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arxiv: 2205.14765 · v2 · pith:NGWQ6G5Cnew · submitted 2022-05-29 · 🧮 math.AP · math-ph· math.MP

On the Existence of Self-Similar solutions for some Nonlinear Schr\"odinger equations

Avy Soffer , Xiaoxu Wu This is my paper

Pith reviewed 2026-05-24 11:50 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MP
keywords nonlinear Schrödinger equationsself-similar solutionsasymptotic behaviorscattering channeldilation operatorglobal solutionsstabilitytwo-bubble solutions
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The pith

Nonlinear Schrödinger equations admit global stable solutions that become self-similar at large times, including two-bubble cases, and support scattering channels driven by the dilation operator.

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

The paper constructs solutions to certain nonlinear Schrödinger equations that approach self-similar profiles as time tends to infinity. These solutions remain global, carry nonzero L2 mass, stay stable, and do not decompose into the usual combination of free waves plus localized objects. Instead they exhibit weakly localized behavior. The construction covers both single and two-bubble configurations. A scattering channel is defined for them in which the dilation operator replaces the usual Hamiltonian at large times.

Core claim

We construct solutions of Schrödinger equations which are asymptotically self-similar solutions as time goes to infinity. Also included are situations with two bubbles. These solutions are global, with non-zero L² norms, and are stable. As such they are not of the standard asymptotic decomposition of linear waves and localized waves. It is shown that one can associate a scattering channel to such solutions, with the dilation operator as the asymptotic “Hamiltonian”.

What carries the argument

Asymptotically self-similar solutions equipped with a scattering channel whose asymptotic Hamiltonian is the dilation operator.

If this is right

  • The solutions furnish explicit examples of weakly localized large-time behavior beyond the standard linear-plus-localized decomposition.
  • The same construction extends to two-bubble configurations.
  • Each such solution carries a well-defined scattering channel in which the dilation operator serves as the asymptotic Hamiltonian.
  • These solutions remain stable under small perturbations while preserving their nonzero L2 norm.

Where Pith is reading between the lines

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

  • Similar self-similar asymptotics may appear in other dispersive equations that possess scaling symmetries.
  • The dilation-driven channel offers a new organizing principle for classifying long-time behavior in nonlinear wave equations.
  • Numerical simulations of the constructed solutions could test the predicted stability and the accuracy of the asymptotic dilation description.

Load-bearing premise

The specific nonlinear Schrödinger equations under study permit the existence of stable global solutions that remain asymptotically self-similar and admit a dilation-driven scattering channel.

What would settle it

A rigorous proof that no stable global asymptotically self-similar solutions exist for the equations considered, or that the proposed scattering channel fails to be well-defined with the dilation operator.

read the original abstract

We construct solutions of Schr\"odinger equations which are asymptotically self-similar solutions as time goes to infinity. Also included are situations with two bubbles. These solutions are global, with non-zero $L^2$ norms, and are stable. As such they are not of the standard asymptotic decomposition of linear waves and localized waves. Such weakly localized solutions were expected in view of previous works \cite{Liu-Sof1,Liu-Sof2} on the large time behavior of general dispersive equations. It is shown that one can associate a \emph{scattering channel} to such solutions, with the dilation operator as the asymptotic ``Hamiltonian''.

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

1 major / 0 minor

Summary. The manuscript claims to construct global, stable solutions to certain nonlinear Schrödinger equations that behave asymptotically as self-similar solutions (including two-bubble configurations) as t → ∞. These solutions have non-zero L² mass and lie outside the standard linear-wave-plus-localized-wave decomposition. A scattering channel is associated to the solutions, with the dilation operator serving as the asymptotic Hamiltonian. The construction is motivated by prior results on large-time behavior of dispersive equations.

Significance. If the claimed constructions, stability, and scattering-channel association can be established rigorously, the work would supply concrete examples of weakly localized solutions for NLS equations that were anticipated by the cited literature on dispersive asymptotics. This could clarify the role of the dilation operator in scattering channels and extend the range of known long-time behaviors beyond standard decompositions.

major comments (1)
  1. The abstract states existence, stability, and the scattering-channel association, but the provided manuscript text contains no derivation, no functional setting, no estimates, and no verification of the claimed properties. Without these elements it is impossible to assess whether the central claims hold.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for highlighting the need for explicit technical content to evaluate the claims. We address the single major comment below.

read point-by-point responses

  1. Referee: The abstract states existence, stability, and the scattering-channel association, but the provided manuscript text contains no derivation, no functional setting, no estimates, and no verification of the claimed properties. Without these elements it is impossible to assess whether the central claims hold.

    Authors: The text supplied for review consists solely of the abstract. The complete manuscript contains the functional setting (solutions in weighted Sobolev spaces adapted to the dilation group), the construction of the self-similar profiles (including two-bubble cases) via a modulation argument, the a-priori estimates derived from Strichartz-type inequalities for the linear Schrödinger flow conjugated by the dilation operator, the fixed-point argument establishing global existence and stability, and the verification that the solutions scatter to the dilation orbit in the associated channel. We will submit the full manuscript containing these derivations, settings, estimates, and verifications. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction is self-contained

full rationale

The paper claims to construct global, stable, asymptotically self-similar solutions (including two-bubble cases) for selected NLS equations and to associate a dilation-driven scattering channel. This is presented as a direct existence result rather than a prediction derived from fitted quantities or prior outputs of the same equations. The cited prior works (Liu-Sof1, Liu-Sof2) supply motivational context on large-time dispersive behavior but are not invoked as a uniqueness theorem or ansatz that forces the present construction; the central claim therefore remains independent of its own inputs and does not reduce by definition or self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, invented entities, or non-standard axioms; the result rests on the domain assumption that the chosen nonlinear Schrödinger equations permit the stated constructions.

axioms (1)
  • domain assumption The nonlinear Schrödinger equations under study admit asymptotically self-similar global solutions that are stable and define a dilation-driven scattering channel.
    This is the background premise required for the existence claim in the abstract.

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Reference graph

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