arXiv subjects
Publications and source records attributed to T. V. Dudnikova.
For a system consisting of several Dirac fields and a particle, we study the Cauchy problem with random initial data. We assume that the initial measure has zero mean value, a finite mean charge density, a translation-invariant covariance and satisfies a mixing condition. The main result is the long-time convergence of distributions of the random solutions to a limit Gaussian measure.
We consider an one-dimensional inhomogeneous harmonic chain consisting of two different semi-infinite chains of harmonic oscillators. We study the Cauchy problem with random initial data. Under some restrictions on the interaction between the oscillators of the chain and on the distribution of the initial data, we prove the convergence of space-time statistical solutions to a Gaussian measure.
An infinite irregular harmonic chain of particles is considered. We assume that some particles (``defects'') in the chain have masses and force constants of interaction different from the masses and the interaction constants of the other particles. We study the Cauchy problem for this model. The main goal is to study the long-time behavior and derive the dispersive bounds for the solutions in the energy weighted norms.
The initial-boundary value problem for an infinite one-dimensional chain of harmonic oscillators on the half-line is considered. The large time asymptotic behavior of solutions is studied. The initial data of the system are supposed to be a random function which has some mixing properties. We study the distribution $μ_t$ of the random solution at time moments $t\in\mathbb{R}$. The main result is the convergence of $μ_t$ to a Gaussian probability measure as $t\to\infty$. We find stationary states in which there is a non-zero energy current at origin.
We consider a $d$-dimensional harmonic crystal, $d\ge 1$, and study the Cauchy problem with random initial data. We assume that the random initial function is close to different translation-invariant processes for large values of $x_1,\dots,x_k$ with some $k\in\{1,\dots,d\}$. The distribution $μ_t$ of the solution at time $t\in\mathbb{R}$ is studied. We prove the convergence of correlation functions of the measures $μ_t$ to a limit for large times. The explicit formulas for the limiting correlation functions and for the energy current density (in mean) are obtained in the terms of the initial covariance. We give the application to the case of the Gibbs initial measures with different temperatures. In particular, we find stationary states in which there is a constant non-zero energy current flowing through the harmonic crystal. Furthermore, the weak convergence of $μ_t$ to a limit measure is proved. We also study the initial boundary value problem for the harmonic crystal with zero boundary condition and obtain the similar results.
The mixed initial-boundary value problem for infinite one-dimensional chain of harmonic oscillators on the half-line is considered. We study the large time behavior of solutions and derive the dispersive bounds.
A system of equations consisting of an infinite string coupled to a nonlinear oscillator is considered. The Cauchy problem for the system with the periodic initial data is studied. The main goal is to prove the convergence of the solutions as $t\to\infty$ to a time periodic solution.
Solitary waves of nonlinear Dirac, Maxwell-Dirac and Klein-Gordon-Dirac equations are considered. We prove that the energy-momentum relation for solitary waves coincides with the Einstein energy-momentum relation for point particles.
We consider a linear Hamiltonian system consisting of a classical particle and a scalar field describing by the wave or Klein-Gordon equations with variable coefficients. The initial data of the system are supposed to be a random function which has some mixing properties. We study the distribution μ_t of the random solution at time moments t\in\R. The main result is the convergence of μ_t to a Gaussian probability measure as t\to\infty. The mixing properties of the limit measures are studied. The application to the case of Gibbs initial measures is given.
The lattice dynamics in $\mathbb{Z}^d$, $d\ge1$, is considered. The initial data are supposed to be random function. We introduce the family of initial measures $\{μ_0^ε,ε>0\}$ depending on a small scaling parameter $ε$. We assume that the measures $μ_0^ε$ are locally homogeneous for space translations of order much less than $ε^{-1}$ and nonhomogeneous for translations of order $ε^{-1}$. Moreover, the covariance of $μ_0^ε$ decreases with distance uniformly in $ε$. Given $τ\in\mathbb{R}\setminus 0$, $r\in\mathbb{R}^d$, and $κ>0$, we consider the distributions of random solution in the time moments $t=τ/ε^κ$ and at lattice points close to $[r/ε]\in\mathbb{Z}^d$. The main goil is to study the asymptotics of these distributions as $ε\to0$ and derive the limit hydrodynamic equations of the Euler or Navier-Stokes type. The similar results are obtained for lattice dynamics in the half-space $\mathbb{Z}^d_+$.
