Nonlinear Schrödinger Equations on looping-edge graphs with $δ'$-type interactions

TL;DR

The paper addresses the cubic nonlinear Schrödinger equation on looping-edge quantum graphs with δ′-type vertex interactions, establishing existence and orbital (in)stability of standing waves formed by a dnoidal profile on the circle together with either trivial tails or tail solitons on the attached rays. It develops a local and global well-posedness theory in the energy space, analyzes the linearized operators via Krein-space extension theory and the Grillakis–Shatah–Strauss framework, and derives explicit stability criteria depending on the number of rays $N$, vertex strength $Z$, and frequency $\omega$. A key methodological contribution is a boundary-system approach in Krein spaces to classify self-adjoint Laplacian extensions, yielding δ-type and δ′-type couplings and enabling unitary dynamics on looping-edge networks. The results provide a rigorous foundation for nonlinear dynamics on quantum graphs with singular vertex interactions and supply explicit constructions for tadpole and looping-edge geometries.

Abstract

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. Our main goal is to take the first steps toward understanding the dynamics of the NLS under $δ'$-type interactions. Here we consider a negative $Z$-strength at the vertex, where continuity of the wave function is not mandatory. On the circle, we propose Jacobi elliptic profiles of dnoidal type combined with soliton tail profiles on the half-lines, and we establish the existence and (in)stability of these solutions depending on the relative size of $N$, $Z$, and the phase velocity of the standing wave. Tools from Krein--von Neumann extension theory for symmetric operators play a fundamental role in our stability analysis. We also develop a local and global well-posedness theory for the NLS in the energy space $H^1(\mathcal{G})$. Finally, we present an approach to characterize the domains of self-adjoint extensions of the Laplace operator on a looping-edge graph, which incorporate the continuity of derivatives at the vertex.

Figures (8)

• Figure 1:
• Figure 2: $\Phi_\omega$ dnoidal $+$ trivial soliton profiles.
• Figure 3: $\Phi_\omega$-dnoidal translated $+$ tail profiles.
• Figure 4: For a looping-edge graph with $N=3$, we exhibit profile elements in $D_Z'$ with $\Phi$ as centered periodic dnoidal wave and trivial tail-type profiles on all half-lines.
• Figure 5: Graph of the periodic function $\chi_0(x)$ with $k=\frac{1}{2}$.

Theorems & Definitions (56)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: Local and global well-posedness
• Proposition 3.1
• proof
• Corollary 3.2
• Definition 3.1
• Proposition 3.3
• proof
• Proposition 3.4