Multi-Peak Ground-State Solutions for the Nonlinear Schrödinger Equation with External Potential

Mathematical research on multi-peak ground-state solutions of the stationary NLS with external potential in the high-energy regime ($E \to \infty$), using concentration compactness and Lyapunov–Schmidt reduction. The work establishes existence and classification of ground-state configurations with multiple concentration peaks at a non-degenerate local maximum of the potential $V$.

Authors

• Eduard Kirr — Associate Professor, University of Illinois at Urbana-Champaign
• Ethan Knox — Graduate Student, University of Illinois at Urbana-Champaign

Overview

We study the stationary NLS equation with external potential in the high-energy regime:

$$\left(-\Delta + V + E\right)\psi + \sigma\left|\psi\right|^{2p}\psi = 0$$

as $E \to \infty$. The work is motivated by optical fiber physics — the envelope equation in silica fibers with Kerr nonlinearity ($\chi^{\left(3\right)}$) maps to this stationary Schrödinger equation, where the refractive-index profile becomes an external potential $V\left(\mathbf{x}\right)$.

Hypotheses on the Potential

Following Kirr, Kevrekidis, and Pelinovsky (2011), Floer and Weinstein (1986), and Kirr and Natarajan (2018):

• (V1) $V \in L^\infty\left(\mathbb{R}^n\right)$ — bounded and measurable.
• (V2) $\lim_{\left|x\right|\to\infty} V\left(x\right) = 0$ — vanishes at infinity.
• (V3) $V$ is $C^2$ near $x_0$, a non-degenerate critical point: $\nabla V\left(x_0\right) = 0$ and $H_V\left(x_0\right)$ invertible.
• (V4) All eigenvalues of $H_V\left(x_0\right)$ are strictly negative — $x_0$ is a strict local maximum.

Key Framework

1. Rescaling. Define $u_E\left(x\right) = E^{-\frac{1}{2p}}\psi_E\left(E^{-\frac12}x\right)$, which transforms the problem into a family of equations parametrized by $R = E^{-\frac12} \to 0$.
2. Concentration compactness decomposition. Each solution decomposes into $M$ well-separated profiles $u_R$ centered at distinct points $z_1\left(R\right), \ldots, z_M\left(R\right)$, with a remainder $h_R$:

$$u_E\left(x\right) = \sum_{i=1}^{M} u_R\left(x - z_i\left(R\right)\right) + h_R\left(x\right)$$

3. Lyapunov–Schmidt reduction. Projection onto the kernel of the linearized operator yields a finite-dimensional algebraic system for peak positions:

$$H_V\left(x_0\right) y_i = \chi \sum_{k \in N_i} \alpha_{ki}\left(y_k - y_i\right)$$

where $H_V\left(x_0\right)$ is the Hessian of $V$ at the critical point $x_0$, $\alpha_{ki} = \frac{Q_R\left(\left|z_k - z_i\right|\right)}{Q_R\left(m_R\right)} \in \left[0,1\right]$ are normalized interaction coefficients, and $\chi > 0$ is the overall interaction strength.

4. Perturbation system. The full problem at finite energy $R > 0$ is captured by $F_i\left(\delta y, R\right) = 0$, with $\delta y$ denoting peak displacements from the $R = 0$ limiting configuration.

Main Results

1. Classification of Limiting Configurations ($R = 0$)

Theorem 3.1. Under Hypotheses (V1)–(V4), the admissible solutions of the algebraic system with all $\alpha_{ki} \geq 0$ are:

• (a) Collinear ($M \geq 2$) — Peaks aligned along a single eigenvector $\mathbf{r}_1$ of $H_V\left(x_0\right)$.
• (b) Isosceles triangle ($M = 3$) — Peaks with $\alpha_{12} = \alpha_{13} = -\frac{\lambda_2}{3}$; requires the eigenvalue constraint $\lambda_2 = 3\lambda_1$.
• (c) Equilateral triangle ($M = 3$) — All peaks at unit mutual distance; $\alpha_{12} = \alpha_{13} = -\frac{\lambda_2}{3}$, $\alpha_{23} = \frac{\lambda_2}{6} - \frac{\lambda_1}{2}$; requires $\frac{\left|\lambda_2\right|}{\left|\lambda_1\right|} \geq 3$.
• (d) Rotational equilateral ($M = 3$, angle $\theta$) — Equilateral triangle rotated by $\theta$ relative to the eigenvector basis, with admissible $(\mu, \theta)$ regions given below.

