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Existence and (in)stability of standing waves for the nonlinear Schrödinger Equations on looping-edge graphs with $δ'$-type interactions

Jaime Angulo Pava, Alexander Muñoz

TL;DR

This work addresses the existence and orbital stability of standing waves for the cubic nonlinear Schrödinger equation on a looping-edge graph $\mathcal{G}_N$ with $\delta'$-type vertex conditions. The authors construct local branches of standing waves by bifurcating from the periodic case $Z_2=0$ using the Implicit Function Theorem, producing profiles that couple a dnoidal on the circle with soliton tails on the rays, and they analyze stability via spectral perturbation theory and Krein-extension tools. The stability results are sharp in terms of the frequency parameter $\omega$ relative to the threshold $\frac{2N^2}{Z_1^2}$, with additional symmetry-restricted branches providing further insights for even $N$. The findings extend to other bound states on looping graphs and illustrate how vertex couplings influence spectral properties and orbital stability in quantum-graph NLS models.

Abstract

In this work, we investigate the existence and orbital (in)stability of several branches of 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. The model is endowed with $δ'$--type interaction boundary conditions at the vertex, which enforce continuity of the derivatives of the wave functions, while continuity of the wave function itself is not required. By means of the Implicit Function Theorem, we establish the existence of families of standing--wave profiles that converge, on the circular component of the graph, to Jacobi elliptic solutions of dnoidal type, coupled with soliton--type tail profiles on the half--lines. Tools from perturbation theory and Kreĭn--von Neumann extension theory for symmetric operators play a central role in the (in)stability analysis of such standing wave solutions. Our approach may be extended to other bound states for the NLS on looping graphs or more general non--compact metric graphs.

Existence and (in)stability of standing waves for the nonlinear Schrödinger Equations on looping-edge graphs with $δ'$-type interactions

TL;DR

This work addresses the existence and orbital stability of standing waves for the cubic nonlinear Schrödinger equation on a looping-edge graph with -type vertex conditions. The authors construct local branches of standing waves by bifurcating from the periodic case using the Implicit Function Theorem, producing profiles that couple a dnoidal on the circle with soliton tails on the rays, and they analyze stability via spectral perturbation theory and Krein-extension tools. The stability results are sharp in terms of the frequency parameter relative to the threshold , with additional symmetry-restricted branches providing further insights for even . The findings extend to other bound states on looping graphs and illustrate how vertex couplings influence spectral properties and orbital stability in quantum-graph NLS models.

Abstract

In this work, we investigate the existence and orbital (in)stability of several branches of standing--wave solutions for the cubic nonlinear Schrödinger equation (NLS) posed on a looping--edge graph , consisting of a circle and a finite number of infinite half--lines attached to a common vertex. The model is endowed with --type interaction boundary conditions at the vertex, which enforce continuity of the derivatives of the wave functions, while continuity of the wave function itself is not required. By means of the Implicit Function Theorem, we establish the existence of families of standing--wave profiles that converge, on the circular component of the graph, to Jacobi elliptic solutions of dnoidal type, coupled with soliton--type tail profiles on the half--lines. Tools from perturbation theory and Kreĭn--von Neumann extension theory for symmetric operators play a central role in the (in)stability analysis of such standing wave solutions. Our approach may be extended to other bound states for the NLS on looping graphs or more general non--compact metric graphs.
Paper Structure (11 sections, 17 theorems, 139 equations, 1 figure)

This paper contains 11 sections, 17 theorems, 139 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Review of the pure periodic case $Z_2=0$ and $Z_1<0$.
  4. Existence theory of standing waves
  5. Spectral theory
  6. Existence of standing wave solutions for $Z_2\neq 0$
  7. Spectral analysis for the case $Z_2\neq 0$ and $Z_1<0$
  8. Local and global well-posedness
  9. Stability theory
  10. Orbital (in)stability criterion
  11. Perturbation theory

Key Result

Theorem 1.1

Let $L>0$, $Z_1<0$, and $N\geqslant 1$. Consider $I\subset\mathbb R$ the open set defined in Theorem existence1 below (see also AM). For every fixed $\omega_0\in I$, there exist $\delta,\gamma>0$ and a unique smooth map of solutions to system uv, such that $\mathbf U(\omega_0)=\mathbf U_0$ and $Z_2(\omega_0)=0$, where $\mathbf U_0$ is the dnoidal--translated plus tail profile $\Theta_{\omega_0}$

Figures (1)

  • Figure 1: Orange: homoclinic orbit for the standing NLS equation for positive solutions. Green: Soliton tail to be considered. Red: the line $v=\frac{N}{Z_1}u$. Blue: intersection points of the periodic orbit at the derivative level of $q$. Purple: inner periodic orbit starting and ending at the same $q_i$. Purple + Green: a feasible profile $\Theta\in D_{Z_1, 0, N}$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 3.1
  • proof
  • Remark 1
  • Theorem 3.2
  • proof
  • Definition 3.1
  • Theorem 3.3
  • proof
  • ...and 29 more