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

TL;DR

This work addresses the existence and shape of action ground states for the cubic nonlinear Schrödinger equation on a tadpole graph with a repulsive delta vertex. It combines variational methods on the Nehari manifold with profile decomposition and phase-portrait analysis to characterize ground-state shapes, proving existence for both small and large loop lengths $L$ and classifying nonnegative stationary states via boundary data on phase-plane curves. The results reveal a rich phenomenology where ground-state shape transitions from half-soliton–like to loop-soliton–like as $L$ grows, with precise threshold lengths; non-even states also exist under suitable length constraints. Numerical simulations corroborate the analytical predictions, illustrating the data-curves and confirming the predicted families of ground states across $L$ and $eta$ (vertex strength). The study clarifies how a repulsive delta vertex influences standing waves on metric graphs and lays groundwork for extensions to energy-ground-state problems on similar graph geometries.

Abstract

We consider the stationary nonlinear Schr{ö}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.

Figures (9)

• Figure 1: The tadpole graph $\mathcal{G}_L$.
• Figure 2: Phase portrait of the cubic nonlinear Schrödinger equation. The small dashed-curve is $\mathcal{E}_{\omega, -0.3}$ (dnoidal-type solution), the continuous curve is $\mathcal{E}_{\omega, 0}$ (hyperbolic secant), the large dashed-curve is $\mathcal{E}_{\omega, 0.5}$ (cnoidal-type solution) and the bold blue points are the fixed points, corresponding to $(-\sqrt{\omega},0)$, $(0,0)$ and $(\sqrt{\omega},0)$; the arrows represent the flow of the equation; for $E = -0.3$ and $(p_0, q_0) \in \mathcal{E}_{\omega, E}$, the sets $\mathcal{E}_{\omega, -0.3}^{p_0, 1}$ (red), $\mathcal{E}_{\omega, -0.3}^{p_0, 2}$ (green) and $\mathcal{E}_{\omega, -0.3}^{p_0, 3}$ (yellow) have been represented.
• Figure 3: $\mathcal{E}_\omega$-curve of level $0$ (blue) and $\Gamma_\omega$-level curve of level $\gamma$ (red); $\Gamma_{\omega, \gamma}^1$ is the continuous curve, $\Gamma_{\omega, \gamma}^2$ is the large-dashed curve and $\Gamma_{\omega, \gamma}^3$ is the small-dashed curve. The parameters are $\omega = 1$ and $\gamma = 1/2$.
• Figure 4: $\mathcal{E}_\omega$-curve of level $0$ and $\mathcal{E}_\omega (\tilde{p}, \tilde{q})$ (blue); $\Gamma_{\omega, \gamma}^1$ (red, left) and $\Gamma_{\omega, \gamma}^2$ (red, right); and the solution built in Step $1$ of the proof of Lemma \ref{['lemShapeFinite']} (green).
• Figure 5: $\mathcal{E}_\omega$-curve of level $0$ (blue); $\Gamma_{\omega, \gamma}^4$ (red); and the solution built in Step $1$ of the proof of Lemma \ref{['lemNonEvenExistence']} (green).

Theorems & Definitions (42)

• Theorem 1.1
• Proposition 2.1
• Remark 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Lemma 3.1
• Lemma 3.2
• proof : Proof of Lemma \ref{['lemExistenceCriterion']}
• Lemma 3.3