Newton-based methods for finding the positive ground state of Gross-Pitaevskii equations

TL;DR

This paper introduces two Newton-based algorithms for efficiently computing the positive ground state of Gross-Pitaevskii equations, providing convergence analysis and numerical validation.

Contribution

It adapts Newton-Noda iteration and combines bisection with Newton methods to solve the nonlinear eigenvalue problem from GPE discretization.

Findings

The proposed methods converge reliably for GPE ground state computation.

Explicit convergence and complexity analyses are provided.

Numerical experiments confirm the effectiveness of the algorithms.

Abstract

The discretization of Gross-Pitaevskii equations (GPE) leads to a nonlinear eigenvalue problem with eigenvector nonlinearity (NEPv). In this paper, we use two Newton-based methods to compute the positive ground state of GPE. The first method comes from the Newton-Noda iteration for saturable nonlinear Schr\"odinger equations proposed by Liu, which can be transferred to GPE naturally. The second method combines the idea of the Bisection method and the idea of Newton method, in which, each subproblem involving block tridiagonal linear systems can be solved easily. We give an explicit convergence and computational complexity analysis for it. Numerical experiments are provided to support the theoretical results.