NEWTON-NODA ITERATION FOR COMPUTING THE GROUND STATES OF NONLINEAR SCHRODINGER EQUATIONS

In this paper, we present a Newton-Noda iteration (NNI) to find the ground state of nonlinear Schrodinger (NLS) equations. The ground state of an NLS equation is defined as the minimizer of the energy functional, which satisfies the associated Euler-Lagrange equation. By discretizing the Euler-Lagrange equation, the so-called nonlinear algebraic eigenvalue problem (NAEP) can be established. For any positive initial vector, the proposed method is globally convergent and the convergence rate is quadratic. More specifically, NNI produces a bounded increasing sequence approximating an eigenvalue of NAEP and a positive vector sequence approximating the corresponding positive eigenvector. Finally, a number of numerical experiments are performed to evaluate the performance of NNI for nonlinear-focusing Schrodinger equations, logarithmic Schrodinger equations, and modified Gross-Pitaevskii equations. Also, these results confirm our theory and demonstrate the efficiency, robustness, and wide applicability of the proposed method.