arXiv subjects
Publications and source records attributed to Elena Kopylova.
We consider the 2D Maxwell--Lorentz equations with an extended charged rotating particle. The system admits soliton solutions corresponding to a particle moving with a constant velocity and rotating with a constant angular velocity. Our main result is the asymptotic stability of the solitons.
We improve previous results on dispersive decay for 1D Klein- Gordon equation. We develop a novel approach, which allows us to establish the decay in more strong norms and weaken the assumption on the potential.
We consider 2D Maxwell-Lorentz equations with extended charged rotating particle. The system admits solitons which are solutions corresponding to a particle moving with a constant velocity and rotating with a constant angular velocity. Our main result is asymptotic stability of the solitons.
We consider the 2D Maxwell--Lorentz system which describes a rotating particle coupled to the Maxwell field. The system admits stationary soliton-type solutions. We prove the attraction to solitons for any finite energy solution relying on the conservation of angular momentum.
We consider stability of solitons of 3D Maxwell--Lorentz system with extended charged spinning particle.The solitons are solutions which correspond to a particle moving with a constant velocity $v\in\R^3$ with $|v|<1$ and rotating with a constant angular velocity $ω\in R^3$. Our main results are the orbital stability of moving solitons with $ω=0$ and a {\it linear} orbital stability of rotating solitons with $v=0$. The Hamilton--Poisson structure of the Maxwell--Lorentz system is degenerate and admits the Casimir invariants. We construct the Lyapunov function as a linear combination of the Hamiltonian with a suitable Casimir invariant. The key point is a lower bound for this function. The proof of the bound in the case $\om\ne 0$ relies on angular momentum conservation and suitable spectral arguments including the Heinz inequality and closed graph theorem.
We prove the orbital stability of soliton solutions for 2D Maxwell--Lorentz system with extended charged particle. The solitons corresponds to the uniform motion and rotation of the particle. We reduce the corresponding Hamilton system by the canonical transformation via transition to a comoving frame. The solitons are the critical points of the reduced Hamiltonian. The key point of the proof is a lower bound for the Hamiltonian.
We develop the theory of momentum map for the Maxwell-Lorentz equations with spinning extended charged particle. This theory is indispensable for the study of long-time behaviour and radiation of the solitons of this system. The development relies on the Hamilton-Poisson structure of the system. As an example, we apply the theory to the rotation group of symmetry, calculating the expression for the conserved angular momentum. We check the coincidence of this expression with known classical invariant.
We survey the theory of attractors of nonlinear Hamiltonian partial differential equations since its appearance in 1990. These are results on global attraction to stationary states, to solitons and to stationary orbits, on adiabatic effective dynamics of solitons and their asymptotic stability. Results of numerical simulations are also given. Based on these results, we propose a new general hypothesis on attractors of $G$-invariant nonlinear Hamiltonian partial differential equations. The obtained results suggest a novel dynamical interpretation of basic quantum phenomena: Bohr's transitions between quantum stationary states, wave-particle duality, and probabilistic interpretation.
We derive dispersion estimates for solutions of a one-dimensional discrete Dirac equations with a potential. In particular, we improve our previous result, weakening the conditions on the potential. To this end we also provide new results concerning scattering for the corresponding perturbed Dirac operators which are of independent interest. Most notably, we show that the reflection and transmission coefficients belong to the Wiener algebra.
We survey our recent results on stability of 3D crystals in the Schrödinger-Poisson-Newton model. We establish orbital stability for the ground state in the case of finite crystal and linear stability for infinite crystals under novel Jellium and Wiener conditions on the charge density of ions. The corresponding examples are given. In the case of finite crystals, the proofs rely on positivity of the Hessian of Hamiltonian functional in the directions orthogonal to the manifold of ground states. The problem of spatial periodicity of the ground states is discussed. The non-periodic examples are constructed. In the case of infinite crystals the proofs rely on a novel spectral theory of Hamiltonian operators which is a special version of the Gohberg-Krein-Langer theory of selfadjoint operators in the Hilbert spaces with indefinite metric. We establish the existence of the ground states and the dispersive decay for the linearised dynamics.
