Reduced-order modeling for Ablowitz-Ladik equation

TL;DR

This paper develops reduced-order models for the Ablowitz-Ladik equation, capturing its Hamiltonian structure and stability properties, enabling efficient long-term simulations of soliton solutions.

Contribution

It introduces a structure-preserving POD-Galerkin approach combined with DEIM and tensor techniques for efficient reduced modeling of ALEs with Hamiltonian properties.

Findings

ROMs preserve Hamiltonian and dissipation properties.

Efficient offline-online decomposition achieved.

Long-term stability of soliton solutions demonstrated.

Abstract

In this paper, reduced-order models (ROMs) are constructed for the Ablowitz-Ladik equation (ALE), an integrable semi-discretization of the nonlinear Schr\"odinger equation (NLSE) with and without damping. Both ALEs are non-canonical conservative and dissipative Hamiltonian systems with the Poisson matrix depending quadratically on the state variables, and with quadratic Hamiltonian. The full-order solutions are obtained with the energy preserving midpoint rule for the conservative ALE and exponential midpoint rule for the dissipative ALE. The reduced-order solutions are constructed intrusively by preserving the skew-symmetric structure of the reduced non-canonical Hamiltonian system by applying proper orthogonal decomposition (POD) with the Galerkin projection. For an efficient offline-online decomposition of the ROMs, the quadratic nonlinear terms of the Poisson matrix are approximated by the discrete empirical interpolation method (DEIM). The computation of the reduced-order solutions is further accelerated by the use of tensor techniques. Preservation of the Hamiltonian and momentum for the conservative ALE, and preservation of dissipation properties of the dissipative ALE, guarantee the long-term stability of soliton solutions.