Elena Kopylova - Academia.edu (original) (raw)

Papers by Elena Kopylova

Research paper thumbnail of Dispersive estimates for the 2D wave equation

Russian Journal of Mathematical Physics, 2010

We obtain a dispersive long-time decay with respect to weighted energy norms for solutions of the... more We obtain a dispersive long-time decay with respect to weighted energy norms for solutions of the 2D wave equation with generic potential. The decay extends results obtained by Murata for the 2D Schrödinger equation.

Research paper thumbnail of Dispersion estimates for discrete Schrödinger and Klein–Gordon equations

St Petersburg Mathematical Journal, 2010

ABSTRACT We derive the long-time asymptotics for solutions of the discrete 3D Schrödinger and Kle... more ABSTRACT We derive the long-time asymptotics for solutions of the discrete 3D Schrödinger and Klein-Gordon equations.

Research paper thumbnail of On asymptotic stability of solitary waves in discrete Klein–Gordon equation coupled to a nonlinear oscillator

Applicable Analysis, 2010

The long-time asymptotics is analysed for finite energy solutions of the 1D discrete Klein–Gordon... more The long-time asymptotics is analysed for finite energy solutions of the 1D discrete Klein–Gordon 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

Research paper thumbnail of On Asymptotic Stability of Kink for Relativistic Ginzburg–Landau Equations

Archive for Rational Mechanics and Analysis, 2011

We prove the asymptotic stability of kink for the nonlinear relativistic wave equations of the Gi... more We prove the asymptotic stability of 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.

Research paper thumbnail of Scattering of Solitons for Dirac Equation Coupled to a Particle

We establish soliton-like asymptotics for finite energy solutions to the Dirac equation coupled t... more 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.

Research paper thumbnail of Dispersion Decay and Scattering Theory

Research paper thumbnail of Scattering of solitons for Dirac equation coupled to a particle

Journal of Mathematical Analysis and Applications, 2011

We establish soliton-like asymptotics for finite energy solutions to the Dirac equation coupled t... more 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 symplectic projection onto solitary manifold in the Hilbert phase space.

Research paper thumbnail of On Dispersive Decay for Discrete 3D Schr\"odinger and Klein-Gordon Equations

We derive the long-time decay in weighted norms for solutions of the discrete 3D Schr\"odinger an... more We derive the long-time decay in weighted norms for solutions of the discrete 3D Schr\"odinger and Klein-Gordon equations.

Research paper thumbnail of Long time decay for 2D Klein–Gordon equation

Journal of Functional Analysis, 2010

We obtain a dispersive long-time decay in weighted energy norms for solutions of the 2D Klein–Gor... more We obtain a dispersive long-time decay in weighted energy norms for solutions of the 2D Klein–Gordon equations. The decay extends the results obtained by Jensen, Kato and Murata for the equations of Schrödinger's type by the spectral approach. For the proof we modify the approach to make it applicable to relativistic equations.

Research paper thumbnail of On the asymptotic stability of solitary waves in the discrete Schrödinger equation coupled to a nonlinear oscillator

Nonlinear Analysis-theory Methods & Applications, 2009

a b s t r a c t The long-time asymptotics is analyzed for finite energy solutions of the 1D discr... more a b s t r a c t 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 U(1). For initial states close to a solitary wave, the solution converges to a sum of another solitary wave and a dispersive wave which is a solution to the free Schrödinger equation. The proofs use the strategy of Buslaev-Perelman [V.S. Buslaev, G.S. Perelman, On the stability of solitary waves for nonlinear Schrödinger equations, Amer. Math. Soc. Trans. 2 164 (1995), 75-98]: the linearization of the dynamics on the solitary manifold, the symplectic orthogonal projection, method of majorants, etc.

Research paper thumbnail of On Asymptotic Stability of Solitary Waves in Discrete Schr\"odinger Equation Coupled to Nonlinear Oscillator

The long-time asymptotics is analyzed for finite energy solutions of the 1D discrete Schr\"odinge... more The long-time asymptotics is analyzed for finite energy solutions of the 1D discrete Schr\"odinger 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 Schr\"odinger 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.

Research paper thumbnail of Weighted energy decay for 1D wave equation

Journal of Mathematical Analysis and Applications, 2010

We obtain a dispersive long time decay in weighted energy norms for solutions to the 1D wave equa... more We obtain a dispersive long time decay in weighted energy norms for solutions to the 1D wave equation with generic potential. The decay extends the results obtained by Murata for the 1D Schrödinger equation.

