Paper

Standing waves of the quintic NLS equation on the tadpole graph

Noja, Diego, Pelinovsky, Dmitry E

2020

Overview

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.

• Publisher: Cornell University Library, arXiv.org
• Subject: Boundary conditions / Critical frequencies / Differential equations / Ground state / Nonlinearity / Saddle points / Standing waves / Variational methods