pith. sign in

arxiv: 1907.04644 · v1 · pith:A2YMNWSRnew · submitted 2019-07-10 · 🧮 math.NA · cs.NA· physics.comp-ph

A positivity preserving iterative method for finding the ground states of saturable nonlinear Schr\"odinger equations

Ching-Sung Liu This is my paper

Pith reviewed 2026-05-24 23:45 UTC · model grok-4.3

classification 🧮 math.NA cs.NAphysics.comp-ph
keywords saturable nonlinear Schrödinger equationground statesiterative methodnonlinear algebraic eigenvalue problemglobal convergencepositivity preservingquadratic convergence
0
0 comments X

The pith

An iterative method converges globally with local quadratic rate to positive solutions of the nonlinear algebraic eigenvalue problem from saturable nonlinear Schrödinger equations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes an iterative method to compute the positive ground states of saturable nonlinear Schrödinger equations after discretization produces a nonlinear algebraic eigenvalue problem. It proves global convergence from any positive initial vector to a positive solution, with the rate becoming locally quadratic near the limit. A halving procedure is given for choosing the positive parameter θ_k at each step so that the sequence of scalar approximations is strictly increasing. The construction also yields a proof of existence for positive solutions. Numerical experiments illustrate the convergence behavior.

Core claim

The iterative method, for any positive starting vector, generates a positive vector sequence converging to the eigenvector and a strictly monotonic increasing scalar sequence converging to the eigenvalue of the NAEP; global convergence and local quadratic rate are proved when θ_k is chosen by the halving procedure that begins at 1 and halves until the monotonicity condition holds.

What carries the argument

The positivity-preserving iteration combined with the halving selection rule for the parameter θ_k that enforces strict increase of the eigenvalue approximations at every step.

If this is right

  • The method computes positive ground states from any positive initial guess without requiring a good starting point.
  • The strictly increasing scalar sequence supplies a constructive demonstration that positive solutions exist.
  • Once near the solution the quadratic local rate implies rapid final convergence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same halving technique for enforcing monotonicity could be tested on other nonlinear eigenvalue problems that preserve positivity.
  • The global convergence guarantee may extend to discretizations on different meshes or domains if the monotonicity condition can still be met.
  • The method's behavior under round-off error or for very fine grids remains open and could be checked numerically.

Load-bearing premise

A suitable positive parameter θ_k can always be selected via halving starting from 1 so that the scalar sequence stays strictly monotonic increasing.

What would settle it

A concrete discretization of the saturable nonlinear Schrödinger equation in which the halving procedure never finds a θ_k that keeps the scalar sequence monotonic or in which the vector iterates fail to converge to a positive solution.

Figures

Figures reproduced from arXiv: 1907.04644 by Ching-Sung Liu.

Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
read the original abstract

In this paper, we propose an iterative method to compute the positive ground states of saturable nonlinear Schr\"odinger equations. A discretization of the saturable nonlinear Schr\"odinger 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 $k$th 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 Schr\"odinger equations. Numerical experiments are provided to support the theoretical results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proposes an iterative method to compute positive ground states of saturable nonlinear Schrödinger equations. Discretization yields a nonlinear algebraic eigenvalue problem (NAEP). The authors prove that, for any positive initial vector, the iteration converges globally to a positive solution of the NAEP with locally quadratic rate. A halving procedure selects the parameter θ_k (starting from θ_k=1) at each step so that the scalar sequence remains strictly monotonic increasing while preserving positivity of the vector sequence. Numerical experiments are included to illustrate the theory and the method's ability to demonstrate existence of positive ground states.

Significance. If the convergence analysis holds, the contribution is significant: it supplies a positivity-preserving iteration with a global convergence guarantee from arbitrary positive starts and a quadratic local rate, together with an explicit, implementable procedure for choosing θ_k. Such results are valuable in numerical analysis of nonlinear eigenvalue problems arising from nonlinear Schrödinger equations, and the existence-illustration aspect adds utility beyond computation.

minor comments (3)
  1. [Abstract, §3] The abstract and introduction state that the halving procedure ensures the scalar sequence is strictly monotonic increasing, but the precise condition under which a suitable θ_k is guaranteed to exist (e.g., any restriction on the saturation parameter or mesh size) should be stated explicitly in the theorem statement in §3.
  2. [§5] In the numerical section, the reported iteration counts and residual histories would benefit from a table comparing the proposed method against a standard inverse iteration or Newton method on the same NAEP instances, to quantify the practical advantage of the positivity-preserving property.
  3. [§2] Notation for the discrete Laplacian and the nonlinearity after discretization is introduced without a dedicated preliminary section; adding a short §2 that collects the NAEP formulation, the inner-product notation, and the definition of positivity would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of the manuscript, the recognition of its significance in providing a positivity-preserving iteration with global convergence from arbitrary positive initial data, and the recommendation for minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; proof is self-contained

