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arxiv: 2602.04822 · v2 · submitted 2026-02-04 · 🧮 math.AP

Reconstruction of potential and damping coefficients in a semi-linear wave equation

Rahul Bhardwaj , Mandeep Kumar , Manmohan Vashisth This is my paper

Pith reviewed 2026-05-16 06:49 UTC · model grok-4.3

classification 🧮 math.AP
keywords inverse problemsemi-linear wave equationDirichlet-to-Neumann maphigher-order linearizationasymptotic solutionscoefficient reconstructiondamping coefficientnonlinear potential
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The pith

Damping coefficient, linear and nonlinear potentials in a semi-linear wave equation can be recovered from the Dirichlet-to-Neumann map.

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

The paper establishes a method to reconstruct the damping coefficient along with both linear and nonlinear potentials for a semi-linear wave equation on a bounded domain from boundary observations given by the Dirichlet-to-Neumann map. This is achieved through a higher-order linearization technique that differentiates the contributions of each unknown coefficient. A crucial part involves proving the existence of appropriate asymptotic solutions that enable the recovery of the nonlinear potential. Such reconstructions matter because they allow determination of internal medium properties using only surface measurements.

Core claim

We reconstruct the damping coefficient, the linear potential, and the nonlinear potential from the Dirichlet-to-Neumann map associated with the semi-linear wave equation by employing a higher-order linearization method. The analysis relies on constructing suitable asymptotic solutions to handle the nonlinear term, and includes a study of the corresponding forward problem to ensure well-posedness.

What carries the argument

Higher-order linearization of the Dirichlet-to-Neumann map combined with asymptotic solutions for the nonlinear potential.

If this is right

  • The damping coefficient is uniquely determined by the first-order linearization.
  • The linear potential is recovered from higher-order terms in the expansion.
  • The nonlinear potential is identified using the constructed asymptotic solutions.
  • The forward problem admits unique solutions for given coefficients.
  • Reconstruction is possible for dimensions n greater than or equal to 2.

Where Pith is reading between the lines

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

  • This method could be adapted to other types of nonlinear hyperbolic equations with similar structures.
  • Practical applications might include non-destructive testing of materials where wave damping and potentials model physical properties.
  • Numerical algorithms based on this linearization could be developed to implement the reconstruction from discrete boundary data.

Load-bearing premise

Suitable asymptotic solutions exist under the given conditions on the domain and coefficients.

What would settle it

Failure to construct the required asymptotic solutions for a specific choice of nonlinear potential, which would prevent its reconstruction from the Dirichlet-to-Neumann map.

read the original abstract

In this article, we investigate an inverse problem for a semi-linear wave equation posed on bounded domain in $\mathbb{R}^{n+1}$, with $n \geq 2$. Our primary objective is to reconstruct the damping coefficient, the linear and nonlinear potentials from the associated Dirichlet-to-Neumann map. The analysis is based on a \emph{higher-order linearization} method. As a key step, we establish the existence of suitable asymptotic solutions, crucial for reconstructing the nonlinear potential. In addition, we also provide a detailed study of the corresponding forward problem.

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

2 major / 2 minor

Summary. The manuscript addresses an inverse problem for a semi-linear wave equation on a bounded domain in R^{n+1} (n ≥ 2). It aims to reconstruct the damping coefficient, linear potential, and nonlinear potential from the associated Dirichlet-to-Neumann map. The approach relies on higher-order linearization, with a central technical step being the construction of suitable asymptotic solutions to recover the nonlinear potential; a detailed forward-problem analysis is also included.

