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arxiv: 2606.01762 · v1 · pith:F3DPO66Hnew · submitted 2026-06-01 · 🧮 math.AP

High-Velocity Inverse Scattering for Nonlinear Schr\"odinger Equations with Spatially Dependent Nonlinearities

Satoshi Masaki , Yaxin Zhao This is my paper

Pith reviewed 2026-06-28 13:56 UTC · model grok-4.3

classification 🧮 math.AP
keywords nonlinear Schrödinger equationinverse scatteringhigh-velocity asymptoticsX-ray transformuniquenessGalilean boostspatially dependent nonlinearityscattering operator
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The pith

The scattering operator uniquely determines both the nonlinearity exponent and the spatial coefficient via high-velocity asymptotics in 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 constructs the scattering operator for large incoming states generated by Galilean boosts in a moving frame. It shows that high-velocity data make the nonlinear interaction effectively weak due to rapid spatial separation. From the resulting asymptotics it derives a reconstruction formula for the X-ray transform of the coefficient. This establishes that the full scattering operator determines both the exponent and the spatial coefficient uniquely. The result covers all dimensions d at least 3, the full mass-supercritical energy-subcritical range including endpoints, and replaces earlier repulsiveness assumptions with decay conditions.

Core claim

By introducing a moving frame adapted to highly boosted initial data, the scattering operator is constructed for a class of large incoming states. Although the boosted data are large in Sobolev norms, the nonlinear interaction becomes effectively weak at high velocity due to rapid spatial separation. The high-velocity asymptotics then yield a reconstruction formula for the X-ray transform of the coefficient, from which it follows that the scattering operator uniquely determines both the nonlinearity exponent and the spatial coefficient.

What carries the argument

Moving frame adapted to highly boosted initial data, used to establish high-velocity asymptotics that make the nonlinear interaction effectively weak despite large Sobolev norms.

If this is right

  • A reconstruction formula recovers the X-ray transform of the spatial coefficient directly from high-velocity scattering data.
  • The nonlinearity exponent is uniquely fixed once the scattering operator is known.
  • The spatial coefficient is uniquely fixed once the scattering operator is known.
  • The results apply in every dimension d at least 3 and to all mass-supercritical energy-subcritical powers including the endpoints.
  • Only decay conditions on the coefficient are required; repulsiveness and radial monotonicity are no longer needed.

Where Pith is reading between the lines

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

  • The same high-velocity reduction might apply to other dispersive equations whose nonlinear terms separate under Galilean boosts.
  • Numerical recovery of the coefficient could be tested by simulating scattering for a sequence of increasingly boosted data and inverting the X-ray transform.
  • Stability of the reconstruction with respect to small perturbations in the scattering operator remains open and would be a natural next quantitative question.

Load-bearing premise

The nonlinear interaction becomes effectively weak at high velocity due to rapid spatial separation, even though the boosted data are large in Sobolev norms.

What would settle it

Construct two distinct pairs of exponent and coefficient satisfying the decay conditions that produce identical scattering operators on a sequence of high-velocity boosted states; if such pairs exist the uniqueness claim fails.

read the original abstract

We study a high-velocity inverse scattering problem for nonlinear Schr\"odinger equations with spatially dependent nonlinearities in dimensions $d\ge3$. We consider the whole mass-supercritical and energy-subcritical range, including the endpoint cases. By introducing a moving frame adapted to highly boosted initial data, we construct the scattering operator for a class of large incoming states generated by Galilean boosts. The key observation is that, although the boosted data become large in Sobolev norms, the nonlinear interaction becomes effectively weak at high velocity due to rapid spatial separation. Using the resulting high-velocity asymptotics, we derive a reconstruction formula for the X-ray transform of the coefficient. As a consequence, we prove that the scattering operator uniquely determines both the nonlinearity exponent and the spatial coefficient. Our results extend previous work of Watanabe to all dimensions $d \ge 3$, include the endpoint nonlinearities, and replace the repulsiveness and radial monotonicity assumptions on the coefficient by suitable decay conditions.

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 / 2 minor

Summary. The paper studies high-velocity inverse scattering for the NLS equation i∂_t u + Δu = |u|^{p-1} u V(x) (or similar spatially dependent power nonlinearity) in d ≥ 3. It constructs the scattering operator for large Galilean-boosted incoming states by working in a moving frame, where rapid spatial separation makes the nonlinear interaction effectively weak despite large Sobolev norms. From the resulting high-velocity asymptotics it derives an explicit reconstruction formula for the X-ray transform of the coefficient V and proves that the scattering operator uniquely determines both the exponent p and the function V. The results cover the full mass-supercritical/energy-subcritical range (including endpoints) and replace radial monotonicity/repulsiveness assumptions by suitable decay conditions on V, extending prior work of Watanabe.

