arXiv subjects

Vladimir Georgiev

Publications and source records attributed to Vladimir Georgiev.

65 records · Page 4Linked to original sources

arXiv2012

Decay and scattering of small solutions of pure power NLS in $\R$ with $p>3$ and with a potential

We prove decay and scattering of solutions of the Nonlinear Schröding-er equation (NLS) in ${\mathbf R}$ with pure power nonlinearity with exponent $3<p<5$ when the initial datum is small in $Σ$ (bounded energy and variance), in the presence of a linear inhomogeneity represented by a linear potential which is a real valued Schwarz function. We assume absence of discrete modes. The proof is analogous to the one for the translation invariant equation. In particular we find appropriate operators commuting with the linearization.

math.AP↗

arXiv2011

On the radiality of constrained minimizers to the Schroedinger-Poisson-Slater energy

We study the radial symmetry of minimizers to the Schroedinger-Poisson-Slater (S-P-S) energy.

math.AP↗

arXiv2011

Local energy decay for wave equation in the absence of resonance at zero energy in 3D

In this paper we study spectral properties associated to Schrodinger operator with potential that is an exponential decaying function. As applications we prove local energy decay for solutions to the perturbed wave equation and lack of resonances for the NLS.

math.AP↗

arXiv2010

Symmetry and uniqueness of minimizers of Hartree type equations with external Coulomb potential

In the present article we study the radial symmetry of minimizers of the energy functional, corresponding to the repulsive Hartree equation in external Coulomb potential. To overcome the difficulties, resulting from the "bad" sign of the nonlocal term, we modify the reflection method and then, by using Pohozaev integral identities we get the symmetry result.

math.AP↗

arXiv2010

Nonlinear instability of linearly unstable standing waves for nonlinear Schrödinger equations

We study the instability of standing waves for nonlinear Schrödinger equations. Under a general assumption on nonlinearity, we prove that linear instability implies orbital instability in any dimension. For that purpose, we establish a Strichartz type estimate for the propagator generated by the linearized operator around standing wave.

math.AP↗

arXiv2010

Finite time blow-up for a wave equation with a nonlocal nonlinearity

In this article, we study the local existence of solutions for a wave equation with a nonlocal in time nonlinearity. Moreover, a blow-up results are proved under some conditions on the dimensional space, the initial data and the nonlinear forcing term.

math.AP↗

arXiv2005

Smoothing - Strichartz Estimates for the Schrodinger Equation with small Magnetic Potential

The work treats smoothing and dispersive properties of solutions to the Schrodinger equation with magnetic potential. Under suitable smallness assumption on the potential involving scale invariant norms we prove smoothing - Strichartz estimate for the corresponding Cauchy problem. An application that guarantees absence of pure point spectrum of the corresponding perturbed Laplace operator is discussed too.

math.AT↗

arXiv2005

Scale invariant energy smoothing estimates for the Schrödinger Equation with small Magnetic Potential

We consider some scale invariant generalizations of the smoothing estimates for the free Schrödnger equation obtained by Kenig, Ponce and Vega. Applying these estimates and using appropriate commutator estimates, we obtain similar scale invariant smoothing estimates for perturbed Schrödnger equation with small magnetic potential.

math.AP↗

arXiv2004

Blow Up for the Semilinear Wave Equation in Schwarzschild Metric

We study the semilinear wave equation in Schwarzschild metric (3+1 dimensional space--time). First, we establish that the problem is locally well--posed in $\cs H^σ$ for any $σ\geq 1$; then we prove the blow up of the solution for every real $p \in ]1,1+\sqrt{2}[$ and non--negative non--trivial initial data.

math-ph↗

arXiv2003

Solitary waves for Maxwell-Schrodinger equations

In this paper we study the solitary waves for the coupled Schrödinger - Maxwell equations in three-dimensional space. We prove the existence of a sequence of radial solitary waves for these equations with a fixed $L^2$ norm. We study the asymptotic behavior and the smoothness of these solutions. We show also the fact that the eigenvalues are negative and the first one is isolated.

math.AP↗

arXiv2002

On the continuity of the solution operator to the wave map system

We investigate the continuity properties of the solution operator to the wave map system from the flat Minkowski space to a general nonflat target of arbitrary dimension, and we prove by an explicit class of counterexamples that this map is not uniformly continuous in the critical norms on any neighbourhood of zero.

math.AP↗