arXiv subjects

Nataliia Goloshchapova

Publications and source records attributed to Nataliia Goloshchapova.

9 recordsLinked to original sources

arXiv2023

Variational and stability properties of coupled NLS equations on the star graph

We consider variational and stability properties of a system of two coupled nonlinear Schrödinger equations on the star graph $Γ$ with the $δ$ coupling at the vertex of $Γ$. The first part is devoted to the proof of an existence of the ground state as the minimizer of the constrained energy in the cubic case. This result extends the one obtained recently for the coupled NLS equations on the line. In the second part, we study stability properties of several families of standing waves in the case of a general power nonlinearity. In particular, we study one-component standing waves $e^{iωt}(Φ_1(x), 0)$ and $e^{iωt}(0, Φ_2(x))$. Moreover, we study two-component standing waves $e^{iωt}(Φ(x), Φ(x))$ for the case of power nonlinearity depending on a unique power parameter $p$. To our knowledge, these are the first results on variational and stability properties of coupled NLS equations on graphs.

math.AP↗

arXiv2022

Dynamical and variational properties of the NLS-$δ'_s$ equation on the star graph

We study the nonlinear Schrödinger equation with $δ'_s$ coupling of intensity $β\in\mathbb{R}\setminus\{0\}$ on the star graph $Γ$ consisting of $N$ half-lines. The nonlinearity has the form $g(u)=|u|^{p-1}u, p>1.$ In the first part of the paper, under certain restriction on $β$, we prove the existence of the ground state solution as a minimizer of the action functional $S_ω$ on the Nehari manifold. It appears that the family of critical points which contains a ground state consists of $N$ profiles (one symmetric and $N-1$ asymmetric). In particular, for the attractive $δ'_s$ coupling ($β<0$) and the frequency $ω$ above a certain threshold, we managed to specify the ground state. The second part is devoted to the study of orbital instability of the critical points. We prove spectral instability of the critical points using Grillakis/Jones Instability Theorem. Then orbital instability for $p>2$ follows from the fact that data-solution mapping associated with the equation is of class $C^2$ in $H^1(Γ)$. Moreover, for $p>5$ we complete and concertize instability results showing strong instability (by blow up in finite time) for the particular critical points.

math.AP↗

arXiv2021

A nonlinear Klein-Gordon equation on a star graph

We study local well-posedness and orbital stability/instability of standing waves for a first order system associated with a nonlinear Klein-Gordon equation on a star graph. The proof of the well-posedness uses a classical fixed point argument and the Hille-Yosida theorem. Stability study relies on the linearization approach and recent results for the NLS equation with the $δ$-interaction on a star graph.

math.SP↗

arXiv2021

Instability of ground states for the NLS equation with potential on the star graph

We study the nonlinear Schrödinger equation with an arbitrary real potential $V(x)\in (L^1+L^\infty)(Γ)$ on a star graph $Γ$. At the vertex an interaction occurs described by the generalized Kirchhoff condition with strength $-γ<0$. We show the existence of ground states $φ_ω(x)$ as minimizers of the action functional on the Nehari manifold under additional negativity and decay conditions on $V(x)$. Moreover, for $V(x)=-\dfracβ{x^α}$, in the supercritical case, we prove that the standing waves $e^{iωt}φ_ω(x)$ are orbitally unstable in $H^{1}(Γ)$ when $ω$ is large enough. Analogous result holds for an arbitrary $γ\in\mathbb{R}$ when the standing waves have symmetric profile.

math.AP↗

arXiv2020

Blow-up and strong instability of standing waves for the NLS-$δ$ equation on a star graph

We study strong instability (by blow-up) of the standing waves for the nonlinear Schrödinger equation with $δ$-interaction on a star graph $Γ$. The key ingredient is a novel variational technique applied to the standing wave solutions being minimizers of a specific variational problem. We also show well-posedness of the corresponding Cauchy problem in the domain of the self-adjoint operator which defines $δ$-interaction. This permits to prove virial identity for the $H^1$- solutions to the Cauchy problem. We also prove certain strong instability results for the standing waves of the NLS-$δ'$ equation on the line.

math.AP↗

arXiv2019

Extension theory approach in the stability of the standing waves for the NLS equation with point interactions on a star graph

The aim of this work is to demonstrate the effectiveness of the extension theory of symmetric operators in the investigation of the stability of standing waves for the nonlinear Schrödinger equations with two types of non-linearities (power and logarithmic) and two types of point interactions ($δ$- and $δ'$-) on a star graph. Our approach allows us to overcome the use of variational techniques in the investigation of the Morse index for self-adjoint operators with non-standard boundary conditions which appear in the stability study. We also demonstrate how our method simplifies the proof of the stability results known for the NLS equation with point interactions on the line.

math.SP↗

arXiv2019

On the orbital instability of excited states for the NLS equation with the $δ$-interaction on a star graph

We study the nonlinear Schrödinger equation (NLS) on a star graph $\mathcal{G}$. At the vertex an interaction occurs described by a boundary condition of delta type with strength $α\in \mathbb{R}$. We investigate an orbital instability of the standing waves $e^{iωt}\mathbfΦ(x)$ of NLS-$δ$ equation with attractive power nonlinearity on $\mathcal{G}$ when the profile $Φ(x)$ has mixed structure (i.e. has bumps and tails). In our approach we essentially use the extension theory of symmetric operators by Krein - von Neumann, and the analytic perturbations theory, avoiding the variational techniques standard in the stability study. We also prove orbital stability of the unique standing wave solution of NLS-$δ$ equation with repulsive nonlinearity.

math.SP↗

arXiv2018

On the standing waves of the NLS-log equation with point interaction on a star graph

We study a nonlinear Schrödinger equation with logarithmic nonlinearity on a star graph $\mathcal{G}$. At the vertex an interaction occurs described by a boundary condition of delta type with strength $α\in \mathbb{R}$. We investigate orbital stability and spectral instability of the standing wave solutions $e^{iωt}\mathbfΦ(x)$ to the equation when the profile $\mathbfΦ(x)$ has mixed structure (i.e. has bumps and tails). In our approach we essentially use the extension theory of symmetric operators by Krein - von Neumann, and the analytic perturbations theory.

math.SP↗

arXiv2017

Stability of standing waves for NLS-log equation with $δ$-interaction

We study analytically the orbital stability of the standing waves with a peak-Gausson profile for a nonlinear logarithmic Schrödinger equation with $δ$-interaction (attractive and repulsive). A major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing wave. This is overcome by the perturbation method, the continuation arguments, and the theory of extensions of symmetric operators.

math.SP↗