On scattering of solitons for the Klein-Gordon equation coupled to a particle

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.Comment: 47 pages, 2 figure

Convergence to equilibrium distribution. The Klein-Gordon equation coupled to a particle

We consider the Hamiltonian system consisting of a Klein-Gordon vector field and a particle in $\R^3$. The initial date of the system is a random function with a finite mean density of energy which also satisfies a Rosenblatt- or Ibragimov-type mixing condition. Moreover, initial correlation functions are translation-invariant. We study the distribution $\mu_t$ of the solution at time $t\in\R$. The main result is the convergence of $\mu_t$ to a Gaussian measure as $t\to\infty$, where $\mu_\infty$ is translation-invariant.Comment: 22 page

Relativistic dynamics of accelerating particles derived from field equations

In relativistic mechanics the energy-momentum of a free point mass moving without acceleration forms a four-vector. Einstein's celebrated energy-mass relation E=mc^2 is commonly derived from that fact. By contrast, in Newtonian mechanics the mass is introduced for an accelerated motion as a measure of inertia. In this paper we rigorously derive the relativistic point mechanics and Einstein's energy-mass relation using our recently introduced neoclassical field theory where a charge is not a point but a distribution. We show that both the approaches to the definition of mass are complementary within the framework of our field theory. This theory also predicts a small difference between the electron rest mass relevant to the Penning trap experiments and its mass relevant to spectroscopic measurements.Comment: A few typos were correcte