Admissibility regions (where $\mu = \frac{\left|\lambda_2\right|}{\left|\lambda_1\right|}$):

$\mu$-Range	$\theta$-Range
$0 < \mu < \frac{1}{3}$	$\left[0, \frac{\pi}{6} - \frac{\varphi}{2}\right] \cup \left[\frac{\pi}{6} + \frac{\varphi}{2}, \frac{\pi}{2} - \frac{\varphi}{2}\right]$
$\frac{1}{3} \leq \mu \leq 3$	$\left[0, \frac{\pi}{2}\right]$ (all orientations)
$\mu > 3$	$\left[\frac{\varphi}{2}, \frac{\pi}{3} - \frac{\varphi}{2}\right] \cup \left[\frac{\pi}{3} + \frac{\varphi}{2}, \frac{\pi}{2}\right]$

where $\nu\left(\mu\right) = \frac{\mu+1}{2\left|\mu-1\right|}$ and $\varphi\left(\mu\right) = \arccos\left(\nu\left(\mu\right)\right)$.

The peak positions satisfy $\text{span}\lbrace y_1, \ldots, y_M\rbrace \subseteq U$, the unstable subspace of $H_V\left(x_0\right)$.

2. Perturbation System ($R > 0$)

• Explicit solution at $R = 0$ (Proposition 4.1). For the equilateral triangle at $\theta = 0$, define $\beta = \frac{1}{2}\ln\left(\frac{\alpha_{12}}{\alpha_{23}}\right)$ and $\gamma = \frac{2\beta}{3\sqrt{3}}$. Then:

$$\delta y_1^0 = -\gamma, \mathbf{r}_2, \qquad \delta y_{2,3}^0 = \mp\beta, \mathbf{r}_1 + \frac{\gamma}{2},\mathbf{r}_2$$

solves $F_i\left(\delta y, 0\right) = 0$ and satisfies the center-of-mass condition $\sum_i \delta y_i^0 = 0$.

• Four-dimensional kernel (Proposition 4.2). The linearization $D_{\delta y} F_i\left(\delta y^0, 0\right)$ has kernel $\Omega = \lbrace\left(w_1, w_2, w_3\right) \in \left(\mathbb{R}^2\right)^3 : \left \langle w_k - w_i, y_k - y_i \right \rangle = 0\rbrace$, decomposing as:
  • 2 translational modes $\lbrace\left(v,v,v\right) : v \in \mathbb{R}^2\rbrace$
  • 1 rotational mode $w_{\mathrm{rot}}$
  • 1 breathing mode $w_{\mathrm{br}} = \left(y_1^\perp, y_2^\perp, y_3^\perp\right)$

3. Kernel Resolution and Gauge-Fixing

• Projection decomposition. The system is split into perpendicular ($P_\perp F$) and parallel ($P_w F$, $w \in \Omega$) projections.
• Gauge-fixing constraints:
  • (G1) $\delta y_1 \perp \mathbf{r}_1$ — first peak has no displacement along $\mathbf{r}_1$.
  • (G2) $\left(\delta y_3 - \delta y_2\right) \parallel \mathbf{r}_1$ — base of triangle displaces only along $\mathbf{r}_1$.
• Rotational equivariance (Proposition 5.1). The one-parameter family $y_i\left(\theta\right) = R\left(\theta\right) y_i'$ generates a smooth family $\delta y^0\left(\theta\right)$ of solutions at $R = 0$.
• Center-of-mass identity (Proposition 5.2). $\sum_i F_i\left(\delta y, R\right) = H_V \frac{\sum_i \delta y_i}{\tilde{m}_R}$, so $\sum_i F_i = 0$ iff $\sum_i \delta y_i = 0$.

Open Problems

1. Prove Conjecture 5.1 (gauge-fixed invertibility). Compute the $4 \times 4$ linearized operator on the gauge-fixed subspace explicitly and verify nonsingularity for $\mu > 3$.
2. Extend to all admissible $\theta$. The perturbation analysis is at $\theta = 0$; extending to $R > 0$ for all admissible $\theta$ simultaneously requires uniform estimates.
3. Stability. Multi-peak branches with $M \geq 2$ have at least two negative directions, implying orbital instability. Adapting asymptotic stability techniques to multi-peak states remains open.

Repository Structure

nls-research/
├── report/                                  # Main research paper
│   ├── Research.lyx                         # Primary LyX source (source of truth, ~7400 lines)
│   ├── Research.tex                         # LyX-exported LaTeX (~1200 lines)
│   ├── Research.pdf                         # Compiled PDF
│   └── Research.bib                         # BibTeX bibliography (14 entries)
│
├── figures/                                 # Figure generation and output
│   ├── figures.py                           # Single-file script generating all figures (~1400 lines)
│   ├── figure_1_admissibility_binding.png
│   ├── figure_2_peak_configurations.png
│   ├── figure_2b_perturbation.png
│   ├── figure_3_alpha_vs_theta_vs_mu.png
│   ├── figure_4_profile_decomposition.png
│   ├── figure_5_concentration_vs_peaks.png
│   ├── figure_6_interaction_decay.png
│   └── figure_7_methodology_flowchart.png
│
├── requirements.txt                         # Python dependencies (numpy, matplotlib)
├── .forgejo/workflows/                      # CI: Forgejo → GitHub mirror
└── .gitignore

Figures

All figures are generated by figures/figures.py and output as PNG at 300 DPI. The create_figure_2 function produces both Figure 2 and Figure 2b.

