Elena Kopylova

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.

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.

Weighted Energy Decay for 1D Klein–Gordon Equation

We obtain a dispersive long-time decay in weighted energy norms for solutions of the 1D Klein–Gordon equation with generic potential. The decay extends the results obtained by Jensen, Kato and Murata for the equations of Schrödingers type by the spectral approach. For the proof we modify the approach to make it applicable to relativistic equations.

Weighted energy decay for 3D wave equation

We obtain a dispersive long-time decay in weighted energy norms for solutions to the 3D wave equation with generic potential. The decay extends the results obtained by Jensen and Kato for the 3D Schrodinger equation.

Dispersion estimates for one-dimensional Schrödinger and Klein-Gordon equations revisited

It is shown that for a one-dimensional Schrodinger operator with a potential whose first moment is integrable the elements of the scattering matrix are in the unital Wiener algebra of functions with integrable Fourier transforms. This is then used to derive dispersion estimates for solutions of the associated Schrodinger and Klein-Gordon equations. In particular, the additional decay conditions are removed in the case where a resonance is present at the edge of the continuous spectrum. Bibliography: 29 titles.

On the Linear Stability of Crystals in the Schrödinger–Poisson Model

We consider the Schrödinger–Poisson–Newton equations for crystals with one ion per cell. We linearize this dynamics at the periodic minimizers of energy per cell and introduce a novel class of the ion charge densities that ensures the stability of the linearized dynamics. Our main result is the energy positivity for the Bloch generators of the linearized dynamics under a Wiener-type condition on the ion charge density. We also adopt an additional ‘Jellium’ condition which cancels the negative contribution caused by the electrostatic instability and provides the ‘Jellium’ periodic minimizers and the optimality of the lattice: the energy per cell of the periodic minimizer attains the global minimum among all possible lattices. We show that the energy positivity can fail if the Jellium condition is violated, while the Wiener condition holds. The proof of the energy positivity relies on a novel factorization of the corresponding Hamilton functional. The Bloch generators are nonselfadjoint (and even nonsymmetric) Hamilton operators. We diagonalize these generators using our theory of spectral resolution of the Hamilton operators with positive definite energy (Komech and Kopylova in, J Stat Phys 154(1–2):503–521, 2014, J Spectral Theory 5(2):331–361, 2015). The stability of the linearized crystal dynamics is established using this spectral resolution.

Dispersion estimates for discrete Schrödinger and Klein–Gordon equations

The long-time asymptotics is derived for solutions of the discrete 3dimensional Schrödinger and Klein–Gordon equations. §

On convergence to equilibrium distribution for wave equation in even dimensions

Consider a wave equation (WE) with constant coefficients in

Dispersion estimates for one-dimensional discrete Dirac equations☆

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