Nonlinear discrete Schrödinger equations with a point defect

Technical Summary: Nonlinear Discrete Schrödinger Equations with a Point Defect

Problem Statement
This paper investigates the $d$-dimensional discrete nonlinear Schrödinger equation (DNLS) on an infinite lattice $\mathbb{Z}^d$ subject to a point defect modeled by a delta potential. The system is governed by:
$$i\dot{u}_n + \kappa(\Delta u)_n + \gamma|u_n|^{2\sigma}u_n + V_0\delta_{n,0}u_n = 0, \quad n \in \mathbb{Z}^d$$
where $\Delta$ is the discrete Laplacian, $\gamma$ controls the nonlinearity (focusing for $\gamma > 0$, defocusing for $\gamma < 0$), and $V_0$ represents the strength of the delta potential (attractive for $V_0 > 0$, repulsive for $V_0 < 0$).

The central objective is to analyze the interplay between two distinct localization mechanisms:

1. Linear Localization: Caused by the delta potential breaking translational invariance, creating linear bound states (defect modes) with energy outside the continuous spectrum of the Laplacian.
2. Nonlinear Localization: Caused by focusing nonlinearity leading to self-trapping of excitation energy (breathers).

The study focuses on the existence of spatially exponentially localized, time-periodic ground states (breathers), the determination of excitation thresholds (critical $l^2$ norms) required for their creation, and the asymptotic scattering behavior of solutions when these thresholds are not met.

Methodology
The analysis employs a combination of variational methods, spectral theory, and dispersive estimates:

• Variational Approach: The existence of ground states is established by minimizing an action functional $J(x)$ on the Nehari manifold $\mathcal{N}$. The authors prove that the Nehari manifold is non-empty and that minimizers exist, which correspond to critical points of the action functional.
• Equivalence of Minimization Problems: The paper establishes an equivalence between minimizing the action functional (with fixed frequency $\omega$) and minimizing the energy functional (with fixed mass $\nu = \|u\|_{l^2}^2$). This allows the application of Weinstein's criterion to determine the existence of excitation thresholds.
• Concentration Compactness and Explicit Construction: To derive explicit excitation thresholds, the authors utilize the exponential localization of ground states. They approximate solutions using exponentially decaying profiles ($x_n \sim A\eta^{|n|}$) to derive explicit formulas for the energy in terms of the mass $\nu$ and the localization parameter $\eta$.
• Dispersive Estimates and Scattering: For the asymptotic analysis, the authors utilize Strichartz estimates adapted to the discrete setting. They analyze the scattering of solutions to the linear problem by bounding the difference between the nonlinear solution and the linear evolution group, utilizing the decay properties of the linear propagator.

Key Contributions and Results

1. Existence of Ground States (Focusing Nonlinearity, $\gamma > 0$):

   • Attractive Potential ($V_0 > 0$): The paper proves that for any frequency $\omega < 0$ satisfying $V_0 > -\omega$, a ground state exists. Crucially, for attractive potentials, there is no excitation threshold; ground states exist for any $l^2$ norm $\nu > 0$. The linear and nonlinear mechanisms cooperate to lower the total energy.
   • Repulsive Potential ($V_0 < 0$): The existence of ground states depends on the nonlinearity degree $\sigma$.
     • If $\sigma < \min\{1, 2/d\}$, ground states exist for all $\nu > 0$ (no threshold).
     • If $\sigma \ge \max\{1, 2/d\}$, an excitation threshold $\nu_{exc}$ exists. Ground states exist if and only if $\nu > \nu_{exc}$. The explicit threshold is given by:
       $$\nu_{exc} = \left( \frac{(\sigma + 1)(2 - V_0)}{\gamma} \right)^{1/\sigma}$$
     • In this regime, the repulsive potential counteracts nonlinear localization, requiring a minimum mass to sustain a bound state.

2. Defocusing Nonlinearity ($\gamma < 0$) and Linear Mode Persistence:

   • The paper analyzes the durability of linear defect modes under defocusing nonlinearity.
   • For attractive potentials ($V_0 > 0$), defocusing nonlinearity weakens localization. If $\sigma < \min\{1, 2/d\}$, no localized states exist for any $\nu > 0$. If $\sigma \ge \max\{1, 2/d\}$, an upper threshold exists; the linear bound state persists only if the mass $\nu$ is below a critical value.
   • For repulsive potentials ($V_0 < 0$), the paper discusses the conditions under which staggering defect modes survive the impact of focusing nonlinearity.

3. Asymptotic Behavior and Scattering:

   • The authors establish that if the initial mass $\|u_0\|_{l^2}$ is below the excitation threshold $\nu_{exc}$, the solutions scatter to the linear problem in $l^p$ spaces ($p > 2$).
   • Specifically, for $V_0 = 0$ and $\sigma \ge \max\{1, 2/d\}$, or for $V_0 < 0$ and $\sigma \ge \max\{1, 2/d\}$, solutions decay as $|t| \to \infty$ and converge to free linear solutions.
   • This result confirms Weinstein's conjecture for the general power DNLS: subcritical mass solutions decay, while supercritical mass solutions may form non-scattering ground states.

4. Discrete vs. Continuum Distinctions:

   • The paper highlights a unique feature of the discrete system: the simultaneous possibility of localized states both below the lower edge and above the upper edge of the absolutely continuous spectrum (non-staggering and staggering modes, respectively). This phenomenon is attributed to spatial discreteness and is not present in the continuum NLS with a point defect.

Significance and Claims
The paper claims to extend previous studies (specifically those in [40]) in four significant ways:

1. Infinite Lattices: Unlike prior work restricted to finite lattices with Dirichlet boundary conditions, this analysis deals with infinite lattices, requiring advanced variational techniques (Nehari manifold method) rather than simple mountain pass theorems.
2. Ground State Specificity: The derived excitation thresholds are sharp for the existence of ground states, whereas previous thresholds often applied to any solution in $l^2$ (including multi-hump states).
3. Interplay Analysis: The work provides a detailed quantitative analysis of the competition and cooperation between linear (defect-induced) and nonlinear (self-trapping) localization mechanisms.
4. Asymptotic Dynamics: The paper introduces the study of asymptotic scattering behavior for the $\delta$-DNLS, a feature not previously addressed in the context of linear impurities.

The authors emphasize that their results provide explicit, sharp thresholds for the formation of breathers, distinguishing between regimes where localization is guaranteed for any amplitude and regimes where a critical mass is required. The findings are presented as rigorous mathematical proofs regarding the existence, stability, and long-time behavior of solutions to the discrete system.