DCDS
Metric cycles, curves and solenoids
Vladimir Georgiev Eugene Stepanov
Discrete & Continuous Dynamical Systems - A 2014, 34(4): 1443-1463 doi: 10.3934/dcds.2014.34.1443
We prove that every one-dimensional real Ambrosio-Kirchheim current with zero boundary (i.e. a cycle) in a lot of reasonable spaces (including all finite-dimensional normed spaces) can be represented by a Lipschitz curve parameterized over the real line through a suitable limit of Cesàro means of this curve over a subsequence of symmetric bounded intervals (viewed as currents). It is further shown that in such spaces, if a cycle is indecomposable, i.e. does not contain ``nontrivial'' subcycles, then it can be represented again by a Lipschitz curve parameterized over the real line through a limit of Cesàro means of this curve over every sequence of symmetric bounded intervals, that is, in other words, such a cycle is a solenoid.
keywords: Ambrosio-Kirchheim currents solenoids. metric currents Lipschitz curves Normal currents
DCDS
Orbital stability and uniqueness of the ground state for the non-linear Schrödinger equation in dimension one
Daniele Garrisi Vladimir Georgiev
Discrete & Continuous Dynamical Systems - A 2017, 37(8): 4309-4328 doi: 10.3934/dcds.2017184

We prove that standing-waves which are solutions to the non-linear Schrödinger equation in dimension one, and whose profiles can be obtained as minima of the energy over the mass, are orbitally stable and non-degenerate, provided the non-linear term satisfies a Euler differential inequality. When the non-linear term is a combined pure power-type, then there is only one positive, symmetric minimum of prescribed mass.

keywords: Stability uniqueness Schrödinger
CPAA
Focusing nlkg equation with singular potential
Vladimir Georgiev Sandra Lucente
Communications on Pure & Applied Analysis 2018, 17(4): 1387-1406 doi: 10.3934/cpaa.2018068
We study the dynamics for the focusing nonlinear Klein Gordon equation with a positive, singular, radial potential and initial data in energy space. More precisely, we deal with
$u_{tt}-Δ u+m^2 u=|x|^{-a}|u|^{p-1}u$
with
$0 < a < 2$
. In dimension
$d≥3$
, we establish the existence and uniqueness of the ground state solution that enables us to define a threshold size for the initial data that separates global existence and blow-up. We find a critical exponent depending on
$a$
. We establish a global existence result for subcritical exponents and subcritical energy data. For subcritical exponents and critical energy some solutions blow-up, other solutions exist for all time due to the decomposition of the energy space of the initial data into two complementary sets.
keywords: Ground state, critical energy global existence blow up
DCDS
Smoothing-Strichartz estimates for the Schrodinger equation with small magnetic potential
Vladimir Georgiev Atanas Stefanov Mirko Tarulli
Discrete & Continuous Dynamical Systems - A 2007, 17(4): 771-786 doi: 10.3934/dcds.2007.17.771
The work treats smoothing and dispersive properties of solutions to the Schrödinger 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.
keywords: Strichartz estimates smoothing properties. Schrödinger Equation

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