Abstract. We consider a Maxwell field translation invariantly coupled to a single charge. This Hamiltonian system admits soliton-type solutions, where the charge and the co-moving field travel with constant velocity. We prove that a solution of finite energy converges, in suitable local energy seminorms, to a certain soliton-type solution in the long time limit t → ±∞.
We consider the dynamics of a harmonic crystal in $d$ dimensions with $n$
components, $d,n$ arbitrary, $d,n\ge 1$, and study the distribution $\mu_t$ of
the solution at time $t\in\R$. The initial measure $\mu_0$ has a
translation-invariant correlation matrix, zero mean, and finite mean energy
density. It also satisfies a Rosenblatt- resp. Ibragimov-Linnik type mixing
condition. The main result is the convergence of $\mu_t$ to a Gaussian measure
as $t\to\infty$. The proof is based on the long time asymptotics of the Green's
function and on Bernstein's ``room-corridors'' method
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.
SUMMARYWe consider a non-stationary scattering of plane waves by a wedge. We prove the Sommerfeld-type representation and uniqueness of solution to the Cauchy problem in appropriate functional spaces developing the general method of complex characteristics (Math.
We establish the long time soliton asymptotics for the translation invariant nonlinear system consisting of the Klein-Gordon equation coupled to a charged relativistic particle.The coupled system has a six dimensional invariant manifold of the soliton solutions. We show that in the large time approximation any finite energy solution, with the initial state close to the solitary manifold, is a sum of a soliton and a dispersive wave which is a solution of the free Klein-Gordon equation. It is assumed that the charge density satisfies the Wiener condition which is a version of the "Fermi Golden Rule". The proof is based on an extension of the general strategy introduced by Soffer and Weinstein, Buslaev and Perelman, and others: symplectic projection in Hilbert space onto the solitary manifold, modulation equations for the parameters of the projection, and decay of the transversal component.
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