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Persistence of integrable wave dynamics in the Discrete Gross--Pitaevskii equation: the focusing case

G. Fotopoulos, N. I. Karachalios, V. Koukouloyannis

TL;DR

This work addresses whether integrable wave dynamics can persist within a nearby nonintegrable lattice, specifically comparing the focusing DGPE with the integrable AL lattice under a weak harmonic trap. It develops a functional-analytic framework in weighted spaces, proves global existence for the Salerno lattice in $l^2_w$, and derives explicit distance estimates between DGPE and AL solutions, revealing a trapezoidal trapping region that bounds the distance for all times. The results show that, for small initial data and weak traps, the distance growth is at most exponential in the infinite lattice (and linear in the finite lattice), indicating long-time proximity of dynamics and the persistence of small-amplitude AL solitons within DGPE; numerical simulations corroborate these predictions and reveal robust soliton behavior despite the curved orbit induced by the trap. Collectively, the findings extend to a broad class of NLS-type models, highlighting the practical impact of integrable dynamics persisting under small nonintegrable perturbations and suggesting avenues for exploring varied boundary conditions and perturbations.

Abstract

Expanding upon our prior findings on the proximity of dynamics between integrable and non-integrable systems within the framework of nonlinear Schrödinger equations, we examine this phenomenon for the focusing Discrete Gross-Pitaevskii equation in comparison to the Ablowitz-Ladik lattice. The presence of the harmonic trap necessitates the study of the Ablowitz-Ladik lattice in weighted spaces. We establish estimates for the distance between solutions in the suitable metric, providing a comprehensive description of the potential evolution of this distance for general initial data. These results apply to a broad class of nonlinear Schrödinger models, including both discrete and partial differential equations. For the Discrete Gross-Pitaevskii equation, they guarantee the long-term persistence of small-amplitude bright solitons, driven by the analytical solution of the AL lattice, especially in the presence of a weak harmonic trap. Numerical simulations confirm the theoretical predictions about the proximity of dynamics between the systems over long times. They also reveal that the soliton exhibits remarkable robustness, even as the effects of the weak harmonic trap become increasingly significant, leading to the soliton's curved orbit.

Persistence of integrable wave dynamics in the Discrete Gross--Pitaevskii equation: the focusing case

TL;DR

This work addresses whether integrable wave dynamics can persist within a nearby nonintegrable lattice, specifically comparing the focusing DGPE with the integrable AL lattice under a weak harmonic trap. It develops a functional-analytic framework in weighted spaces, proves global existence for the Salerno lattice in , and derives explicit distance estimates between DGPE and AL solutions, revealing a trapezoidal trapping region that bounds the distance for all times. The results show that, for small initial data and weak traps, the distance growth is at most exponential in the infinite lattice (and linear in the finite lattice), indicating long-time proximity of dynamics and the persistence of small-amplitude AL solitons within DGPE; numerical simulations corroborate these predictions and reveal robust soliton behavior despite the curved orbit induced by the trap. Collectively, the findings extend to a broad class of NLS-type models, highlighting the practical impact of integrable dynamics persisting under small nonintegrable perturbations and suggesting avenues for exploring varied boundary conditions and perturbations.

Abstract

Expanding upon our prior findings on the proximity of dynamics between integrable and non-integrable systems within the framework of nonlinear Schrödinger equations, we examine this phenomenon for the focusing Discrete Gross-Pitaevskii equation in comparison to the Ablowitz-Ladik lattice. The presence of the harmonic trap necessitates the study of the Ablowitz-Ladik lattice in weighted spaces. We establish estimates for the distance between solutions in the suitable metric, providing a comprehensive description of the potential evolution of this distance for general initial data. These results apply to a broad class of nonlinear Schrödinger models, including both discrete and partial differential equations. For the Discrete Gross-Pitaevskii equation, they guarantee the long-term persistence of small-amplitude bright solitons, driven by the analytical solution of the AL lattice, especially in the presence of a weak harmonic trap. Numerical simulations confirm the theoretical predictions about the proximity of dynamics between the systems over long times. They also reveal that the soliton exhibits remarkable robustness, even as the effects of the weak harmonic trap become increasingly significant, leading to the soliton's curved orbit.
Paper Structure (14 sections, 6 theorems, 64 equations, 5 figures)

This paper contains 14 sections, 6 theorems, 64 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Functional set up and estimates for the Salerno lattice in weighted sequence spaces
  3. Functional set up: standard and weighted sequence spaces and the discrete Laplacian in weighted spaces
  4. Global existence of solutions for the Salerno lattice on $l^2_w$ and remarks on the solutions of the AL lattice on $l^2_w$
  5. The distance between the solutions of the AL and the DGPE lattice
  6. The distance between the solutions: general initial data
  7. The case of the infinite lattice.
  8. The case of the finite lattice with Dirichlet boundary conditions.
  9. Understanding the dynamics of the distance
  10. Numerical Results
  11. The case of initial amplitude $A=0.05$.
  12. The case of initial amplitude $A=0.2$.
  13. Variations of the parameter $h$.
  14. Conclusions

Key Result

Proposition 2.1

We assume that the weight $w_n$ satisfies the conditions w_cond. Then, the discrete Laplacian dL defines a bounded operator on $l^2_w$, i.e., and

Figures (5)

  • Figure 1: An intuitive example relevant to the estimate \ref{['expest']}. The $l^2_w$-norm (continuous blue curve) of the analytical soliton solution \ref{['ALBS']} with $v>0$ and its corresponding $l^2$-norm (dashed blue curve). (a) $A=0.05$, $\|\psi\|_{l^2}=0.6345$. (b) $A=0.2$, $\|\psi\|_{l^2}=0.3163$.
  • Figure 2: (a) Geometrical interpretation of Theorem \ref{['MR']}. (b) Geometrical interpretation of Corollary \ref{['MRF']}. Details in the text (see Section \ref{['SecIIIB']}).
  • Figure 3: (a) Spatiotemporal evolution of $|\phi_n(0)|$ by the DGPE lattice, where $\phi_n(0)=\psi_n(0)$ is the initial condition defined by the analytical solution \ref{['ALBS']}, against the same evolution by the AL lattice. The curved orbit of the soliton corresponds to the DGPE soliton while the straight orbit to the analytical AL soliton. (b) Contour plot of the evolution in panel (a). (c) The evolution of $\|y(t)\|_{l^2}$. (d) Evolution of the difference of the momenta $|\Delta M(t)|$ of the solutions. Parameters: Amplitude of the initial condition $A=0.05$, $\Omega=0.002$. More details in text (see Section \ref{['SecIV']}-paragraph a.)
  • Figure 4: The same numerical results as those presented in Figure \ref{['Fig2']}, but for $A=0.2$. More details in text (see Section \ref{['SecIV']}-paragraph b.)
  • Figure 5: The same numerical results as those presented in Figure \ref{['Fig2']}-panel (c), but for two different values of $h$. (a) $h=0.1$. (b) $h=10$. More details in text (see Section \ref{['SecIV']}-paragraph c).

Theorems & Definitions (13)

  • Remark 2.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Proposition 3.1
  • proof
  • Theorem 3.1
  • ...and 3 more