes as xd → ±∞ , with the distributions μ± .(More)
Consider the Klein-Gordon equation (KGE) in IR, n ≥ 2, with constant or variable coefficients. We study the distribution μt of the random solution at time t ∈ IR. We assume that the initial probability measure μ0 has zero mean, a translation-invariant covariance, and a finite mean energy density. We also asume that μ0 satisfies a Rosenblattor(More)
1 Mathematics Department, Elektrostal Polytechnical Institute, Elektrostal, 144000 Russia; e-mail: misis@elsite.ru 2 Mechanics and Mathematics Department, Moscow State University, Moscow, 119899 Russia; Institüt für Mathematik, Universität Wien, Vienna 1090, Austria; e-mail: komech@ mat.univie.at 3 Economics Department, Chelyabinsk State University,(More)
The long-time asymptotics is analyzed for finite energy solutions of the 1D Schrödinger equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the phase rotation group U(1). For initial states close to a solitary wave, the solution converges to a sum of another solitary wave and dispersive wave which is a solution to the(More)
We consider the dynamics of a field coupled to a harmonic crystal with n components in dimension d , d, n ≥ 1 . The crystal and the dynamics are translation-invariant with respect to the subgroup ZZ of IR . The initial data is a random function with a finite mean density of energy which also satisfies a Rosenblattor Ibragimov-Linniktype mixing condition.(More)
We consider the Dirac equation in IR with constant coefficients and study the distribution μt of the random solution at time t ∈ IR . It is assumed that the initial measure μ0 has zero mean, a translation-invariant covariance, and finite mean charge density. We also assume that μ0 satisfies a mixing condition of Rosenblattor Ibragimov-Linnik-type. The main(More)
We consider the Schrödinger-Poisson-Newton equations for crystals with one ion per cell. We linearize this dynamics at the periodic minimizers of energy per cell and introduce a novel class of the ion charge densities that ensures the stability of the linearized dynamics. Our main result is the energy positivity for the Bloch generators of the linearized(More)
We establish soliton-like asymptotics for finite energy solutions to the Dirac equation coupled to a relativistic particle. Any solution with initial state close to the solitary manifold, converges in long time limit to a sum of traveling wave and outgoing free wave. The convergence holds in global energy norm. The proof uses spectral theory and symplectic(More)