Research paper thumbnail of On dispersive properties of discrete 2D Schrödinger and Klein–Gordon equations

Journal of Functional Analysis, 2008

We derive the long-time asymptotics for solutions of the discrete 2D Schrödinger and Klein–Gordon... more We derive the long-time asymptotics for solutions of the discrete 2D Schrödinger and Klein–Gordon equations.

Research paper thumbnail of Weighted Energy Decay for 1D Klein–Gordon Equation

Communications in Partial Differential Equations, 2010

We obtain a dispersive long-time decay in weighted energy norms for solutions of the 1D Klein–Gor... more 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 Schrödinger's type by the spectral approach. For the proof we modify the approach to make it applicable to relativistic equations.

Research paper thumbnail of On Asymptotic Stability of Solitary Waves in a Nonlinear Schr\"odinger Equation

The long-time asymptotics is analyzed for finite energy solutions of the 1D Schrödinger equation ... more 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.

Research paper thumbnail of Weighted energy decay for 3D Klein–Gordon equation

Journal of Differential Equations, 2010

We obtain a dispersive long-time decay in weighted energy norms for solutions of the 3D Klein–Gor... more 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 Schrödinger equation. For the proof we modify the spectral approach of Jensen and Kato to make it applicable to relativistic equations.

Research paper thumbnail of On asymptotic stability of moving kink for relativistic Ginzburg-Landau equation

We prove the asymptotic stability of standing kink for the nonlinear relativistic wave equations ... more 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.

Research paper thumbnail of On long-time decay for modified Klein-Gordon equation

We obtain a dispersive long-time decay in weighted energy norms for solutions of the Klein-Gordon... more 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.

Research paper thumbnail of Dispersive estimates for 1D discrete Schrödinger and Klein–Gordon equations

Applicable Analysis, 2006

We derive the long-time asymptotics for solutions of the discrete 1D Schrödinger and Klein–Gordon... more We derive the long-time asymptotics for solutions of the discrete 1D Schrödinger and Klein–Gordon equations.

Research paper thumbnail of On nonlinear wave equations with parabolic potentials

Journal of Spectral Theory, 2013

ABSTRACT We introduce a new class of piece-wise quadratic potentials for nonlinear wave equations... more ABSTRACT 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.

Research paper thumbnail of Dispersive estimates for the 2D wave equation

Russian Journal of Mathematical Physics, 2010

We obtain a dispersive long-time decay with respect to weighted energy norms for solutions of the... more We obtain a dispersive long-time decay with respect to weighted energy norms for solutions of the 2D wave equation with generic potential. The decay extends results obtained by Murata for the 2D Schrödinger equation.

Research paper thumbnail of Dispersion estimates for discrete Schrödinger and Klein–Gordon equations

St Petersburg Mathematical Journal, 2010

ABSTRACT We derive the long-time asymptotics for solutions of the discrete 3D Schrödinger and Kle... more ABSTRACT We derive the long-time asymptotics for solutions of the discrete 3D Schrödinger and Klein-Gordon equations.

Research paper thumbnail of On asymptotic stability of solitary waves in discrete Klein–Gordon equation coupled to a nonlinear oscillator

Applicable Analysis, 2010

The long-time asymptotics is analysed for finite energy solutions of the 1D discrete Klein–Gordon... more The long-time asymptotics is analysed for finite energy solutions of the 1D discrete Klein–Gordon 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

Research paper thumbnail of On Asymptotic Stability of Kink for Relativistic Ginzburg–Landau Equations

Archive for Rational Mechanics and Analysis, 2011

We prove the asymptotic stability of kink for the nonlinear relativistic wave equations of the Gi... more We prove the asymptotic stability of 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.

Research paper thumbnail of Scattering of Solitons for Dirac Equation Coupled to a Particle

We establish soliton-like asymptotics for finite energy solutions to the Dirac equation coupled t... more 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.

Research paper thumbnail of Dispersion Decay and Scattering Theory

Research paper thumbnail of Scattering of solitons for Dirac equation coupled to a particle

Journal of Mathematical Analysis and Applications, 2011

We establish soliton-like asymptotics for finite energy solutions to the Dirac equation coupled t... more 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 symplectic projection onto solitary manifold in the Hilbert phase space.

Research paper thumbnail of On Dispersive Decay for Discrete 3D Schr\"odinger and Klein-Gordon Equations

We derive the long-time decay in weighted norms for solutions of the discrete 3D Schr\"odinger an... more We derive the long-time decay in weighted norms for solutions of the discrete 3D Schr\"odinger and Klein-Gordon equations.