full rationale

The paper's central result is a mathematical proof of global convergence (with local quadratic rate) for a positivity-preserving iteration on the discretized NAEP, for arbitrary positive initial vectors. The halving procedure for selecting θ_k is defined explicitly to enforce monotonicity of the scalar sequence and is part of the algorithm whose convergence is proved; it does not presuppose the target result. No self-citations, fitted parameters renamed as predictions, or ansatzes imported from prior work appear in the derivation chain. The claim that the method illustrates existence follows directly from the convergence theorem rather than assuming it. The argument is therefore independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on the existence of positive solutions to the NAEP from discretization and the algorithmic selection of θ_k; no data-fitted parameters or new entities are introduced.

axioms (1)
  • domain assumption Discretization of the saturable NLS equation produces a NAEP that admits positive solutions.
    This is invoked to justify targeting positive solutions via the iteration.

pith-pipeline@v0.9.0 · 5700 in / 1268 out tokens · 29498 ms · 2026-05-24T23:45:42.964236+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

  1. [1]

    Berestycki and P

    H. Berestycki and P. L. Lions , Nonlinear scalar field equations, I existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), no. 4, 313345

    work page 1983

  2. [2]

    Berman and R

    A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Vol. 9 of Classics in Applied Mathematics, SIAM, Philadelphia, PA (1 994)

    work page

  3. [3]

    Cazenave, P.L

    T. Cazenave, P.L. Lions , Orbital stability of standing waves for some nonlinear Schr dinger equations, Comm. Math. Phys. 85 (1982) 549561

    work page 1982

  4. [4]

    Elsner , Inverse iteration for calculating the spectral radius of a n on-negative irreducible matrix, Linear Algebra and Appl., 15 (1976), pp

    L. Elsner , Inverse iteration for calculating the spectral radius of a n on-negative irreducible matrix, Linear Algebra and Appl., 15 (1976), pp. 235–242

    work page 1976

  5. [5]

    Gatz and J

    S. Gatz and J. Herrmann , Propagation of optical beams and the properties of two dimen sional spatial solitons in media with a local saturable nonlinear r efractive index , J. Opt. Soc. Amer. B, 14 (1997), pp. 1795–1806

    work page 1997

  6. [6]

    R. A. Horn and C. R. Johnson, Matrix Analysis, The Cambrid ge University Press, Cambridge, UK (1985)

    work page 1985

  7. [7]

    Karlsson , Optical beams in saturable self-focusing media , Phys

    M. Karlsson , Optical beams in saturable self-focusing media , Phys. Rev. A, 46 (1992), pp. 2726– 2734

    work page 1992

  8. [8]

    Kelley , Self-focusing of optical beams , Phys

    P.L. Kelley , Self-focusing of optical beams , Phys. Rev. Lett. 15 (1965), 1005

    work page 1965

  9. [9]

    T.-C. Lin, X. W ang, Z.-Q. W ang , Orbital stability and energy estimate of ground states of saturable nonlinear Schr¨ odinger equations with intensit y functions in R2, J. Differential Equations, 263 (2017), pp. 47504786

    work page 2017

  10. [10]

    L.A. Maia, E. Montefusco, and B. Pellacc , Weakly coupled nonlinear Schr¨ odinger systems: the saturation effect , Calculus of Variations and Partial Differential Equations 46 (2013), Issue 1-2, pp. 325–351

    work page 2013

  11. [11]

    J. H. Marburger and E. Dawesg , Dynamical formation of a small-scale filament , Phys. Rev. Lett., 21(8), pp. 556–558 (1968)

    work page 1968

  12. [12]

    I. M. Merhasin, B. A. Malomed, K. Senthilnathan, K. Nakkeeran, P. K. A. W ai and K. W. Chow , Solitons in Bragg gratings with saturable nonlinearities , J. Opt. Soc. Amer. B,Vol. 24 (2007), pp. 1458–1468

    work page 2007

  13. [13]

    Noda , Note on the computation of the maximal eigenvalue of a non-ne gative irreducible matrix, Numer

    T. Noda , Note on the computation of the maximal eigenvalue of a non-ne gative irreducible matrix, Numer. Math., 17 (1971), pp. 382–386

    work page 1971

  14. [14]

    Rabinowitz , On a class of nonlinear Schrdinger equations , Z

    P.H. Rabinowitz , On a class of nonlinear Schrdinger equations , Z. Angew. Math. Phys. 43 (1992), no. 2, 270291

    work page 1992