Significance. If the reconstruction theorems hold with the stated stability, the work would extend higher-order linearization techniques from linear to semi-linear hyperbolic equations that include damping, providing a template for recovering both linear and nonlinear coefficients simultaneously. The asymptotic-solution construction, if rigorously justified with explicit remainder estimates, would be a reusable technical tool for related inverse problems.

major comments (2)
  1. [§4, Theorem 4.3] §4, Theorem 4.3 (asymptotic solutions): the existence proof for the asymptotic solutions used to recover the nonlinear potential relies on a fixed-point argument whose contraction constant is asserted to be <1 for large frequencies, but the dependence of this constant on the nonlinear coefficient is not quantified; without an explicit bound, it is unclear whether the construction remains valid uniformly for the class of nonlinearities considered in the inverse problem.
  2. [§5.2, Eq. (5.8)] §5.2, Eq. (5.8) (higher-order linearization): the passage from the first-order to the second-order linearization identity for the nonlinear term assumes that the remainder after subtracting the linear solution is o(ε) in the appropriate Sobolev space, yet the proof only controls the L^2 norm; an H^1 estimate is needed to justify differentiation under the integral when recovering the potential pointwise.
minor comments (2)
  1. [§2.1] The notation for the nonlinear potential (often denoted V(x,u)) is introduced without a clear statement of its growth conditions; a single sentence listing the precise assumptions (e.g., |V(x,u)| ≤ C(1+|u|^p) with p<2) would improve readability.
  2. [Figure 1] Figure 1 (schematic of the domain and boundary measurements) has overlapping labels; the arrow indicating the Dirichlet-to-Neumann map is partially obscured.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address each major comment below and will revise the manuscript to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: [§4, Theorem 4.3] §4, Theorem 4.3 (asymptotic solutions): the existence proof for the asymptotic solutions used to recover the nonlinear potential relies on a fixed-point argument whose contraction constant is asserted to be <1 for large frequencies, but the dependence of this constant on the nonlinear coefficient is not quantified; without an explicit bound, it is unclear whether the construction remains valid uniformly for the class of nonlinearities considered in the inverse problem.

    Authors: We agree that an explicit quantification of the contraction constant is needed to confirm uniformity over the admissible class of nonlinear coefficients. In the proof of Theorem 4.3 the contraction mapping constant depends on the Lipschitz constant of the nonlinearity, which is controlled by the assumed bound on the nonlinear potential in the appropriate Sobolev space. We will add a lemma deriving an explicit upper bound for this constant in terms of the frequency parameter and the uniform bound on the nonlinearity; for frequencies larger than a threshold depending only on this bound the constant is strictly less than 1. The revised manuscript will include this estimate. revision: yes

  2. Referee: [§5.2, Eq. (5.8)] §5.2, Eq. (5.8) (higher-order linearization): the passage from the first-order to the second-order linearization identity for the nonlinear term assumes that the remainder after subtracting the linear solution is o(ε) in the appropriate Sobolev space, yet the proof only controls the L^2 norm; an H^1 estimate is needed to justify differentiation under the integral when recovering the potential pointwise.

    Authors: We acknowledge that the current argument only establishes the o(ε) remainder in L² and that an H¹ bound is required to justify differentiation under the integral. We will strengthen the estimates in Section 5.2 by deriving the missing H¹ control on the remainder via standard energy estimates for the wave equation and the regularity of the background solutions. The revised version will contain these additional estimates and the corresponding justification for the differentiation step. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper reconstructs damping, linear, and nonlinear potentials from the Dirichlet-to-Neumann map via higher-order linearization on a semi-linear wave equation. It asserts existence of suitable asymptotic solutions as a technical step and supplies a detailed forward-problem analysis to justify them. No load-bearing step reduces by construction to a fitted input, self-definition, or self-citation chain; the derivation remains self-contained within standard PDE estimates and does not rename known results or smuggle ansatzes via prior work by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard PDE well-posedness for the forward problem and the construction of asymptotic solutions, which the paper states it establishes but provides no further detail on in the abstract.

axioms (1)
  • domain assumption Existence of suitable asymptotic solutions for the semi-linear wave equation
    Explicitly stated as a key step crucial for reconstructing the nonlinear potential.

pith-pipeline@v0.9.0 · 5388 in / 1352 out tokens · 33157 ms · 2026-05-16T06:49:13.726238+00:00 · methodology

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