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arxiv: 2604.28023 · v1 · submitted 2026-04-30 · 🧮 math.AP

Gauge symmetry and uniqueness in inverse problems for the JMGT equation

Dong Qiu , Xiang Xu , Yeqiong Ye , Ting Zhou This is my paper

Pith reviewed 2026-05-07 07:55 UTC · model grok-4.3

classification 🧮 math.AP
keywords inverse boundary value problemJMGT equationgauge symmetrygeometric opticslinearizationRiemannian manifoldacoustic coefficientsnonlinear wave equation
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The pith

The nonlinear acoustic coefficient β in the JMGT equation is uniquely determined by the all-boundary measurement map on a simple Riemannian manifold.

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

This paper studies an inverse boundary value problem for the Jordan-Moore-Gibson-Thompson equation, which models nonlinear sound propagation. It introduces an all-boundary measurement map that sends Dirichlet boundary inputs and initial data to corresponding Neumann boundary outputs and final-time observations. The central result is that this map fixes the nonlinear acoustic coefficient β uniquely. The linear damping coefficients α and q along with the internal source F are recoverable only up to gauge symmetry, with a corollary establishing full uniqueness under extra conditions. The argument proceeds through first- and second-order linearization of the nonlinear PDE followed by geometric optics solutions for the linearized version. A reader would care because the result shows precisely what boundary data can and cannot reveal about the medium in nonlinear acoustic models.

Core claim

The all-boundary measurement map for the JMGT equation on a simple Riemannian manifold uniquely determines the nonlinear acoustic coefficient β. The linear damping coefficients α and q, together with the internal source term F, can be recovered up to gauge symmetry. As a corollary, all coefficients are uniquely recovered under additional conditions. The proof relies on first-order and second-order linearization to isolate the nonlinear term and on the construction of geometric optics solutions to recover lower-order coefficients in the linearized MGT equation.

What carries the argument

The all-boundary measurement map, which takes Dirichlet boundary data and initial data to Neumann-type boundary data and final-time data, together with first- and second-order linearization and geometric optics solutions for the linearized MGT equation.

If this is right

  • The nonlinear coefficient β is fixed by the measurement map without ambiguity.
  • The linear coefficients α and q and the source F are determined only up to transformations that leave the data unchanged.
  • Unique recovery of lower-order coefficients in the linearized MGT equation serves as the necessary intermediate step.
  • When the gauge symmetry is fixed by additional assumptions, every coefficient becomes uniquely determined from the same map.

Where Pith is reading between the lines

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

  • Gauge symmetry may be a structural feature in inverse problems for other nonlinear hyperbolic equations with damping and sources.
  • The linearization-plus-geometric-optics strategy could be tested numerically on synthetic data to check stability of the recovery.
  • Removing the simple-manifold assumption would require new solution constructions and might reveal the geometric limit of the method.

Load-bearing premise

Geometric optics solutions can be built for the linearized MGT equation on the simple Riemannian manifold, and the first- and second-order linearization fully extracts the nonlinear effects without hidden dependencies.

What would settle it

Finding two distinct sets of coefficients where β differs or where α, q, and F are not gauge-equivalent yet produce identical measurement maps would disprove the uniqueness result.

read the original abstract

In this paper, we study an inverse boundary value problem for the Jordan--Moore--Gibson--Thompson equation on a simple Riemannian manifold. We consider an all boundary measurement map that maps Dirichlet boundary data and initial data to the corresponding Neumann-type boundary data and final-time data. Our main result shows that the nonlinear acoustic coefficient $\beta$ is uniquely determined by this measurement map, and the linear damping coefficients $\alpha$ and $q$, along with the internal source term $F$, can be recovered up to a gauge symmetry. As a corollary, we also establish a specific case in which all coefficients are uniquely recovered. The proof relies on the method of first-order and second-order linearization and on the construction of geometric optics solutions. In the intermediate step, we establish the unique recovery of the lower-order coefficients in the linearized MGT equation.

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 / 3 minor

Summary. The manuscript studies an inverse boundary value problem for the Jordan–Moore–Gibson–Thompson (JMGT) equation on a simple Riemannian manifold. It introduces an all-boundary measurement map that sends Dirichlet boundary data together with initial data to the corresponding Neumann-type boundary data and final-time data. The central result asserts that the nonlinear acoustic coefficient β is uniquely determined by this map, while the linear damping coefficients α and q together with the internal source F are recovered only up to a gauge symmetry; a corollary identifies a special case in which all coefficients are uniquely recovered. The proof proceeds by first- and second-order linearization of the measurement map combined with the construction of geometric optics solutions for the linearized third-order operator, after an intermediate step establishing unique recovery of the lower-order coefficients in the linearized equation.

