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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., Comment: 19 pages. arXiv admin note: text overlap with arXiv:2306.00508

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., Comment: 14 pages

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 $\omega\in R^3$. Our main results are the orbital stability of moving solitons with $\omega=0$ 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., Comment: 25 pages

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., Comment: 13 pages, 0 figures

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., Comment: 42 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:1907.06998

We survey our recent results on stability of 3D crystals in the Schr\"odinger-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., Comment: 97 pages, 0 figures. arXiv admin note: substantial text overlap with arXiv:1808.10385, arXiv:1310.3084, arXiv:1711.02938, arXiv:1808.08167

We obtain explicit characterization of spectral and orbital stability of solitary wave solutions to the $\mathbf{U}(1)$-invariant Klein--Gordon equation in one spatial dimension coupled to an anharmonic oscillator. We also give the complete analysis of the spectrum of the linearization at a solitary wave., Comment: 23 pages

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., Comment: 17 pages

We prove global well-posedness for 3D Dirac equation with a concentrated nonlinearity., Comment: 13 pages

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., Comment: 69 pages, 8 figures. arXiv admin note: text overlap with arXiv:1409.2009

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 $(\psi_1(x)e^{-i\omega_1 t},\psi_2(x)e^{-i\omega_2 t})$. 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 $\omega_j\in[-m,m]$, $j=1,2$., Comment: 22 pages. arXiv admin note: text overlap with arXiv:math/0609013

We consider the Schr\"odinger-Poisson-Newton equations for finite crystals under periodic boundary conditions with one ion per cell of a lattice. The electrons are described by one-particle Schr\"odinger equation. Our main results are i) the global dynamics with moving ions; ii) the orbital stability of periodic ground state under a novel Jellium and Wiener-type conditions on the ion charge density. Under the Jellium condition both ionic and electronic charge densities for the ground state are uniform., Comment: 21 pages. arXiv admin note: substantial text overlap with arXiv:1711.02938

The Schr\"odinger-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., Comment: 17 pages

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 $\omega\in[-m,m]$.

We prove global well-posedness for the 3D Klein-Gordon equation with a concentrated nonlinearity., Comment: Journal of Mathematical Analysis and Applications 2016

We derive dispersion estimates for solutions of the one-dimensional discrete perturbed Dirac equation. To this end we develop basic scattering theory and establish a limiting absorption principle for discrete perturbed Dirac operators., Comment: 16 pages

We consider the Schr\"odinger--Poisson--Newton equations for crystals with a cubic lattice and one ion per cell. We linearize this dynamics at the ground state and introduce a novel class of the ion charge densities which provide the stability of the linearized dynamics. This is the first result on linear stability for crystals. Our key result is the {\it energy positivity} for the Bloch generators of the linearized dynamics under a Wiener-type condition on the ion charge density. We also assume an additional condition which cancels the negative contribution caused by electrostatic instability. The proof of the energy positivity relies on a special factorization of the corresponding Hamilton functional. We show that the energy positivity can fail if the additional condition breaks down while the Wiener condition holds. The Bloch generators are nonselfadjoint (and even nonsymmetric) Hamilton operators. We diagonalize these generators using our theory of spectral resolution of the Hamilton ope\-rators {\it with positive definite energy} \ci{KK2014a,KK2014b}. Using this spectral resolution, we establish the stability of the linearized crystal dynamics., Comment: 23 pages

We show that for a one-dimensional Schr\"odinger 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 Schr\"odinger 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., Comment: 21 pages

We derive dispersion estimates for solutions of the one-dimensional discrete perturbed Schr\"odinger and wave equations. In particular, we improve upon previous works and weaken the conditions on the potentials. To this end we also provide new results concerning scattering for one-dimensional discrete perturbed Schr\"odinger operators which are of independent interest. Most notably we show that the reflection and transmission coefficients belong to the Wiener algebra., Comment: 23 pages

We obtain a dispersive long-time decay in weighted energy norms for solutions of 3D Klein-Gordon equation with magnetic and scalar potentials. The decay extends the results obtained by Jensen and Kato for the Schroedinger equation with scalar potential. For the proof we develop the spectral theory of Agmon, Jensen and Kato and minimal escape velocities estimates of Hunziker, Sigal and Soffer., Comment: 15 pages, 0 figures

We establish a spectral representation for solutions to linear Hamilton equations with positive definite energy in a Hilbert space. Our approach is a special version of M. Krein's spectral theory of J-selfadjoint operators is the Hilbert spaces with an indefinite metric. Our main result is an application to the eigenfunction expansion for the linearized relativistic Ginzburg-Landau equation., Comment: 18 pages, 0 figures

We introduce a new class of piece-wise quadratic potentials for nonlinear wave equations with a kink solutions. The potentials allow an exact description of the spectral properties for the linearized equation at the kink. This description is necessary for the study of the stability properties of the kinks., Comment: 13 pages, 6 figures

We obtain a dispersive long-time decay in weighted norms for solutions of 3D Schroedinger equation with generic magnetic and scalar potentials. The decay extends the results obtained by Jensen and Kato for the Schroedinger equation without magentic potentials. For the proof we develop the spectral theory of Agmon, Jensen and Kato, extending the high energy decay of the resolvent to the magnetic Schroedinger equation., Comment: 14 pages, 0 figures

We consider the Dirac equation in $\R^3$ with a potential, and study the distribution $\mu_t$ of the random solution at time $t\in\R$. The initial measure $\mu_0$ has zero mean, a translation-invariant covariance, and a finite mean charge density. We also assume that $\mu_0$ satisfies a mixing condition of Rosenblatt- or Ibragimov-Linnik-type. The main result is the long time convergence of projection of $\mu_t$ onto the continuous spectral space. The limiting measure is Gaussian., Comment: 15 pages, 0 figures

We obtain a dispersive long-time decay in weighted energy norms for solutions of the Klein-Gordon equation in a moving frame. The decay extends the results of Jensen, Kato and Murata for the equations of the Schr\"odinger type. We modify the approach to make it applicable to relativistic equations., Comment: 17 pages, 0 figures

We prove the asymptotic stability of standing kink for the nonlinear relativistic wave equations of the Ginzburg-Landau type in one space dimension: for any odd initial condition in a small neighborhood of the kink, the solution, asymptotically in time, is the sum of the kink and dispersive part described by the free Klein-Gordon equation. The remainder converges to zero in a global norm. Crucial role in the proofs play our recent results on the weighted energy decay for the Klein-Gordon equations.

We prove the asymptotic stability of the moving kinks for the nonlinear relativistic wave equations in one space dimension with a Ginzburg-Landau potential: starting in a small neighborhood of the kink, the solution, asymptotically in time, is the sum of a uniformly moving kink and dispersive part described by the free Klein-Gordon equation. The remainder decays in a global energy norm. Crucial role in the proofs play our recent results on the weighted energy decay for the Klein-Gordon equations., Comment: 30 pages, 0 figures

We establish soliton-like asymptotics for finite energy solutions to the Schr\"odinger equation coupled to a nonrelativistic classical particle. Any solution with initial state close to the solitary manifold, converges to a sum of traveling wave and outgoing free wave. The convergence holds in global energy norm. The proof uses spectral theory and the symplectic projection onto solitary manifold in the Hilbert phase space., Comment: 32 pages