We consider the lattice dynamics in the half-space. The initial data are random according to a probability measure which enforces slow spatial variation on the linear scale $\varepsilon^{-1}$. We establish two time regimes. For times of order $\varepsilon^{-γ}$, $0<γ<1$, locally the measure converges to a Gaussian measure which is time stationary with a covariance inherited from the initial measure (non-Gaussian, in general). For times of order $\varepsilon^{-1}$, this covariance changes in time and is governed by a semiclassical transport equation.
We consider the dynamics of a harmonic crystal in the half-space with zero boundary condition. It is assumed that the initial date is a random function with zero mean, finite mean energy density which also satisfies a mixing condition of Rosenblatt or Ibragimov type. We study the distribution $μ_t$ of the solution at time $t\in\R$. The main result is the convergence of $μ_t$ to a Gaussian measure as $t\to\infty$ which is time stationary with a covariance inherited from the initial (in general, non-Gaussian) measure.
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 $μ_t$ of the solution at time $t\in\R$. The main result is the convergence of $μ_t$ to a Gaussian measure as $t\to\infty$, where $μ_\infty$ is translation-invariant.
We consider the dynamics of a field coupled to a harmonic crystal with $n$ components in dimension $d$, $d,n\ge 1$. The crystal and the dynamics are translation-invariant with respect to the subgroup $\Z^d$ of $\R^d$. The initial data is a random function with a finite mean density of energy which also satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. Moreover, initial correlation functions are translation-invariant with respect to the discrete subgroup $\Z^d$. We study the distribution $μ_t$ of the solution at time $t\in\R$. The main result is the convergence of $μ_t$ to a Gaussian measure as $t\to\infty$, where $μ_\infty$ is translation-invariant with respect to the subgroup $\Z^d$.
We consider the Dirac equation in $\R^3$ with constant coefficients and study the distribution $μ_t$ of the random solution at time $t\in\R$. It is assumed that the initial measure $μ_0$ has zero mean, a translation-invariant covariance, and finite mean charge density. We also assume that $μ_0$ satisfies a mixing condition of Rosenblatt- or Ibragimov-Linnik-type. The main result is the convergence of $μ_t$ to a Gaussian measure as $t\to\infty$. The proof uses the study of long time asymptotics of the solution and S.N. Bernstein's ``room-corridor'' method.
Solitary waves of relativistic invariant nonlinear wave equation with symmetry group U(1) are considered. We prove that the energy-momentum relation for spherically symmetric solitary waves coincides with the Einstein energy-momentum relation for point particles.
Consider the wave equation with constant or variable coefficients in $\R^3$. The initial datum is a random function with a finite mean density of energy that also satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. The random function converges to different space-homogeneous processes as $x_3\to\pm\infty$, with the distributions $μ_\pm$. We study the distribution $μ_t$ of the random solution at a time $t\in\R$. The main result is the convergence of $μ_t$ to a Gaussian translation-invariant measure as $t\to\infty$ that means central limit theorem for the wave equation. The proof is based on the Bernstein `room-corridor' argument. The application to the case of the Gibbs measures $μ_\pm=g_\pm$ with two different temperatures $T_{\pm}$ is given. Limiting mean energy current density formally is $-\infty\cdot (0,0,T_+ -T_-)$ for the Gibbs measures, and it is finite and equals to $-C(0,0,T_+ -T_-)$ with $C>0$ for the convolution with a nontrivial test function.
Consider the Klein-Gordon equation (KGE) in $\R^n$, $n\ge 2$, with constant or variable coefficients. We study the distribution $μ_t$ of the random solution at time $t\in\R$. 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 Rosenblatt- or Ibragimov-Linnik-type mixing condition. The main result is the convergence of $μ_t$ to a Gaussian probability measure as $t\to\infty$ 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.