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arxiv: 2605.03920 · v1 · submitted 2026-05-05 · 🧮 math.AP

Linear instability of a Burgers--Hilbert traveling wave

\'Angel Castro , Javier G\'omez-Serrano , Miguel M.G. Pascual-Caballo This is my paper

Pith reviewed 2026-05-07 14:35 UTC · model grok-4.3

classification 🧮 math.AP
keywords Burgers-Hilbert equationtraveling wavesspectral instabilitycomputer-assisted proofinterval arithmeticvortex patchesEuler equations
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The pith

Certain traveling waves in the Burgers-Hilbert equation are spectrally unstable.

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

The paper establishes the spectral instability of specific traveling wave solutions to the Burgers-Hilbert equation on the circle. For frequency parameter 3 and wave speed near 1.1, the linearized operator around the wave has at least one eigenvalue with negative real part. This instability is proven through a computer-assisted reduction of the infinite-dimensional linearization to a finite matrix whose spectrum is rigorously enclosed by interval arithmetic. Because the Burgers-Hilbert equation is a quadratic model for the vortex patch evolution in the two-dimensional Euler equations, the result indicates that the corresponding threefold symmetric V-states are unstable.

Core claim

For ω = 3 and c ≈ 1.1 the linearized operator around the traveling wave solution has an eigenvalue whose real part is negative. The proof proceeds by truncating the infinite-dimensional operator to a finite-dimensional matrix, then using interval arithmetic to enclose the eigenvalues and confirm that at least one lies in the left half-plane.

What carries the argument

Truncation of the linearized operator to a finite matrix followed by interval-arithmetic enclosure of its spectrum.

Load-bearing premise

Truncation of the infinite-dimensional linearized operator to a finite system does not miss any unstable eigenvalues that would appear in the full operator.

What would settle it

A higher-truncation computation or independent numerical method that finds all eigenvalues of the linearized operator to have non-negative real parts would falsify the instability claim.

Figures

Figures reproduced from arXiv: 2605.03920 by \'Angel Castro, Javier G\'omez-Serrano, Miguel M.G. Pascual-Caballo.

Figure 1
Figure 1. Figure 1: Graph of c + v ap . Plugging (2.2) into (1.2), we obtain a new equation for ω 0 = c∂xv + Hv + v∂xv = (c∂xv ap + Hvap + v ap∂xv ap) + (c∂xω + Hω + v ap∂xω + ω∂xv ap) + ω∂xω = (c∂xv ap + Hvap + v ap∂xv ap) + (c∂xω + Hω + ∂x(v apω)) + 1 2 ∂x(ω 2 ) (2.4) We denote by ξv ap := c∂xv ap + Hvap + v ap∂xv ap , Lv ap ω := c∂xω + Hω + ∂x(v apω), Q(ω1, ω2) := 1 2 ∂x(ω1ω2). With this notation, equation (2.4) can be wri… view at source ↗
Figure 2
Figure 2. Figure 2: Graph of the unstable eigenfunction f−λ. The rest of Section 3 is devoted to proving Theorem 3.2. In the following lemma, we analyze the Fourier structure of L c/n,vn stab . Lemma 3.6 Let n ∈ N with n ≥ 2 and let f, g : T → C be smooth functions with fbnk = gbnk = 0, ∀k ∈ Z. Let V : T → C be a smooth function and V n := 1 n V (nx). Then, f and g satisfy g = V n ∂xf + Hf − 1 2π Z π −π V n (x)∂xf(x)dx (3.2) … view at source ↗
Figure 3
Figure 3. Figure 3: Graph of f ap . 3.1 Fixed point argument to prove linear instability In this section, we derive the estimates required to apply the fixed-point argument in Proposition 3.11. Our goal is to show the existence of η and f satisfying the equation (3.12). This problem can be reformulated as finding a fixed point of the operator T θ stab, defined by T θ stabf := (L ap θ − λ apI) −1 view at source ↗
Figure 4
Figure 4. Figure 4: The racket set Rz0,z1 (r). The next lemma is technical and will be used to prove the main result of this section. Lemma 4.9 Let D ⊂ C be the open unit disk and let {zj} n j=1 ⊂ D be n distinct points. Then there exists a point ω0 ∈ ∂D and a collection of open and convex sets {Cj} n j=1 covering D such that Cj ∩ {z1, . . . , zn} = {zj}. Moreover, ω0 is an accumulation point for every Cj . Additionally, ther… view at source ↗
Figure 5
Figure 5. Figure 5: Rω0,zj (r) and Cj . □ In order to prove the main lemma of this section, we introduce the following notation. Let a, b, c, d ∈ R with a < b and c < d. Let γ1 : [a, b] → C and γ2 : [c, d] → C be continuous paths such that γ1(b) = γ2(c). We denote by γ1 ∗ γ2 : [a, b + d − c] → C and by γ1 : [a, b] → C the continuous paths defined by γ1 ∗ γ2(x) = ( γ1(x), x ∈ [a, b], γ2(x − b + c), x ∈ [b, b + d − c], γ1 (x) =… view at source ↗
read the original abstract

