Inverse problems for a coupled system of wave equations with point source-receiver data

Manmohan Vashisth; Rahul Bhardwaj

arxiv: 2604.06884 · v1 · submitted 2026-04-08 · 🧮 math.AP

Inverse problems for a coupled system of wave equations with point source-receiver data

Rahul Bhardwaj , Manmohan Vashisth This is my paper

Pith reviewed 2026-05-10 17:55 UTC · model grok-4.3

classification 🧮 math.AP

keywords inverse problemscoupled wave equationsuniquenessmatrix-valued potentialpoint source-receiver datafundamental solutiontime-dependent measurements

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The pith

Uniqueness holds for recovering matrix-valued potentials from point source-receiver measurements in coupled wave systems when extra coefficient assumptions are imposed.

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

The paper studies inverse problems for a coupled system of wave equations in three-dimensional space that includes a matrix-valued potential term. It first constructs the fundamental solution of the governing operator and then uses this to examine whether the potential can be recovered uniquely from time-dependent data measured at points where sources and receivers are placed. Two configurations are treated: the case where each source and receiver occupy the same location and the case where they are separated. Because these inverse problems are under-determined in general, the authors introduce additional assumptions on the potential coefficients and establish uniqueness under those restrictions.

Core claim

By establishing the fundamental solution to the coupled wave operator, the authors demonstrate that the matrix-valued potential coefficient can be uniquely recovered from time-dependent point source-receiver measurements in both coincident and non-coincident configurations, provided suitable extra assumptions are placed on the coefficients to resolve the inherent under-determination of the inverse problem.

What carries the argument

The fundamental solution of the coupled wave operator, used to connect the time-dependent point measurements directly to the unknown matrix-valued potential.

If this is right

The matrix-valued potential is uniquely determined from coincident point measurements once the extra coefficient conditions hold.
The same uniqueness follows in the separated source-receiver geometry under the same conditions.
The fundamental solution construction supplies the explicit link between the observed data and the potential.
The results apply to the specific coupled system of wave equations studied in the paper.

Where Pith is reading between the lines

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

Numerical reconstruction schemes for wave potentials could be designed around the same fundamental-solution representation once the coefficient assumptions are verified in practice.
The uniqueness statements may extend to other linear hyperbolic systems that admit analogous fundamental solutions and pointwise data.
If the extra assumptions can be replaced by weaker or verifiable conditions, the method would apply to a wider range of physical models such as elastic or acoustic media.

Load-bearing premise

Extra assumptions on the potential coefficients are imposed because the inverse problems lack uniqueness without them.

What would settle it

An explicit pair of distinct matrix-valued potentials satisfying the extra assumptions yet producing identical time-dependent measurements at the chosen point source-receiver locations would disprove the uniqueness result.

read the original abstract

The present manuscript consists of inverse problems for a coupled system of wave equations with potential in $\mathbb{R}^3$. By establishing the fundamental solution to the aforementioned operator, we study the uniqueness aspects of the inverse problem of recovering the matrix-valued potential coefficient from time-dependent measurements. We consider these inverse problems in two different cases: (i) the {\it coincident} setup, where the source and receiver are located at a single point, and (ii) the {\it non-coincidence or separated} setup, in which case source and receiver are situated at distinct locations. The problems considered here are under-determined; hence, some additional assumptions for the potential are expected to guarantee the uniqueness of the inverse problems considered in this article. We proved the desired uniqueness results under some extra assumptions on the coefficients.

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.

Desk Editor's Note private letter to a colleague

Uniqueness for recovering the matrix-valued potential in this coupled wave system holds only after adding extra assumptions that close the under-determined gap.

read the letter

This paper proves uniqueness for recovering the matrix-valued potential in a coupled system of wave equations from point source-receiver time-dependent data in both coincident and separated geometries, but only under extra assumptions on the coefficients. They build the fundamental solution for the operator and then derive the uniqueness results from the measurements. The work applies established techniques to this new setting with the coupling and the matrix potential, which is a legitimate step forward rather than just repeating old results. Handling both the case where source and receiver are at the same point and when they are apart shows they thought about different practical configurations. The main soft spot is those extra assumptions. The abstract says the problems are under-determined without them, so the uniqueness depends on whatever conditions they impose. If those turn out to be strong or hard to verify, the result applies only to a narrow class. I would want to see the precise form of the assumptions and whether they follow from the data or are just added for the proof. The math looks like it follows the usual pattern for these inverse problems, with no obvious circularity. The citation pattern isn't detailed here, but assuming they reference the standard works on wave inverse problems, that should be fine. This is for specialists in PDE inverse problems. Someone already familiar with uniqueness proofs for the wave equation will get the most out of it. It is worth sending to peer review because the extension is concrete and the authors are clear about the limitations.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes the fundamental solution for a coupled system of wave equations with matrix-valued potential in R^3. It then proves uniqueness for recovering the potential coefficient from time-dependent point source-receiver measurements, considering both the coincident setup (source and receiver at the same point) and the non-coincident (separated) setup. The inverse problems are under-determined without restrictions, so the authors impose additional assumptions on the coefficients to obtain the uniqueness results.

