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Discrete Nonlinear Schrödinger versus Ablowitz-Ladik: Existence and dynamics of generalized NLS-type lattices over a nonzero background

Dirk Hennig, Nikos I. Karachalios, Dionyssios Mantzavinos, Dimitrios Mitsotakis

TL;DR

This work provides a rigorous treatment of generalized Ablowitz–Ladik and generalized DNLS lattices with nonzero backgrounds, proving local well-posedness on the infinite lattice and deriving explicit minimal lifespan bounds that depend on initial data, background, and nonlinearity. It establishes a precise proximity framework between gAL and gDNLS, including global existence on finite lattices for gDNLS and a potential blow-up for gAL, and demonstrates an asymptotic equivalence to the continuous NLS in the small-lattice-spacing limit. Numerical simulations corroborate the theory, revealing a collapse/quasi-collapse dichotomy for p=2 and strong AL–DNLS proximity for p=1, with nuanced behavior when comparing discrete and continuous settings. The results illuminate how nonzero backgrounds and discretization influence well-posedness, long-time dynamics, and the transition to continuum models in nonlinear lattice systems.

Abstract

The question of well-posedness of the generalized Ablowitz-Ladik and Discrete Nonlinear Schrödinger equations with \textit{nonzero} boundary conditions on the infinite lattice is far less understood than in the case where the models are supplemented with vanishing boundary conditions. This question remains largely unexplored even in the standard case of cubic nonlinearities in which, in particular, the Ablowitz-Ladik equation is completely integrable while the Discrete Nonlinear Schrödinger equation is not (in contrast with its continuous counterpart). We establish local well-posedness for both of these generalized nonlinear systems supplemented with a broad class of nonzero boundary conditions and, in addition, derive analytical upper bounds for the minimal guaranteed lifespan of their solutions. These bounds depend explicitly on the norm of the initial data, the background, and the nonlinearity exponents. In particular, they suggest the possibility of finite-time collapse (blow-up) of solutions. Furthermore, by comparing models with different nonlinearity exponents, we prove estimates for the distance between their respective solutions (measured in suitable metrics), valid up to their common minimal guaranteed lifespan. Highly accurate numerical studies illustrate that solutions of the generalized Ablowitz-Ladik equation may collapse in finite time. Importantly, the numerically observed blow-up time is in excellent agreement with the theoretically predicted order of the minimal guaranteed lifespan. Furthermore, in the case of the Discrete Nonlinear Schrödinger equation on a finite lattice we prove global existence of solutions; this is consistent with our numerical observations of the phenomenon of \textit{quasi-collapse}, manifested by narrow oscillatory spikes that nevertheless persist throughout time -- continued in pdf ...

Discrete Nonlinear Schrödinger versus Ablowitz-Ladik: Existence and dynamics of generalized NLS-type lattices over a nonzero background

TL;DR

This work provides a rigorous treatment of generalized Ablowitz–Ladik and generalized DNLS lattices with nonzero backgrounds, proving local well-posedness on the infinite lattice and deriving explicit minimal lifespan bounds that depend on initial data, background, and nonlinearity. It establishes a precise proximity framework between gAL and gDNLS, including global existence on finite lattices for gDNLS and a potential blow-up for gAL, and demonstrates an asymptotic equivalence to the continuous NLS in the small-lattice-spacing limit. Numerical simulations corroborate the theory, revealing a collapse/quasi-collapse dichotomy for p=2 and strong AL–DNLS proximity for p=1, with nuanced behavior when comparing discrete and continuous settings. The results illuminate how nonzero backgrounds and discretization influence well-posedness, long-time dynamics, and the transition to continuum models in nonlinear lattice systems.

Abstract

The question of well-posedness of the generalized Ablowitz-Ladik and Discrete Nonlinear Schrödinger equations with \textit{nonzero} boundary conditions on the infinite lattice is far less understood than in the case where the models are supplemented with vanishing boundary conditions. This question remains largely unexplored even in the standard case of cubic nonlinearities in which, in particular, the Ablowitz-Ladik equation is completely integrable while the Discrete Nonlinear Schrödinger equation is not (in contrast with its continuous counterpart). We establish local well-posedness for both of these generalized nonlinear systems supplemented with a broad class of nonzero boundary conditions and, in addition, derive analytical upper bounds for the minimal guaranteed lifespan of their solutions. These bounds depend explicitly on the norm of the initial data, the background, and the nonlinearity exponents. In particular, they suggest the possibility of finite-time collapse (blow-up) of solutions. Furthermore, by comparing models with different nonlinearity exponents, we prove estimates for the distance between their respective solutions (measured in suitable metrics), valid up to their common minimal guaranteed lifespan. Highly accurate numerical studies illustrate that solutions of the generalized Ablowitz-Ladik equation may collapse in finite time. Importantly, the numerically observed blow-up time is in excellent agreement with the theoretically predicted order of the minimal guaranteed lifespan. Furthermore, in the case of the Discrete Nonlinear Schrödinger equation on a finite lattice we prove global existence of solutions; this is consistent with our numerical observations of the phenomenon of \textit{quasi-collapse}, manifested by narrow oscillatory spikes that nevertheless persist throughout time -- continued in pdf ...

