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arxiv: 1906.11274 · v1 · pith:QOUPWWF6new · submitted 2019-06-26 · 🧮 math.AP

Decay of small odd solutions for long range Schr\"odinger and Hartree equations in one dimension

Mar\'ia E. Mart\'inez This is my paper

Pith reviewed 2026-05-25 15:31 UTC · model grok-4.3

classification 🧮 math.AP
keywords nonlinear Schrödinger equationHartree equationodd solutionsvirial identitieslocal decaylong-range nonlinearityone dimension
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The pith

Odd solutions in H^1 decay to zero in any compact spatial region for long-range one-dimensional nonlinear Schrödinger and Hartree equations.

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

The paper establishes decay to zero in compact regions of space as time tends to infinity for odd solutions belonging to the energy space H^1(R) of the semilinear nonlinear Schrödinger equation, the nonlinear Schrödinger equation with a potential, and the defocusing Hartree equation. The results cover scattering subcritical, critical, and supercritical long-range nonlinearities and require no spectral assumptions on the linear operator. The proofs rely on virial identities that close because the solutions remain odd. A reader would care because the statements give long-time spatial localization for solutions that are neither small nor subject to extra structural hypotheses on the potential.

Core claim

We prove decay to zero in compact regions of space as time tends to infinity for odd solutions in H^1(R) to the semilinear NLS, NLS with potential, and defocusing Hartree equations, covering scattering sub, critical and supercritical long-range nonlinearities without spectral assumptions.

What carries the argument

Virial identities applied to odd functions, which close without extra boundary or spectral remainder terms.

If this is right

  • Decay holds uniformly across scattering subcritical, critical, and supercritical long-range nonlinearities.
  • No spectral assumptions on the linear operator are required even when a potential is present.
  • The statements apply to solutions that need not be small in any norm.
  • The same virial argument works for both local semilinear and nonlocal Hartree nonlinearities.

Where Pith is reading between the lines

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

  • Odd symmetry may serve as a substitute for spectral-gap assumptions in other one-dimensional dispersive models.
  • The method could be tested numerically by evolving odd initial data and measuring the L^2 mass remaining inside a fixed interval.
  • If oddness is preserved only approximately, quantitative versions of the decay might still hold with an error controlled by the symmetry defect.

Load-bearing premise

The solutions remain odd for all positive times.

What would settle it

An explicit odd initial datum in H^1 whose corresponding solution fails to tend to zero inside some fixed compact interval as t goes to infinity would falsify the claim.

read the original abstract

We consider the long time asymptotics of (not necessarily small) odd solutions to the nonlinear Schr\"odinger equation with semi-linear and nonlocal Hartree nonlinearities, in one dimension of space. We assume data in the energy space $H^1(\mathbb{R})$ only, and we prove decay to zero in compact regions of space as time tends to infinity. We give three different results where decay holds: semilinear NLS, NLS with a suitable potential, and defocusing Hartree. The proof is based on the use of suitable virial identities, in the spirit of nonlinear Klein-Gordon models as in Kowalczyk-Martel-Mu\~noz, and covers scattering sub, critical and supercritical (long range) nonlinearities. No spectral assumptions on the NLS with potential are needed.

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 manuscript proves decay to zero in compact spatial regions as t → ∞ for (not necessarily small) odd solutions in H¹(ℝ) to the 1D semilinear NLS, NLS with a suitable even potential, and defocusing Hartree equation. The results cover scattering subcritical, critical, and supercritical long-range nonlinearities and require no spectral assumptions on the potential; the proofs rely on virial identities that close due to the preserved odd symmetry of the solutions.

Significance. If the central claims hold, the work is significant for extending local decay results to long-range regimes in one dimension without spectral hypotheses, by exploiting oddness to eliminate boundary and spectral terms in the virial identities. The unified treatment of semilinear, potential, and nonlocal Hartree cases, together with the absence of smallness assumptions on the data, strengthens the applicability of virial techniques originally developed for nonlinear Klein-Gordon equations.

minor comments (3)
  1. The title refers to 'small odd solutions' while the abstract and stated theorems explicitly treat solutions that are 'not necessarily small.' This mismatch should be resolved for consistency.
  2. The abstract states that the potential is 'suitable' and even (implicitly) to preserve oddness; the precise assumptions on the potential (e.g., regularity, decay, or evenness) should be stated explicitly in the introduction or in the statements of the theorems.
  3. The manuscript invokes that odd initial data remain odd for all t > 0 because the nonlinearities are odd maps and the potential is even; a brief verification of this invariance (perhaps in a preliminary section) would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive summary, significance assessment, and recommendation of minor revision. The referee's description of the results is accurate.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on standard preservation of odd symmetry under the given nonlinearities and even potentials, combined with virial identities drawn from external references (Kowalczyk-Martel-Muñoz). No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or ansatz defined within the paper itself. The central decay statements for odd H^1 data are independent of the target conclusions and rest on externally verifiable symmetry and identity closures.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the oddness assumption and the applicability of virial identities are the only structural ingredients visible.

pith-pipeline@v0.9.0 · 5665 in / 1203 out tokens · 19732 ms · 2026-05-25T15:31:58.601198+00:00 · methodology

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