Significance. If the central estimates hold, the work is a meaningful advance in inverse scattering for nonlinear dispersive equations. It supplies a concrete reconstruction procedure via the X-ray transform and establishes uniqueness for both the power and the spatial coefficient under decay-only hypotheses, which broadens the applicability beyond previous radial settings. The moving-frame/high-velocity approach is technically natural for this regime and may serve as a template for related problems. No machine-checked proofs or fully parameter-free derivations are present, but the explicit reconstruction formula is a clear strength.

major comments (1)
  1. [§3 and §4] §3 (moving-frame construction) and §4 (high-velocity asymptotics): the claim that the Duhamel integral of the nonlinearity vanishes as |v|→∞ uniformly enough to yield a clean linear asymptotics plus an exact X-ray term rests on the nonlinear term becoming o(1) despite ||u_v(0)||_{H^s} growing with |v|. For polynomial decay of V and endpoint p = 1 + 4/d, the time scale of interaction ∼1/|v| together with the |v|-dependent Strichartz constants in the boosted frame may leave a remainder that is not o(1) uniformly; this directly affects whether the reconstruction formula isolates the X-ray transform without contamination and therefore underpins the uniqueness statement for both p and V.
minor comments (2)
  1. [Introduction] The abstract and introduction state that radial monotonicity is replaced by decay, but the precise decay rate (e.g., |x|^{-α} with α > ?) required for the endpoint estimates is not stated in the introduction; it should be made explicit early.
  2. [§2] Notation for the boosted data u_v and the moving-frame variable should be introduced once and used consistently; several paragraphs in §2 mix Galilean boost notation with the frame shift.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive summary and for identifying a key technical point in the high-velocity analysis. We address the concern regarding the uniformity of the Duhamel remainder in §§3–4 below.

read point-by-point responses

  1. Referee: [§3 and §4] §3 (moving-frame construction) and §4 (high-velocity asymptotics): the claim that the Duhamel integral of the nonlinearity vanishes as |v|→∞ uniformly enough to yield a clean linear asymptotics plus an exact X-ray term rests on the nonlinear term becoming o(1) despite ||u_v(0)||_{H^s} growing with |v|. For polynomial decay of V and endpoint p = 1 + 4/d, the time scale of interaction ∼1/|v| together with the |v|-dependent Strichartz constants in the boosted frame may leave a remainder that is not o(1) uniformly; this directly affects whether the reconstruction formula isolates the X-ray transform without contamination and therefore underpins the uniqueness statement for both p and V.

    Authors: The moving-frame change of variables is constructed precisely so that the Galilean boost cancels the leading |v|-growth in the Strichartz constants while the spatial separation induced by the boost localizes the nonlinearity to an interval of length O(1/|v|). Under the assumed polynomial decay of V, the integral of the nonlinearity over this short interval is controlled by a product of an L^{p+1} norm that remains bounded (by conservation and the boost) and a factor that decays as |v|^{-α} for some α>0 coming from the spatial decay; the endpoint Strichartz estimates in d≥3 close the argument without additional |v|-dependent losses. The same estimates yield the exact X-ray term plus o(1) remainder uniformly in the relevant range, which is what permits both the reconstruction formula and the uniqueness for p and V. We therefore maintain that the stated hypotheses on V suffice and that the asymptotics hold as claimed. revision: no

Circularity Check

0 steps flagged

Derivation self-contained via external estimates; no circular reductions

full rationale

The paper derives the scattering operator and high-velocity asymptotics from moving-frame analysis and the observation that nonlinear interactions weaken due to spatial separation under Galilean boosts, then extracts an X-ray transform reconstruction and uniqueness for the exponent and coefficient. These steps rely on analytic estimates (Strichartz, Duhamel control) under stated decay conditions that replace prior radial/repulsive assumptions, extending Watanabe's external work rather than reducing to self-citations or fitted inputs. No equation or claim is shown to equal its inputs by construction, and the central uniqueness follows from the derived asymptotics without tautological redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on abstract only; the work rests on standard functional-analytic assumptions for scattering theory and decay at infinity for the coefficient. No free parameters or invented entities are visible.

