A Hyperbolic Approximation of the Nonlinear Schr\"odinger Equation

TL;DR

This paper introduces a hyperbolic approximation to the nonlinear Schrödinger equation, demonstrating its mathematical properties, solution convergence, and effective numerical discretizations that preserve key physical quantities.

Contribution

It presents a novel hyperbolic system approximating NLS, with proven properties, explicit solutions, and asymptotic-preserving discretizations that accurately model NLS behavior.

Findings

The hyperbolic system is strictly hyperbolic and has a modified Hamiltonian structure.

Explicit standing-wave solutions converge to NLS ground states in the relaxation limit.

Numerical discretizations accurately approximate NLS solutions while conserving mass.

Abstract

We study a first-order hyperbolic approximation of the nonlinear Schr\"odinger (NLS) equation. We show that the system is strictly hyperbolic and possesses a modified Hamiltonian structure, along with at least three conserved quantities that approximate those of NLS. We provide families of explicit standing-wave solutions to the hyperbolic system, which are shown to converge uniformly to ground-state solutions of NLS in the relaxation limit. The system is formally equivalent to NLS in the relaxation limit, and we develop asymptotic preserving discretizations that tend to a consistent discretization of NLS in that limit, while also conserving mass. Examples for both the focusing and defocusing regimes demonstrate that the numerical discretization provides an accurate approximation of the NLS solution.