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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.10821 |
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| _version_ | 1866913043634454528 |
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| author | Pava, Jaime Angulo Munoz, Alexander |
| author_facet | Pava, Jaime Angulo Munoz, Alexander |
| contents | In this work, we study the existence and orbital (in)stability of certain standing-wave solutions for the cubic nonlinear Schrödinger equation (NLS) posed on a looping-edge graph $\mathcal{G}$, consisting of a circle and a finite number $N$ of infinite half-lines attached to a common vertex. We consider the self-adjoint realization $(\mathcal{H}_Z, D(\mathcal{H}_Z))$ of the Laplacian, where the domain $D(\mathcal{H}_Z)$ encodes on the half-lines a $δ'$-type vertex conditions (continuity of derivatives at the vertex, without requiring continuity of the wave function) and $Z \in \mathbb{R}\setminus\{0\}$. On the circle, we propose Jacobian elliptic profiles of dnoidal type combined with either trivial (zero) or soliton tail profiles on the half-lines with full derivative matching at the boundary. For the trivial tail case we establish orbital stability for all $Z \in \mathbb{R}\setminus\{0\}$, while for the non-trivial tail case (which requires $Z < 0$) we establish both existence and orbital (in)stability depending on the relative size of $N$, $Z$, and the phase velocity of the standing wave. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10821 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonlinear Schrödinger Equations on looping-edge graphs with $δ'$-type interactions Pava, Jaime Angulo Munoz, Alexander Analysis of PDEs In this work, we study the existence and orbital (in)stability of certain standing-wave solutions for the cubic nonlinear Schrödinger equation (NLS) posed on a looping-edge graph $\mathcal{G}$, consisting of a circle and a finite number $N$ of infinite half-lines attached to a common vertex. We consider the self-adjoint realization $(\mathcal{H}_Z, D(\mathcal{H}_Z))$ of the Laplacian, where the domain $D(\mathcal{H}_Z)$ encodes on the half-lines a $δ'$-type vertex conditions (continuity of derivatives at the vertex, without requiring continuity of the wave function) and $Z \in \mathbb{R}\setminus\{0\}$. On the circle, we propose Jacobian elliptic profiles of dnoidal type combined with either trivial (zero) or soliton tail profiles on the half-lines with full derivative matching at the boundary. For the trivial tail case we establish orbital stability for all $Z \in \mathbb{R}\setminus\{0\}$, while for the non-trivial tail case (which requires $Z < 0$) we establish both existence and orbital (in)stability depending on the relative size of $N$, $Z$, and the phase velocity of the standing wave. |
| title | Nonlinear Schrödinger Equations on looping-edge graphs with $δ'$-type interactions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.10821 |