Research paper thumbnail of Long time decay for 2D Klein–Gordon equation

Journal of Functional Analysis, 2010

We obtain a dispersive long-time decay in weighted energy norms for solutions of the 2D Klein–Gor... more We obtain a dispersive long-time decay in weighted energy norms for solutions of the 2D Klein–Gordon equations. The decay extends the results obtained by Jensen, Kato and Murata for the equations of Schrödinger's type by the spectral approach. For the proof we modify the approach to make it applicable to relativistic equations.

Research paper thumbnail of On the asymptotic stability of solitary waves in the discrete Schrödinger equation coupled to a nonlinear oscillator

Nonlinear Analysis-theory Methods & Applications, 2009

a b s t r a c t The long-time asymptotics is analyzed for finite energy solutions of the 1D discr... more a b s t r a c t 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 U(1). For initial states close to a solitary wave, the solution converges to a sum of another solitary wave and a dispersive wave which is a solution to the free Schrödinger equation. The proofs use the strategy of Buslaev-Perelman [V.S. Buslaev, G.S. Perelman, On the stability of solitary waves for nonlinear Schrödinger equations, Amer. Math. Soc. Trans. 2 164 (1995), 75-98]: the linearization of the dynamics on the solitary manifold, the symplectic orthogonal projection, method of majorants, etc.

Research paper thumbnail of On Asymptotic Stability of Solitary Waves in Discrete Schr\"odinger Equation Coupled to Nonlinear Oscillator

The long-time asymptotics is analyzed for finite energy solutions of the 1D discrete Schr\"odinge... more The long-time asymptotics is analyzed for finite energy solutions of the 1D discrete Schr\"odinger 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 Schr\"odinger 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.

Research paper thumbnail of Weighted energy decay for 1D wave equation

Journal of Mathematical Analysis and Applications, 2010

We obtain a dispersive long time decay in weighted energy norms for solutions to the 1D wave equa... more We obtain a dispersive long time decay in weighted energy norms for solutions to the 1D wave equation with generic potential. The decay extends the results obtained by Murata for the 1D Schrödinger equation.

Research paper thumbnail of On dispersive properties of discrete 2D Schrödinger and Klein–Gordon equations

Journal of Functional Analysis, 2008

We derive the long-time asymptotics for solutions of the discrete 2D Schrödinger and Klein–Gordon... more We derive the long-time asymptotics for solutions of the discrete 2D Schrödinger and Klein–Gordon equations.

Research paper thumbnail of Weighted Energy Decay for 1D Klein–Gordon Equation

Communications in Partial Differential Equations, 2010

We obtain a dispersive long-time decay in weighted energy norms for solutions of the 1D Klein–Gor... more 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 Schrödinger's type by the spectral approach. For the proof we modify the approach to make it applicable to relativistic equations.

Research paper thumbnail of On Asymptotic Stability of Solitary Waves in a Nonlinear Schr\"odinger Equation

The long-time asymptotics is analyzed for finite energy solutions of the 1D Schrödinger equation ... more 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.

Research paper thumbnail of Weighted energy decay for 3D Klein–Gordon equation

Journal of Differential Equations, 2010

We obtain a dispersive long-time decay in weighted energy norms for solutions of the 3D Klein–Gor... more 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 Schrödinger equation. For the proof we modify the spectral approach of Jensen and Kato to make it applicable to relativistic equations.

Research paper thumbnail of On asymptotic stability of moving kink for relativistic Ginzburg-Landau equation

We prove the asymptotic stability of standing kink for the nonlinear relativistic wave equations ... more 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.

Research paper thumbnail of On long-time decay for modified Klein-Gordon equation

We obtain a dispersive long-time decay in weighted energy norms for solutions of the Klein-Gordon... more 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.

Research paper thumbnail of Dispersive estimates for 1D discrete Schrödinger and Klein–Gordon equations

Applicable Analysis, 2006

We derive the long-time asymptotics for solutions of the discrete 1D Schrödinger and Klein–Gordon... more We derive the long-time asymptotics for solutions of the discrete 1D Schrödinger and Klein–Gordon equations.

Research paper thumbnail of On nonlinear wave equations with parabolic potentials

Journal of Spectral Theory, 2013

ABSTRACT We introduce a new class of piece-wise quadratic potentials for nonlinear wave equations... more ABSTRACT 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.