#	Filename	Description
1	figure_1_admissibility_binding	$(\mu, \theta)$ admissibility phase diagram with binding constraints
2	figure_2_peak_configurations	Schematic configurations: collinear ($M=2$, $M=5$), isosceles, and equilateral triangles
2b	figure_2b_perturbation	Perturbation displacement $\delta y^0$ for the equilateral triangle at $\theta = 0$
3	figure_3_alpha_vs_theta_vs_mu	Interaction coefficients $\alpha_{12}$, $\alpha_{13}$, $\alpha_{23}$ vs $\theta$ for various $\mu$
4	figure_4_profile_decomposition	Concentration-compactness decomposition for $M=3$ peaks with remainder $h_R$
5	figure_5_concentration_vs_peaks	Evolution of peak structure with increasing $E$; $Rz_i \to x_0$ convergence
6	figure_6_interaction_decay	Semilog plot of $Q_R(d)$ decay and normalized $\alpha_{ki}$ vs distance ratio
7	figure_7_methodology_flowchart	Analytical pipeline: Kirr–Natarajan (2018) framework → Kirr–Knox (2026) contributions

Paper Outline

The report (Research.lyx, exported as Research.tex) uses the extarticle document class and is organized as follows:

1. Introduction — Physical motivation from optical fiber physics (NLS, Kerr nonlinearity, refractive-index profiles as external potentials); overview of contributions.
2. Setup and Preliminaries
   • 2.1 Hypotheses (V1)–(V4) on the potential
   • 2.2 Rescaling and concentration compactness decomposition
   • 2.3 Lyapunov–Schmidt reduction (two lemmas)
   • 2.4 The reduced algebraic system; interaction functional $Q_R$; coefficients $\alpha_{ki}$; interaction strength $\chi$; corollary on unstable subspace
3. Classification of Limiting Configurations — Theorem 3.1 with proofs for each configuration type
   • 3.1 Collinear configurations
   • 3.2 Isosceles triangle
   • 3.3 Equilateral triangle
   • 3.4 Rotational equilateral triangle
   • 3.5 Admissibility analysis with unified constraint; summary table
4. The Perturbation System
   • 4.1 Formulation of $F_i\left(\delta y, R\right) = 0$
   • 4.2 Explicit solution at $R = 0$ (Proposition 4.1)
   • 4.3 Kernel of the linearization (Proposition 4.2): 4D kernel with translational, rotational, and breathing modes
5. Kernel Resolution and Gauge-Fixing
   • 5.1 Projection decomposition ($P_\perp F$, $P_w F$)
   • 5.2 Gauge-fixing constraints (G1)–(G2)
   • 5.3 Rotational equivariance at $R = 0$ (Proposition 5.1, Corollary 5.1)
   • 5.4 Extension to $R > 0$: Conjecture 5.1 (gauge-fixed invertibility), Proposition 5.2 (center-of-mass identity)
6. Conclusion, Open Problems, and Acknowledgements — Summary of contributions; three concrete open problems
7. Notation Reference — Comprehensive notation tables

Prerequisites

Python

• Python 3.8+
• numpy
• matplotlib

LaTeX / LyX

• LyX 2.4+ (for editing the .lyx source)
• A standard LaTeX distribution (TeX Live or MiKTeX) with packages: extarticle, geometry, amsmath, amsthm, amssymb, graphicx, booktabs, tabularx, units, float

Setup and Usage

Generating Figures

cd figures
pip install -r ../requirements.txt
python figures.py

Figures are written to figures/ as PNG files at 300 DPI.

Compiling the Paper

From report/:

pdflatex Research.tex
bibtex Research
pdflatex Research.tex
pdflatex Research.tex

Or open Research.lyx in LyX and use the built-in export/compile. Note that Research.lyx is the source of truth — Research.tex is generated from it via LyX export.

