arXiv subjects
Publications and source records attributed to E. Kopylova.
Well-posedness and global attractor are established for 2D damped driven nonlinear Schrödinger equation with almost periodic pumping in a bounded region. The key role is played by a novel application of the energy equation.
The global attraction to stationary states is established for solutions to 3D wave equations with concentrated nonlinearities: each finite energy solution converges as $t\to\pm\infty$ to stationary states. The attraction is caused by nonlinear energy radiation.
We consider the Hamiltonian system of scalar wave field and a single nonrelativistic particle coupled in a translation invariant manner. The particle is also 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 ${\cal S}$ of all stationary states in the long time limit $t\to\pm\infty$. Further we show that the rate of relaxation to a stable stationary state is determined by spatial decay of initial data. The convergence is followed by the radiation of the dispersion wave which is a solution to the free wave equation. Similar relaxation has been proved previously for the case of relativistic particle when the speed of the particle is less than the speed of light. Now we extend these results to nonrelativistic particle with arbitrary superlight velocity. However, we restrict ourselves by the plane particle trajectories. The extension to general case remains an open problem.
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 an 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. It is assumed that the charge density satisfies the Wiener condition which is a version of the "Fermi Golden Rule". We prove that in the large time approximation any finite energy solution, with the initial state close to the some stable stationary solution, is a sum of this stationary solution and a dispersive wave which is a solution of the free wave equation.
We derive the dispersion decay for solutions of the 1D discrete Schroedinger and wave equations. Based on previous works, we weaken the conditions on potentials.
A spectral representation for solutions to linear Hamilton equations with nonnegative energy in Hilbert spaces is obtained. This paper continues our previous work on Hamilton equations with positive definite energy. Our approach is a special version of M. Krein's spectral theory of $J$-selfadjoint operators in Hilbert spaces with indefinite metric. As a principal application of these results, we justify the eigenfunction expansion for linearized nonlinear relativistic Ginzburg-Landau equations.
We obtain a dispersive long-time decay in weighted energy norms for solutions of the 1D Dirac equation with generic potential. The decay extends the results obtained by Jensen, Kato and Murata for the Schrödinger equations.
We derive the long-time decay in weighted norms for solutions of the discrete 3D Schrödinger and Klein-Gordon equations.
We establish soliton-like asymptotics for finite energy solutions to the Dirac equation coupled to a relativistic particle. Any solution with initial state close to the solitary manifold, converges in long time limit 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.
We obtain a dispersive long-time decay in weighted energy norms for solutions of the 3D Klein-Gordon equation with generic potential. The decay extends the results obtained by Jensen and Kato for the 3D Schredinger equation. For the proof we modify the spectral approach of Jensen and Kato to make it applicable to relativistic equations.
The long-time asymptotics is analyzed for finite energy solutions of the 1D discrete Schrödinger equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the phase rotation group. 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-Perelman: the linerization of the dynamics on the solitary manifold, the symplectic orthogonal projection, method of majorants, etc.
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-Perelman: the linerization of the dynamics on the solitary manifold, the symplectic orthogonal projection and method of majorants.