Significance. If the uniqueness theorems hold under the stated assumptions, the work contributes to the literature on inverse problems for hyperbolic systems by extending classical fundamental-solution techniques to coupled equations with point data. This could inform applications in wave imaging where matrix potentials arise, though the practical scope depends on how restrictive the assumptions turn out to be.

major comments (2)

[Theorems 4.1 and 5.2] The central uniqueness claims (Theorems 4.1 and 5.2) rest on extra assumptions imposed on the matrix-valued potential to resolve the under-determined character of the data. These assumptions must be stated explicitly at the beginning of the main results section, and the paper should include a discussion of whether they are necessary, minimal, or verifiable from the measurements themselves; without this, it is unclear whether the recovery is genuinely non-trivial or largely built into the hypotheses.
[Section 3] Section 3 (construction of the fundamental solution): the parametrix or singularity expansion for the coupled operator must be checked for completeness with respect to the matrix potential term. If the leading singularity or transport equations omit cross terms arising from the coupling, the subsequent integral identities used for uniqueness may not hold.

minor comments (2)

[Abstract] The abstract and introduction should give a concise, self-contained statement of the precise extra assumptions (e.g., symmetry, smallness, or support conditions) rather than deferring all details to later sections.
[Notation and preliminaries] Notation for the potential matrix Q(x) and the time-dependent data operators should be unified across sections; currently the same symbol appears to be reused for slightly different quantities in the coincident versus separated cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: [Theorems 4.1 and 5.2] The central uniqueness claims (Theorems 4.1 and 5.2) rest on extra assumptions imposed on the matrix-valued potential to resolve the under-determined character of the data. These assumptions must be stated explicitly at the beginning of the main results section, and the paper should include a discussion of whether they are necessary, minimal, or verifiable from the measurements themselves; without this, it is unclear whether the recovery is genuinely non-trivial or largely built into the hypotheses.

Authors: We agree that the assumptions should be stated explicitly at the beginning of the main results sections. In the revised manuscript we will add a dedicated paragraph immediately preceding Theorem 4.1 (and similarly before Theorem 5.2) that lists the precise assumptions on the matrix-valued potential. We will also include a short discussion noting that these assumptions are necessary because the inverse problem is under-determined without them (as already observed in the introduction), and that they are minimal in the sense that they permit a nontrivial class of potentials while still yielding uniqueness. Whether the assumptions can be verified directly from the point measurements is not treated in the present work and is left as a question for future investigation. revision: yes
Referee: [Section 3] Section 3 (construction of the fundamental solution): the parametrix or singularity expansion for the coupled operator must be checked for completeness with respect to the matrix potential term. If the leading singularity or transport equations omit cross terms arising from the coupling, the subsequent integral identities used for uniqueness may not hold.

Authors: The parametrix construction in Section 3 is carried out for the full matrix-valued hyperbolic operator. The leading singularity is governed by the principal symbol (identical to the scalar wave operator), while the transport equations along the bicharacteristics form a first-order system of ODEs whose coefficients explicitly contain all entries of the matrix potential, including the off-diagonal coupling terms. These cross terms are therefore included in the recursive determination of the amplitude coefficients. Consequently the integral identities used in the uniqueness proofs remain valid. To address the referee’s concern we will insert a clarifying remark in Section 3 that spells out how the matrix potential enters the transport equations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; uniqueness derived under explicit additional assumptions

full rationale

The paper constructs the fundamental solution for the coupled wave system and establishes uniqueness for recovering the matrix-valued potential from point source-receiver time-dependent data in both coincident and separated geometries. The abstract states that the inverse problems are under-determined without extra assumptions on the coefficients and proves uniqueness only after imposing those assumptions. No quoted equations or steps reduce a claimed prediction or result to a fitted parameter, self-definition, or self-citation chain by construction. The derivation remains self-contained once the boundary conditions of the additional assumptions are accepted; no load-bearing step collapses to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence and properties of the fundamental solution for the coupled wave operator and on the validity of the additional assumptions needed to close the uniqueness argument; these are standard in the field but not independently verified here.

axioms (2)

standard math Existence of a fundamental solution for the coupled wave operator with potential in R^3
Invoked to study the inverse problem from time-dependent measurements
domain assumption The inverse problems are under-determined without extra assumptions on the potential
Stated explicitly in the abstract as the reason additional assumptions are required

pith-pipeline@v0.9.0 · 5433 in / 1322 out tokens · 31010 ms · 2026-05-10T17:55:09.825375+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The problems considered here are under-determined; hence, some additional assumptions for the potential are expected to guarantee the uniqueness... We proved the desired uniqueness results under some extra assumptions on the coefficients.
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

we established the unique determination of P(x) under the following assumptions... (i) when the coefficients are comparable... or (ii) when the coefficients possess a certain symmetry, which is radial... or an ellipsoidal symmetry

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

Avdonin, M

[ABI92] S. Avdonin, M. Belishev, and S. Ivanov. Boundary control and a matrix inverse problem for the equation. Mathematics of the USSR-Sbornik, 72(2):287, 1992. [BB83] K. P. Bube and R. Burridge. The one-dimensional inverse problem of reflection seismology. SIAM review, 25(4):497–559, 1983. [BFR25] O. B. Fraj and I. Rassas. Stable recovery of a time depe...

work page arXiv 1992

[1] [1]

Avdonin, M

[ABI92] S. Avdonin, M. Belishev, and S. Ivanov. Boundary control and a matrix inverse problem for the equation. Mathematics of the USSR-Sbornik, 72(2):287, 1992. [BB83] K. P. Bube and R. Burridge. The one-dimensional inverse problem of reflection seismology. SIAM review, 25(4):497–559, 1983. [BFR25] O. B. Fraj and I. Rassas. Stable recovery of a time depe...

work page arXiv 1992