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 free Schrödinger equation. The proofs use the strategy of 3]: the linerization of the dynamics on the solitary manifold, the symplectic orthogonal projection, method of majorants, etc.

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We prove the asymptotic stability of standing kink for the nonlinear relativistic wave equations of the Ginzburg-Landau type in one space dimension: for any odd initial condition in a small neighborhood of the kink, the solution, asymptotically in time, is the sum of the kink and dispersive part described by the free Klein-Gordon equation. The remainder converges to zero in a global norm. Crucial role in the proofs play our recent results on the weighted energy decay for the Klein-Gordon equations.We consider the Cauchy problem for the Hamilton system (2.15) which we write aṡ(2.15) Here Y (t) = (ψ(t), π(t)), Y 0 = (ψ 0 , π 0 ), and all derivatives are understood in the sense of distributions. To formulate our results precisely, let us first we introduce a suitable phase space

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Consider the Klein-Gordon equation (KGE) in IR n , n ≥ 2, with constant or variable coefficients. We study the distribution µ t of the random solution at time t ∈ IR. We assume that the initial probability measure µ 0 has zero mean, a translation-invariant covariance, and a finite mean energy density. We also asume that µ 0 satisfies a Rosenblattor Ibragimov-Linnik-type mixing condition. The main result is the convergence of µ t to a Gaussian probability measure as t → ∞ which gives a Central Limit Theorem for the KGE. The proof for the case of constant coefficients is based on an analysis of long time asymptotics of the solution in the Fourier representation and Bernstein's 'room-corridor' argument. The case of variable coefficients is treated by using an 'averaged' version of the scattering theory for infinite energy solutions, based on Vainberg's results on local energy decay.