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

TL;DR

This paper investigates the existence, characterization, and stability of ground state solutions for the nonlinear Schrödinger equation with a delta prime coupling on a star graph, revealing conditions for stability and instability.

Contribution

It establishes the existence of ground states under certain conditions, characterizes their profiles, and analyzes their spectral and orbital instability, including blow-up behavior for specific nonlinearities.

Findings

Existence of ground states as minimizers on the Nehari manifold.

Identification of symmetric and asymmetric profiles of critical points.

Spectral and orbital instability results, including finite-time blow-up for strong nonlinearities.

Abstract

We study the nonlinear Schr\"odinger equation with $\delta'_s$ coupling of intensity $\beta\in\mathbb{R}\setminus\{0\}$ on the star graph $\Gamma$ 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 $\beta$, we prove the existence of the ground state solution as a minimizer of the action functional $S_\omega$ 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 $\delta'_s$ coupling ($\beta<0$) and the frequency $\omega$ 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 $H^1(\Gamma)$. Moreover, for $p>5$ we complete and concertize instability results showing strong instability (by blow up in finite time) for the particular critical points.