axioms (2)
  • domain assumption The nonlinearity lies in the mass-supercritical and energy-subcritical range (including endpoints) for d ≥ 3
    Explicitly stated as the setting considered; required for the scattering construction to hold.
  • domain assumption Suitable decay conditions at spatial infinity on the coefficient replace earlier repulsiveness/monotonicity assumptions
    Invoked to obtain the reconstruction and uniqueness; location: abstract paragraph on assumptions.

pith-pipeline@v0.9.1-grok · 5700 in / 1448 out tokens · 24541 ms · 2026-06-28T13:56:53.170875+00:00 · methodology

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Reference graph

Works this paper leans on

12 extracted references · 10 canonical work pages

  1. [1]

    Gong Chen and Jason Murphy,Recovery of the nonlinearity from the modified scattering map, Int. Math. Res. Not. IMRN8(2024), 6632–6655, DOI 10.1093/imrn/rnad243. MR4735639

    work page doi:10.1093/imrn/rnad243 2024

  2. [2]

    Volker Enss and Ricardo Weder,The geometrical approach to multidimensional inverse scattering, J. Math. Phys.36(1995), no. 8, 3902–3921, DOI 10.1063/1.530937. MR1341964

    work page doi:10.1063/1.530937 1995

  3. [3]

    Rowan Killip, Jason Murphy, and Monica Visan,The scattering map determines the nonlinearity, Proc. Amer. Math. Soc.151(2023), no. 6, 2543–2557, DOI 10.1090/proc/16297. MR4576319

    work page doi:10.1090/proc/16297 2023

  4. [4]

    5, Birkh¨ auser, Boston, 1999

    Sigurdur Helgason,The Radon Transform, 2nd ed., Progress in Mathematics, vol. 5, Birkh¨ auser, Boston, 1999

    1999

  5. [5]

    1, Paper No

    Rowan Killip, Jason Murphy, and Monica Visan,Determination of Schr¨ odinger nonlinearities from the scat- tering map, Nonlinearity38(2025), no. 1, Paper No. 015021, 20, DOI 10.1088/1361-6544/ada1bf. MR4844579

    work page doi:10.1088/1361-6544/ada1bf 2025

  6. [6]

    Partial Differential Equations48(2023), no

    Jason Murphy,Recovery of a spatially-dependent coefficient from the NLS scattering map, Comm. Partial Differential Equations48(2023), no. 7-8, 991–1007, DOI 10.1080/03605302.2023.2241546. MR4645492

    work page doi:10.1080/03605302.2023.2241546 2023

  7. [7]

    Partial Differential Equations33(2008), no

    Hironobu Sasaki,Inverse scattering for the nonlinear Schr¨ odinger equation with the Yukawa potential, Comm. Partial Differential Equations33(2008), no. 7-9, 1175–1197, DOI 10.1080/03605300701790245. MR2450155

    work page doi:10.1080/03605300701790245 2008

  8. [8]

    Differential Equations252(2012), no

    ,Inverse scattering problems for the Hartree equation whose interaction potential decays rapidly, J. Differential Equations252(2012), no. 2, 2004–2023, DOI 10.1016/j.jde.2011.07.022. MR2853569

    work page doi:10.1016/j.jde.2011.07.022 2012

  9. [9]

    Strauss,Nonlinear scattering theory, Scattering Theory in Mathematical Physics (1974), 53–78

    Walter A. Strauss,Nonlinear scattering theory, Scattering Theory in Mathematical Physics (1974), 53–78

    1974

  10. [10]

    Math.24(2001), no

    Michiyuki Watanabe,Inverse scattering for the nonlinear Schr¨ odinger equation with cubic convolution non- linearity, Tokyo J. Math.24(2001), no. 1, 59–67, DOI 10.3836/tjm/1255958311. MR1844417

    work page doi:10.3836/tjm/1255958311 2001

  11. [11]

    Partial Differential Equa- tions22(1997), no

    Ricardo Weder,Inverse scattering for the nonlinear Schr¨ odinger equation, Comm. Partial Differential Equa- tions22(1997), no. 11-12, 2089–2103, DOI 10.1080/03605309708821332. MR1629534

    work page doi:10.1080/03605309708821332 1997

  12. [12]

    ,Inverse scattering for the nonlinear Schr¨ odinger equation. II. Reconstruction of the potential and the nonlinearity in the multidimensional case, Proc. Amer. Math. Soc.129(2001), no. 12, 3637–3645, DOI 10.1090/S0002-9939-01-06016-6. MR1860498 Department of mathematics, Hokkaido University, Sapporo Hokkaido, 060-0810, Japan Email address:masaki@math.sci...

    work page doi:10.1090/s0002-9939-01-06016-6 2001