Paper Structure

This paper contains 13 sections, 13 theorems, 145 equations, 6 figures, 2 tables.

Table of Contents

  1. Motivation and main results
  2. Local well-posedness
  3. Modified equations with zero boundary conditions at infinity
  4. Local well-posedness of gDNLS
  5. Local well-posedness of gAL
  6. Order of the minimum guaranteed solution lifespan
  7. Distance between the gAL and gDNLS solutions
  8. Numerical analysis and simulations
  9. Numerical scheme
  10. The case $p=1$: integrable AL versus non-integrable DNLS
  11. The case $p=2$: collapse for gAL and quasi-collapse for gDNLS
  12. Discrete versus continuous: gAL against NLS on a nonzero background
  13. Asymptotic equivalence between AL and DNLS

Key Result

Theorem 1.1

For any $p\geq 1$, initial data in the discrete $\ell^2$ space and nonzero boundary conditions at infinity of the form uU-bc, the Cauchy problems for the gAL and the gDNLS equations gal and gdnls on the infinite lattice are locally well-posed in the sense of Hadamard, namely they each admit a unique

Figures (6)

  • Figure 5.1: Top: Spatiotemporal evolution of the density $|u_n(t)|^2$ of the solution to the AL lattice \ref{['eq:al2']} with $h=1$ and the initial condition \ref{['eq:num']} with $q_0=0.1$. Middle: The same evolution for the DNLS lattice \ref{['eq:dnls2']}. Bottom: Time evolution of the distance $\delta(t)$ given by \ref{['delta-def']} in $\ell^r$ for $r=2,3,4,\infty$.
  • Figure 5.2: Top: Spatiotemporal evolution of the density $|u_n(t)|^2$ of the solution to the AL lattice \ref{['eq:al2']} with $h=1$ and the initial condition \ref{['eq:num']} with $q_0=0.4$. Middle: The same evolution for the DNLS lattice \ref{['eq:dnls2']}. Bottom: Time evolution of the distance $\delta(t)$ given by \ref{['delta-def']} in $\ell^r$ for $r=2,3,4,\infty$.
  • Figure 5.3: Top: Spatiotemporal evolution over $t\in [0,26]$ of the density $|u_n(t)|^2$ of the solution to the gAL equation \ref{['eq:al2']} with $p=2$, $h=1$ and the initial condition \ref{['eq:num']} with $q_0=0.4$. Middle: The same evolution for the gDNLS equation \ref{['eq:dnls2']} with $p=2$, $h=1$. Bottom: Time evolution of the distance $\delta(t)$ given by \ref{['delta-def']} in $\ell^r$ for $r=2,3,4,\infty$.
  • Figure 5.4: Spatiotemporal evolution over $t\in [0,100]$ of the density $|u_n(t)|^2$ of the solution to the gDNLS lattice \ref{['eq:dnls2']} with $p=2$, $h=1$ and the initial condition \ref{['eq:num']} with $q_0=0.4$.
  • Figure 5.5: Top: Spatiotemporal evolution of the density $|u_n(t)|^2$ of the solution to the gAL lattice \ref{['eq:al2']} with $p=2$, $h=1$ and the initial condition \ref{['eq:num']} with $q_0=0.14$, for $t\in [0, 5.435]$. Bottom: The same evolution for the gDNLS lattice \ref{['eq:dnls2']} with $p=2$, $h=1$, for $t\in [0,10000]$.
  • ...and 1 more figures

Theorems & Definitions (35)

  • Remark 1.1
  • Theorem 1.1: Local well-posedness
  • Theorem 1.2: Proximity
  • Remark 1.2: Defocusing case
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Theorem 2.1: Local well-posedness of modified gDNLS
  • Remark 2.2
  • proof
  • ...and 25 more