Background and Context

Physical Motivation

The propagation of intense light through optical fibers is governed by the interplay between linear dispersion and Kerr nonlinearity ($\chi^{(3)}$). Starting from Maxwell's equations in a source-free, non-magnetic dielectric, the electric field is written as a slowly varying envelope modulating a carrier:

$$\mathbf{E}\left(\mathbf{r},t\right) = \frac12\hat{\mathbf{x}}\left[F\left(x,y\right),A\left(z,t\right),e^{i\left(\beta_0 z - \omega_0 t\right)} + \text{c.c.}\right]$$

The slowly varying envelope approximation yields the NLS equation:

$$i\partial_z A - \frac{\beta_2}{2}\partial_T^2 A + \gamma \left|A\right|^2 A = 0$$

Spatial variations in the refractive index (fiber Bragg gratings, graded-index fibers, photonic-crystal fibers) enter as an external potential $V\left(\mathbf{x}\right)$. The transverse modal equation is mathematically identical to the stationary Schrödinger equation $-\Delta\psi + V\left(x\right)\psi = E\psi$, making guided modes correspond to bound states of the effective potential.

Mathematical Framework

The analysis builds on:

• Concentration compactness (Lions, 1984) — classifying minimizing sequences into compactness, vanishing, or dichotomy on unbounded domains.
• Global bifurcation analysis (Kirr and Natarajan, 2018) — rescaling and multi-peak decomposition of ground states via equivariant bifurcation theory; the interaction functional $Q_R$ and its exponential decay.
• Lyapunov–Schmidt reduction — reduction to a finite-dimensional algebraic system governing peak positions, with projection onto the translational kernel.
• Grillakis–Shatah–Strauss stability theory — orbital instability of multi-peak states in Hamiltonian systems with symmetry.
• Floer–Weinstein semiclassical analysis (1986) — concentration near non-degenerate critical points for bounded potentials.
• Symmetry-breaking bifurcation (Kirr, Kevrekidis, Pelinovsky, 2011) — formal hypotheses (V1)–(V4) for symmetric potentials.

References

1. [Hasegawa and Tappert, 1973a] A. Hasegawa and F. Tappert. Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Applied Physics Letters, 23(3):142–144, 1973.
2. [Hasegawa and Tappert, 1973b] A. Hasegawa and F. Tappert. Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion. Applied Physics Letters, 23(4):171–172, 1973.
3. [Lions, 1984a] P. L. Lions. The concentration-compactness principle in the Calculus of Variations. The locally compact case, part 1. Annales de l'Institut Henri Poincaré. Analyse non linéaire, 1(2):109–145, 1984.
4. [Lions, 1984b] P. L. Lions. The concentration-compactness principle in the Calculus of Variations. The locally compact case, part 2. Annales de l'Institut Henri Poincaré. Analyse non linéaire, 1(4):223–283, 1984.
5. [Kirr and Natarajan, 2018] E. Kirr and V. Natarajan. The global bifurcation picture for ground states in nonlinear Schrödinger equations. arXiv preprint arXiv:1811.05716, 2018.
6. [Berestycki and Lions, 1983] H. Berestycki and P. L. Lions. Nonlinear scalar field equations, I existence of a ground state. Archive for Rational Mechanics and Analysis, 82(4):313–345, 1983.
7. [Floer and Weinstein, 1986] A. Floer and A. Weinstein. Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. Journal of Functional Analysis, 69(3):397–408, 1986.
8. [Kirr, Kevrekidis, and Pelinovsky, 2011] E. Kirr, P. G. Kevrekidis, and D. E. Pelinovsky. Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials. Communications in Mathematical Physics, 308(3):795–844, 2011.
9. [Kirr, 2016] E. Kirr. Long time dynamics and coherent states in nonlinear wave equations. arXiv preprint arXiv:1605.08167, 2016.
10. [Grillakis, Shatah, and Strauss, 1987] M. Grillakis, J. Shatah, and W. Strauss. Stability theory of solitary waves in the presence of symmetry, I. Journal of Functional Analysis, 74(1):160–197, 1987.
11. [Grillakis, Shatah, and Strauss, 1990] M. Grillakis, J. Shatah, and W. Strauss. Stability theory of solitary waves in the presence of symmetry, II. Journal of Functional Analysis, 94(2):308–348, 1990.
12. [Kirr and Zarnescu, 2007] E. Kirr and A. Zarnescu. On the asymptotic stability of bound states in 2D cubic Schrödinger equation. Communications in Mathematical Physics, 272(2):443–468, 2007.
13. [Kirr and Zarnescu, 2009] E. Kirr and A. Zarnescu. Asymptotic stability of ground states in 2D nonlinear Schrödinger equation including subcritical cases. arXiv preprint arXiv:0805.3888, 2009.
14. [Kirr and Mizrak, 2008] E. Kirr and O. Mizrak. Asymptotic stability of ground states in 3D nonlinear Schrödinger equation including subcritical cases. arXiv preprint arXiv:0803.3377, 2008.

License

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