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arxiv: 2604.06340 · v1 · submitted 2026-04-07 · 🧮 math.AP

On the Jordan-Moore-Gibson-Thompson equation of nonlinear acoustics

Barbara Kaltenbacher This is my paper

Pith reviewed 2026-05-10 18:38 UTC · model grok-4.3

classification 🧮 math.AP
keywords Jordan-Moore-Gibson-Thompson equationnonlinear acousticswell-posednesssingular limitsmemory attenuationfractional derivativescontrol problemsinverse problems
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The pith

The JMGT equation amends the infinite speed of sound paradox in nonlinear acoustics models.

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

This paper provides a systematic overview of the mathematical literature on the Jordan-Moore-Gibson-Thompson equation, introduced to resolve the unphysical infinite propagation speed in classical nonlinear acoustic models such as the Westervelt and Kuznetsov equations. It focuses on well-posedness results for initial value and time-periodic problems, the incorporation of memory and fractional attenuation, the analysis of singular limits, and examples of control and inverse problems. A sympathetic reader would care because the overview organizes an extensive body of work on a model with clear physical motivation, making the mathematical tools for analyzing finite-speed wave propagation more accessible and usable.

Core claim

The JMGT equation was put forward to amend the infinite speed of sound paradox of classical models of nonlinear acoustics such as the Westervelt and Kuznetsov equations. This has given rise to a substantial body of mathematical literature, and the paper supplies a systematic though inevitably incomplete overview focusing on well-posedness analysis of initial value and time periodic problems, memory and fractional attenuation as well as singular limits and with one example each control and inverse problems.

What carries the argument

The Jordan-Moore-Gibson-Thompson equation, a nonlinear partial differential equation that adds a relaxation term to enforce finite sound propagation speed.

If this is right

  • Well-posedness holds for initial value problems under suitable conditions on initial data and nonlinearity.
  • Time-periodic solutions exist for periodically forced JMGT equations.
  • Memory kernels and fractional derivatives permit modeling of more general dissipation mechanisms.
  • Singular limits recover the classical Westervelt and Kuznetsov models as a relaxation parameter tends to zero.
  • Control and inverse problem techniques can be applied to steer or identify parameters in the JMGT framework.

Where Pith is reading between the lines

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

  • Analytical techniques for the JMGT equation may transfer to other nonlinear wave models that incorporate relaxation or memory.
  • The singular-limit results point toward possible asymptotic reductions for high-frequency or weak-nonlinearity regimes.
  • Numerical schemes for the JMGT equation could be benchmarked against the existence and uniqueness statements collected here.
  • The control examples suggest that the equation could support optimized wave propagation designs in applications such as focused ultrasound.

Load-bearing premise

The overview is inevitably incomplete and its accuracy rests on the correctness of the cited prior works by Jordan, Moore, Gibson, Thompson and subsequent authors.

What would settle it

Discovery of a major well-posedness result or counterexample for the JMGT equation with memory terms that directly contradicts one of the surveyed theorems would undermine the reliability of the presented overview.

read the original abstract

The JMGT equation was put forward by Pedro Jordan~\cite{jordan2008nonlinear,jordan2014second}, also referring to earlier work by Moore and Gibson~\cite{moore1960propagation}, as well as Thompson~\cite{thompson} to amend the infinite speed of sound paradox of classical models of nonlinear acoustics such as the Westervelt and Kuznetsov's equation. Additionally to its physical significance (and of course related to it), it has given rise to a substantial body of mathematical literature -- possibly even more than the above mentioned classical models. In this paper, we aim to provide a systematic (though inevitably incomplete) overview %and indicate some potential open questions. thereby focusing on well-posedness analysis of initial value and time periodic problems, memory and fractional attenuation as well as singular limits and -- with one example each -- control and inverse problems.

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

0 major / 2 minor

Summary. The manuscript provides a systematic (though explicitly incomplete) overview of the mathematical literature on the Jordan-Moore-Gibson-Thompson (JMGT) equation of nonlinear acoustics. It focuses on well-posedness results for initial-value and time-periodic problems, memory and fractional attenuation models, singular limits, and one example each of control and inverse problems, building on the original contributions of Jordan, Moore, Gibson, and Thompson.

Significance. If the cited results are faithfully summarized, the survey consolidates a growing body of work on a model that resolves the infinite propagation speed issue in classical nonlinear acoustics equations such as Westervelt and Kuznetsov. It can serve as a useful reference for researchers, highlighting connections across well-posedness, attenuation, limits, and applications without advancing new theorems.

minor comments (2)
  1. [Abstract] The abstract states the scope clearly but the parenthetical remark on indicating potential open questions is commented out in the provided text; if open questions are addressed in the body, ensure they are explicitly flagged as such to avoid reader confusion.
  2. Verify that all citations to prior works (e.g., Jordan 2008, Moore-Gibson 1960, Thompson) are consistently formatted and that the references section is complete, as the survey's value rests on accurate attribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment. We are pleased that the survey is viewed as a useful reference consolidating results on the JMGT equation, and we appreciate the recommendation to accept.

Circularity Check

0 steps flagged

Literature survey with no internal derivations or self-referential predictions

full rationale

This manuscript is explicitly framed as a systematic (and incomplete) overview of existing results on the JMGT equation, covering well-posedness, memory/fractional models, singular limits, and selected control/inverse problems. No new theorems, derivations, equations, or quantitative predictions are advanced by the author. All claims rest on citations to prior external works (Jordan, Moore, Gibson, Thompson, and subsequent authors), with the text stating its scope and limitations upfront. No load-bearing step reduces by construction to a self-citation, fitted parameter, or ansatz imported from the author's own prior work. The central content is compilation and organization of independent literature, making circularity analysis inapplicable.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

As a survey paper the central claim rests on the validity of the summarized literature rather than new free parameters, axioms, or invented entities introduced by the author.

pith-pipeline@v0.9.0 · 5440 in / 1004 out tokens · 38340 ms · 2026-05-10T18:38:54.635995+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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