Standing waves of the quintic NLS equation on the tadpole graph

TL;DR

This paper studies standing wave solutions of the quintic nonlinear Schrödinger equation on a tadpole graph, characterizing their existence, stability, and variational properties across different frequencies and masses.

Contribution

It provides a comprehensive analysis of standing waves on a tadpole graph, including their variational characterization and stability properties, extending understanding of NLS on complex networks.

Findings

Standing waves exist for all negative frequencies with a broader mass range.

Ground states are only for specific mass intervals, while standing waves exist more broadly.

Existence of critical frequencies determines the stability and variational nature of the solutions.

Abstract

The tadpole graph consists of a circle and a half-line attached at a vertex. We analyze standing waves of the nonlinear Schr\"{o}dinger equation with quintic power nonlinearity equipped with the Neumann-Kirchhoff boundary conditions at the vertex. The profile of the standing wave with the frequency $\omega\in (-\infty,0)$ is characterized as a global minimizer of the quadratic part of energy constrained to the unit sphere in $L^6$. The set of minimizers includes the set of ground states of the system, which are the global minimizers of the energy at constant mass ($L^2$-norm), but it is actually wider. While ground states exist only for a certain interval of masses, the standing waves exist for every $\omega \in (-\infty,0)$ and correspond to a bigger interval of masses. It is shown that there exist critical frequencies $\omega_0$ and $\omega_1$ such that the standing waves are the ground states for $\omega \in [\omega_0,0)$, local minimizers of the energy at constant mass for $\omega \in (\omega_1,\omega_0)$, and saddle points of the energy at constant mass for $\omega \in (-\infty,\omega_1)$. Proofs make use of both the variational methods and the analytical theory for differential equations.