Abstract
In this paper, we propose an iterative method to compute the positive ground states of saturable nonlinear Schrödinger equations. A discretization of the saturable nonlinear Schrödinger equation leads to a nonlinear algebraic eigenvalue problem (NAEP). For any initial positive vector, we prove that this method converges globally with a locally quadratic convergence rate to a positive solution of NAEP. During the iteration process, the method requires the selection of a positive parameter \(\theta _k\) in the kth iteration, and generates a positive vector sequence approximating the eigenvector of NAEP and a scalar sequence approximating the corresponding eigenvalue. We also present a halving procedure to determine the parameters \(\theta _k\), starting with \(\theta _k=1\) for each iteration, such that the scalar sequence is strictly monotonic increasing. This method can thus be used to illustrate the existence of positive ground states of saturable nonlinear Schrödinger equations. Numerical experiments are provided to support the theoretical results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bao, W., Jacksch, D.: An explicit unconditionaly stable numerical method for solving damped nonlinear Schrödinger equation with focusing nonlinearity. SIAM J. Numer. Anal. 41, 1406–1426 (2003)
Bao, W., Tang, W.: Ground state solution of Bose-Einstein condensate by directly minimizing the energy functional. J. Comput. Phys. 187, 230–254 (2003)
Bao, W., Du, Q.: Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow. SIAM J. Sci. Comput. 25, 1674–1697 (2004)
Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. Classics in Applied Mathematics, vol. 9. SIAM, Philadelphia, PA (1994)
Cazenave, T., Lions, P.L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85, 549–561 (1982)
Cuevas, J., Eilbeck, J.C.: Discrete soliton collisions in a waveguide array with saturable nonlinearity. Phys. Lett. A 358, 15–20 (2006)
Coleman, T.F., Li, Y.: An interior trust region approach for nonlinear minimization subject to bounds. SIAM J. Optim. 6, 418–445 (1996)
Coleman, T.F., Li, Y.: On the convergence of interior-reflective Newton methods for nonlinear minimization subject to bounds. Math. Program. 67(2), 189–224 (1994)
Cuevas, J., Eilbeck, J.C., Karachalios, N.I.: Thresholds for breather solutions of the discrete nonlinear Schrödinger equation with saturable and power nonlinearity. Discrete Contin. Dyn. Syst. 21, 445–475 (2008)
Gatz, S., Herrmann, J.: Propagation of optical beams and the properties of two dimensional spatial solitons in media with a local saturable nonlinear refractive index. J. Opt. Soc. Am. B 14, 1795–1806 (1997)
Horn, R.A., Johnson, C.R.: Matrix Analysis. The Cambridge University Press, Cambridge (1985)
Karlsson, M.: Optical beams in saturable self-focusing media. Phys. Rev. A 46, 2726–2734 (1992)
Karachalios, N., Yannacopoulos, A.: Global existence and compact attractors for the discrete Schrödinger equations. J. Differ. Equ. 217, 88–123 (2005)
Kelley, P.L.: Self-focusing of optical beams. Phys. Rev. Lett. 15, 1005–1008 (1965)
Lin, T.-C., Wang, X., Wang, Z.-Q.: Orbital stability and energy estimate of ground states of saturable nonlinear Schrödinger equations with intensity functions in \( {\mathbb{R}}^2\). J. Differ. Equ. 263, 4750–4786 (2017)
Liu, C.-S., Guo, C.-H., Lin, W.-W.: A positivity preserving inverse iteration for finding the Perron pair of an irreducible nonnegative third order tensor. SIAM J. Matrix Anal. Appl. 37, 911–932 (2016)
Liu, C.-S., Guo, C.-H., Lin, W.-W.: Newton-Noda iteration for finding the Perron pair of a weakly irreducible nonnegative tensor. Numer. Math. 137, 63–90 (2017)
Maia, L.A., Montefusco, E., Pellacci, B.: Weakly coupled nonlinear Schrödinger systems: the saturation effect. Calc. Var. 46, 325–351 (2013)
Marburger, J.H., Dawesg, E.: Dynamical formation of a small-scale filament. Phys. Rev. Lett. 21, 556–558 (1968)
Merhasin, I.M., Malomed, B.A., Senthilnathan, K., Nakkeeran, K., Wai, P.K.A., Chow, K.W.: Solitons in Bragg gratings with saturable nonlinearities. J. Opt. Soc. Am. B 24, 1458–1468 (2007)
Noda, T.: Note on the computation of the maximal eigenvalue of a non-negative irreducible matrix. Numer. Math. 17, 382–386 (1971)
Powell, M.J.D.: A fortran subroutine for solving systems of nonlinear algebraic equations. In: Rabinowitz, P. (ed.) Numerical Methods for Nonlinear Algebraic Equations, Ch. 7 (1970)
Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43(2), 270–291 (1992)
Wu, X., Wen, Z., Bao, W.: A regularized Newton method for computing ground states of Bose-Einstein condensates. J. Sci. Comput. 73, 303–329 (2017)
Yan, Y.: Attractors and dimensions for discretizations of a weakly damped Schrödinger equations and a sine-Gordon equation. Nonlinear Anal. 20, 1417–1452 (1993)
Acknowledgements
I would like to thank Prof. Chun-Hua Guo, Wen-Wei Lin and Tai-Chia Lin for their valuable discussion and the two anonymous referees for their valuable comments. This work is supported by Ministry of Science and Technology in Taiwan.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Liu, CS. A Positivity Preserving Iterative Method for Finding the Ground States of Saturable Nonlinear Schrödinger Equations. J Sci Comput 84, 41 (2020). https://doi.org/10.1007/s10915-020-01297-8
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-020-01297-8
Keywords
- Schrödinger equations
- Saturable nonlinearity
- Ground states
- M-matrix
- Quadratic convergence
- Positivity preserving