We survey the theory of attractors of nonlinear Hamiltonian partial differential equations since its appearance in 1990. These are results on global attraction to stationary states, to solitons and to stationary orbits, on adiabatic effective dynamics of solitons and their asymptotic stability. Results of numerical simulation are given. The obtained results allow us to formulate a new general conjecture on attractors of $G$ -invariant nonlinear Hamiltonian partial differential equations. This conjecture suggests a novel dynamical interpretation of basic quantum phenomena: Bohr's transitions between quantum stationary states, wave-particle duality and probabilistic interpretation.
We prove global well-posedness for 3D Dirac equation with a concentrated nonlinearity.
The long-time asymptotics is analyzed for all finite energy solutions to a model $\mathbf{U}(1)$-invariant nonlinear Dirac equation in one dimension, coupled to a nonlinear oscillator: {\it each finite energy solution} converges as $t\to\pm\infty$ to the set of all `nonlinear eigenfunctions' of the form $(ψ_1(x)e^{-iω_1 t},ψ_2(x)e^{-iω_2 t})$. The {\it global attraction} is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation. We justify this mechanism by the strategy based on \emph{inflation of spectrum by the nonlinearity}. We show that any {\it omega-limit trajectory} has the time-spectrum in the spectral gap $[-m,m]$ and satisfies the original equation. This equation implies the key {\it spectral inclusion} for spectrum of the nonlinear term. Then the application of the Titchmarsh convolution theorem reduces the spectrum of $j$-th component of the omega-limit trajectory to a single harmonic $ω_j\in[-m,m]$, $j=1,2$.
The global attraction is proved for the nonlinear 3D Klein-Gordon equation with a nonlinearity concentrated at one point. Our main result is the convergence of each "finite energy solution" to the manifold of all solitary waves as $t\to\pm\infty$. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersion radiation. We justify this mechanism by the following strategy based on inflation of spectrum by the nonlinearity. We show that any omega-limit trajectory has the time-spectrum in the spectral gap $[-m,m]$ and satisfies the original equation. Then the application of the Titchmarsh Convolution Theorem reduces the spectrum of each omega-limit trajectory to a single frequency $ω\in[-m,m]$.
The Schrödinger-Poisson-Newton equations for crystals with a cubic lattice and one ion per cell are considered. The ion charge density is assumed i) to satisfy the Wiener and Jellium conditions introduced in our previous paper [28], and ii) to be exponentially decaying at infinity. The corresponding examples are given. We study the linearized dynamics at the ground state. The dispersion relations are introduced via spectral resolution for the non-selfadjoint Hamilton generator using the positivity of the energy established in [28]. Our main result is the dispersion decay in the weighted Sobolev norms for solutions with initial states from the space of continuous spectrum of the Hamilton generator. We also prove the absence of singular spectrum and limiting absorption principle. The multiplicity of every eigenvalue is shown to be infinite. The proofs rely on novel exact bounds and compactness for the inversion of the Bloch generators and on uniform asymptotics for the dispersion relations. We derive the bounds by the energy positivity from [28]. We also use the theory of analytic sets.
We consider the Schrödinger--Poisson--Newton equations for finite crystals under periodic boundary conditions with one ion per cell of a lattice. The electron field is described by the $N$-particle Schrödinger equation with antisymmetric wave function. Our main results are i) the global dynamics with moving ions, and ii) the orbital stability of periodic ground state under a novel Jellium and Wiener-type conditions on the ion charge density. Under Jellium condition both ionic and electronic charge densities of the ground state are uniform.
We prove global well-posedness for the 3D Klein-Gordon equation with a concentrated nonlinearity.
We show that for a one-dimensional Schrödinger operator with a potential whose first moment is integrable the scattering matrix is in the unital Wiener algebra of functions with integrable Fourier transforms. Then we use this to derive dispersion estimates for solutions of the associated Schrödinger and Klein-Gordon equations. In particular, we remove the additional decay conditions in the case where a resonance is present at the edge of the continuous spectrum.