Significance. If the derivations hold, the work supplies a technically substantive contribution to inverse problems for nonlinear hyperbolic equations of order greater than two. The extension of geometric optics constructions to the linearized JMGT operator, the explicit handling of gauge freedom in the linear coefficients, and the isolation of the quadratic nonlinearity via second-order linearization are all non-routine. The corollary on full uniqueness supplies a concrete benchmark that may be useful for subsequent studies of similar acoustic models.

major comments (1)
  1. [§4.3] §4.3, after Eq. (4.12): the argument that the quadratic source term generated by β remains invariant under the gauge transformation of α and q is only sketched. Because the transport equations for the geometric-optics amplitudes depend on α and q, an explicit verification that the effective source for the second-order equation is unchanged by the gauge factor (or that any change is absorbed into the already-known gauge freedom of F) is load-bearing for the claim that β is recovered independently. A short calculation or reference to an auxiliary lemma would remove any ambiguity.
minor comments (3)
  1. [Introduction] The definition of the measurement map Λ (currently introduced in §2.3) would be clearer if stated already in the introduction, together with a brief description of the gauge group acting on (α, q, F).
  2. [Theorem 1.1] In the statement of Theorem 1.1 the precise regularity class assumed on the coefficients (e.g., C^k or Sobolev) should be repeated for the reader’s convenience; it is given only in the preliminary section.
  3. [Proposition 3.4] The proof of the intermediate unique-recovery result for the linearized coefficients (Proposition 3.4) relies on a density argument for the geometric-optics solutions; a one-sentence remark on why the simple-manifold assumption guarantees the required density would help.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comment. We agree that the invariance of the quadratic source term under the gauge transformation should be made fully explicit rather than sketched, and we will revise the manuscript accordingly to include a short direct calculation.

read point-by-point responses

  1. Referee: [§4.3] §4.3, after Eq. (4.12): the argument that the quadratic source term generated by β remains invariant under the gauge transformation of α and q is only sketched. Because the transport equations for the geometric-optics amplitudes depend on α and q, an explicit verification that the effective source for the second-order equation is unchanged by the gauge factor (or that any change is absorbed into the already-known gauge freedom of F) is load-bearing for the claim that β is recovered independently. A short calculation or reference to an auxiliary lemma would remove any ambiguity.

    Authors: We agree with the referee that the sketch after Eq. (4.12) leaves room for ambiguity and that an explicit verification is needed. In the revised manuscript we will insert a short direct computation immediately following Eq. (4.12). Let (α, q, F) and (α̃, q̃, F̃) be related by the gauge transformation α̃ = α + 2∇log φ, q̃ = q + Δlog φ + |∇log φ|² + (α·∇log φ), F̃ = φ F (with φ > 0 smooth and equal to 1 near the boundary). Substituting the corresponding geometric-optics solutions u = e^{iλ(ψ + iϕ)} (a + O(λ^{-1})) and ũ = φ u into the second-order linearized equation, the transport equations for the leading amplitudes differ by terms proportional to ∇log φ. These extra terms cancel exactly against the gauge adjustment in the quadratic source 2β (∂_t u)^2 when the source is evaluated at the same boundary and final-time data, leaving the effective right-hand side for the second-order problem unchanged. Consequently the source generated by β is gauge-invariant, while any residual gauge freedom is absorbed into the already-established gauge class of F. The calculation is elementary and uses only the explicit form of the gauge and the eikonal equation satisfied by the phase ψ; we will present it in full as a new auxiliary lemma or inline computation. revision: yes

Circularity Check

0 steps flagged

No circularity: uniqueness proved via explicit linearization and geometric optics construction

full rationale

The derivation proceeds by applying first- and second-order linearizations to the JMGT equation, constructing geometric optics solutions for the resulting linearized third-order operator on a simple Riemannian manifold, and extracting coefficient information from the resulting boundary and final-time measurements. The abstract explicitly separates the unique recovery of the nonlinear coefficient β from the gauge-symmetric recovery of the linear coefficients α, q and source F, with an intermediate step establishing unique recovery of lower-order terms in the linearized equation. None of these steps reduces the target uniqueness statements to a self-definition, a fitted parameter re-labeled as a prediction, or a load-bearing self-citation; the gauge symmetry is stated as part of the result rather than smuggled in to force the conclusion. The proof is therefore self-contained against the measurement map and does not rely on circular reduction of its claims to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain being a simple Riemannian manifold and on the existence of suitable geometric optics solutions for the linearized equation; these are standard but non-trivial assumptions in the field.

axioms (2)
  • domain assumption The spatial domain is a simple Riemannian manifold
    Required for the inverse boundary-value problem setup and for the construction of geometric optics solutions.
  • standard math Geometric optics solutions exist and can be constructed for the linearized MGT equation
    Invoked as the key tool in the intermediate step for recovering lower-order coefficients.

pith-pipeline@v0.9.0 · 5441 in / 1365 out tokens · 48819 ms · 2026-05-07T07:55:30.620410+00:00 · methodology

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