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Traveling waves in periodic metric graphs via spatial dynamics

Stefan Le Coz, Dmitry E. Pelinovsky, Guido Schneider

TL;DR

This work develops a spatial-dynamics framework to define and analyze traveling waves for the focusing NLS on a periodic metric graph (the necklace). It proves that true spatially decaying travelling waves are generically obstructed by an infinite-dimensional center manifold, but travelling modulating pulses with small oscillatory tails exist near a Bloch-band bifurcation; the envelope satisfies a stationary NLS with leading-order amplitude $A$ governed by $(1/2)\omega''_{m_0}(\ell_0)A''-A+2\gamma|A|^2A=0$, and the authors justify this normal-form reduction via near-identity transformations and a local center–stable manifold construction under reversibility. Numerical simulations corroborate the existence and propagation of these modulated travelling waves on the necklace, including tail behavior, while a ladder-graph variant exhibits true solitary waves due to extra symmetry. The results connect spectral Floquet theory, center-manifold methods, and nonlinear envelope reductions to advance understanding of wave propagation on periodic quantum graphs and their potential applications in optics and quantum networks.

Abstract

The purpose of this work is to introduce a concept of traveling waves in the setting of periodic metric graphs. It is known that the nonlinear Schr{ö}dinger (NLS) equation on periodic metric graphs can be reduced asymptotically on long but finite time intervals to the homogeneous NLS equation, which admits traveling solitary wave solutions. In order to address persistence of such traveling waves beyond finite time intervals, we formulate the existence problem for traveling waves via spatial dynamics. There exist no spatially decaying (solitary) waves because of an infinite-dimensional center manifold in the spatial dynamics formulation. Existence of traveling modulating pulse solutions which are solitary waves with small oscillatory tails at very long distances from the pulse core is proven by using a local center-saddle manifold. We show that the variational formulation fails to capture existence of such modulating pulse solutions even in the singular limit of zero wave speeds where true (standing) solitary waves exist. Propagation of a traveling solitary wave and formation of a small oscillatory tail outside the pulse core is shown in numerical simulations of the NLS equation on the periodic graph.

Traveling waves in periodic metric graphs via spatial dynamics

TL;DR

This work develops a spatial-dynamics framework to define and analyze traveling waves for the focusing NLS on a periodic metric graph (the necklace). It proves that true spatially decaying travelling waves are generically obstructed by an infinite-dimensional center manifold, but travelling modulating pulses with small oscillatory tails exist near a Bloch-band bifurcation; the envelope satisfies a stationary NLS with leading-order amplitude governed by , and the authors justify this normal-form reduction via near-identity transformations and a local center–stable manifold construction under reversibility. Numerical simulations corroborate the existence and propagation of these modulated travelling waves on the necklace, including tail behavior, while a ladder-graph variant exhibits true solitary waves due to extra symmetry. The results connect spectral Floquet theory, center-manifold methods, and nonlinear envelope reductions to advance understanding of wave propagation on periodic quantum graphs and their potential applications in optics and quantum networks.

Abstract

The purpose of this work is to introduce a concept of traveling waves in the setting of periodic metric graphs. It is known that the nonlinear Schr{ö}dinger (NLS) equation on periodic metric graphs can be reduced asymptotically on long but finite time intervals to the homogeneous NLS equation, which admits traveling solitary wave solutions. In order to address persistence of such traveling waves beyond finite time intervals, we formulate the existence problem for traveling waves via spatial dynamics. There exist no spatially decaying (solitary) waves because of an infinite-dimensional center manifold in the spatial dynamics formulation. Existence of traveling modulating pulse solutions which are solitary waves with small oscillatory tails at very long distances from the pulse core is proven by using a local center-saddle manifold. We show that the variational formulation fails to capture existence of such modulating pulse solutions even in the singular limit of zero wave speeds where true (standing) solitary waves exist. Propagation of a traveling solitary wave and formation of a small oscillatory tail outside the pulse core is shown in numerical simulations of the NLS equation on the periodic graph.
Paper Structure (18 sections, 9 theorems, 141 equations, 9 figures)

This paper contains 18 sections, 9 theorems, 141 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Formulation of the evolution model
  3. Main results
  4. Organization of the paper
  5. Spatial dynamics formulation
  6. Linear theory
  7. Bifurcation of double imaginary eigenvalues
  8. Numerical approximation of eigenvalues
  9. Bifurcations of traveling modulating pulses
  10. Formal derivation of the normal form
  11. Justification of the normal form
  12. Near-identity transformations
  13. Construction of a local stable-center manifold
  14. Finding a homoclinic orbit under the reversibility constraint
  15. Numerical simulations
  16. ...and 3 more sections

Key Result

Theorem 1

Pick $m_0 \in {\mathbb N}$ and $\ell_0 \in \mathbb{B}$ with $\omega_{m_0}"(\ell_0) > 0$ and define Assume that the set of purely imaginary roots $\lambda \in i {\mathbb R}$ of the characteristic equations admit only simple nonzero roots, with the exception of a double zero root for $m = m_0$ and $\ell = \ell_0$. For every $N_0 > 0$, there exist $\varepsilon_0 > 0$ and $C_0 > 0$ such that for eve

Figures (9)

  • Figure 1: Schematic representation of a necklace graph.
  • Figure 2: Schematic representation of a ladder graph.
  • Figure 3: Eigenvalues $\lambda$ in the homogeneous case $L_1 = 2\pi$ and $L_2 = 0$. Blue dots show the pair of eigenvalues bifurcating from the origin for $k = 0$ and $\sigma > \frac{c^2}{4}$.
  • Figure 4: Numerical calculations of eigenvalues obtained from the characteristic equations \ref{['char-eq-omega']} for $L_1 = L_2 = \pi$ by the two different numerical approaches.
  • Figure 5: Real (left) and imaginary (right) parts of two pairs of eigenvalues coalescing in the parameter continuation in $L_1$ with $L_2 = 2\pi - L_1$.
  • ...and 4 more figures

Theorems & Definitions (23)

  • Theorem 1
  • Proposition 3.1
  • proof
  • Example 3.1
  • Lemma 3.1
  • proof
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Example 3.2
  • ...and 13 more