Ground States of the Nonlinear Schr{\"o}dinger Equation on the Tadpole Graph with a Repulsive Delta Vertex Condition

TL;DR

This paper investigates the existence and shape of ground states for the nonlinear Schrödinger equation on a tadpole graph with a repulsive delta condition, using variational, differential equations, and numerical methods.

Contribution

It establishes conditions for the existence of ground states on a tadpole graph with a repulsive delta vertex, and analyzes their shape through combined analytical and numerical approaches.

Findings

Existence of ground states for very small or large loop sizes.

Characterization of ground state shapes via differential equations.

Numerical validation of theoretical results.

Abstract

We consider the stationary nonlinear Schr{\"o}dinger equation set on a tadpole graph with a repulsive delta vertex condition between the loop and the tail of the tadpole. We establish the existence of an action ground state when the size of the loop is either very small or very large. Our analysis relies on variational arguments, such as profile decomposition. When it exists, we study the shape of the ground state using ordinary differential equations arguments, such as the study of period functions. The theoretical results are completed with a numerical study.