Alexander Komech

Long—time asymptotics for the coupled maxwell—lorentz equations

We determine the long time behavior of solutions to the Maxwell-Lorentz equations, which describe a charge coupled to the electromagnetic eld and subject to external time-independent potentials. The stationary solutions have vanishing magnetic eld and a Coulomb type electrostatic eld centered at the points of the set Z at which the external force vanishes. We prove that solutions of nite energy with bounded charge trajectory converge, in suitable local energy seminorms, to the set of stationary states in the long time limit t ! 1. If the set Z is discrete, this implies the convergence to a deenite stationary state. For an unbounded particle trajectory, at least, the acceleration vanishes as t ! 1 and the Maxwell eld as seen from the particle converges to the stationary Coulomb eld.

Effective Dynamics for a Mechanical Particle Coupled to a Wave Field

Abstract:We consider a particle coupled to a scalar wave field and subject to the slowly varying potential V(ɛq) with small ɛ. We prove that if the initial state is close, order ɛ2, to a soliton (=dressed particle), then the solution stays forever close to the soliton manifold. This estimate implies that over a time span of order ɛ−2 the radiation losses are negligible and that the motion of the particle is governed by the effective Hamiltonian Heff(q,P)=E(P)+V(ɛq) with energy-momentum relation E(P).

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.

Soliton-like asymptotics for a classical particle interacting with a scalar wave field

We consider the Hamiltonian system consisting of scalar wave field and a single particle coupled in a translation invariant manner. The point particle is subject to a confining external potential. The stationary solutions of the system are a Coulomb type wave field centered at those particle positions for which the external force vanishes. We prove that solutions of finite energy converge, in suitable local energy seminorms, to the set S of stationary states in the long time limit t → ±∞. This implies the convergence to some stationary states S ± as t → ±∞ if the set S is discrete. The rate of relaxation to a stable stationary state is determined by spatial decay of initial data. The investigation is inspired by N.Bohrs postulate on transitions to stationary states in quantum systems.

On the Convergence to Statistical Equilibrium for Harmonic Crystals

We consider the dynamics of a harmonic crystal in d dimensions with n components, d,n arbitrary, d,n⩾1, and study the distribution μt of the solution at time t∈R. The initial measure μ0 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 μt to a Gaussian measure as t→∞. The proof is based on the long time asymptotics of the Green’s function and on Bernstein’s “room-corridors” method.

Dispersive Estimates for 1D Discrete Schrodinger and Klein-Gordon Equations

We derive the long-time asymptotics for solutions of the discrete 1D Schrödinger and Klein–Gordon equations.

Scattering of solitons of the Klein–Gordon equation coupled to a classical particle

Long-time asymptotics are established for finite energy solutions of the scalar Klein–Gordon equation coupled to a relativistic classical particle: any “scattering” solution is asymptotically a sum of a soliton and of a dispersive free wave packet as t→±∞. These asymptotics mean the nonlinear scattering of free wave packets by the soliton.

On asymptotic stability of stationary solutions to nonlinear wave and Klein-Gordon equations

We consider nonlinear wave and Klein-Gordon equations with general nonlinear terms, localized in space. Conditions are found which provide asymptotic stability of stationary solutions in local energy norms. These conditions are formulated in terms of spectral properties of the Schrödinger operator corresponding to the linearized problem. They are natural extensions to partial differential equations of the known Lyapunov condition. For the nonlinear wave equation in three-dimensional space we find asymptotic expansions, as t→∞, of the solutions which are close enough to a stationary asymptotically stable solution.

On Asymptotic Stability of Solitary Waves in Schrödinger Equation Coupled to Nonlinear Oscillator

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 Buslaev and Perelman (1993): the linerization of the dynamics on the solitary manifold, the symplectic orthogonal projection, method of majorants, etc.

On Asymptotic Stability of Kink for Relativistic Ginzburg–Landau Equations

It is known that the three dimensional Navier-Stokes system for an incompressible fluid in the whole space has a one parameter family of explicit stationary solutions, which are axisymmetric and homogeneous of degree −1. We show that these solutions are asymptotically stable under any L2-perturbation. Mathematics Subject Classification (2000): 76D07, 76D05, 35Q30, 35B40.