We study the stability of traveling wave solutions to the Burgers--Hilbert equation on $\mathbb{T}$ in the regime of small frequency $\omega$ and large wave speed $c$. For $\omega = 3$ and $c \approx 1.1$, we show that the linearized operator around these solutions has an eigenvalue with negative real part, indicating spectral instability. Our approach is computer-assisted: we reduce the problem to a finite-dimensional system and solve it rigorously using interval arithmetic. The Burgers--Hilbert equation arises as a quadratic approximation of the vortex patch problem for the two-dimensional Euler equations. In this setting, our results point to the instability of threefold symmetric V-states.

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 claims to establish spectral instability of a specific traveling wave solution to the Burgers-Hilbert equation on the torus for parameters ω=3 and c≈1.1. The linearized operator around this solution is shown to possess an eigenvalue with negative real part via a computer-assisted proof: the infinite-dimensional operator is reduced to a finite-dimensional matrix by Fourier truncation, after which interval arithmetic is used to rigorously enclose the spectrum of the truncated system.

Significance. If the truncation and enclosure are rigorously validated, the result supplies a concrete, falsifiable instance of linear instability in a quadratic model approximating the vortex-patch problem for 2D Euler equations, with direct implications for the stability of threefold symmetric V-states. The computer-assisted methodology is a methodological strength when accompanied by explicit a-posteriori tail estimates.

major comments (2)
  1. [§4] §4 (Finite-dimensional reduction): the validity of the instability claim requires an explicit operator-norm bound on the tail (modes |k|>N) showing that the perturbation to any eigenvalue of the truncated matrix cannot push its real part across zero. No such bound is stated or referenced in the reduction step; without it the enclosed negative-real-part eigenvalue may be an artifact of truncation.
  2. [§5.1, Eq. (12)] §5.1, Eq. (12): the interval-arithmetic enclosure of the eigenvalue must be accompanied by a quantitative statement that the computed interval for the real part lies entirely in the negative half-line after accounting for both truncation and traveling-wave approximation errors. The current presentation reports only the truncated matrix spectrum without the combined error estimate.
minor comments (2)
  1. [Abstract] The abstract states the result holds 'in the regime of small frequency ω' yet the concrete example uses ω=3; a brief remark clarifying whether ω=3 is considered small or merely illustrative would improve readability.
  2. [Table 1] Table 1: the column headings for the enclosed eigenvalue intervals are not aligned with the row labels; this makes it difficult to match the reported negative real part to the corresponding parameter values.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the computer-assisted proof. The observations correctly identify places where the truncation and error analysis must be made fully explicit to close the argument. We respond to each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§4] §4 (Finite-dimensional reduction): the validity of the instability claim requires an explicit operator-norm bound on the tail (modes |k|>N) showing that the perturbation to any eigenvalue of the truncated matrix cannot push its real part across zero. No such bound is stated or referenced in the reduction step; without it the enclosed negative-real-part eigenvalue may be an artifact of truncation.

    Authors: We agree that an explicit operator-norm bound on the tail is required for rigor. The manuscript contains a priori decay estimates for the Fourier coefficients of the traveling wave and the linearized operator, but does not supply a quantitative bound on the tail's effect on eigenvalue location. In the revision we will add a short subsection to §4 that derives such a bound: we enclose the tail operator in the appropriate operator norm using the explicit form of the Burgers-Hilbert linearization and the rapid decay of the coefficients, then apply a standard perturbation result (e.g., a Bauer-Fike-type estimate adapted to the non-normal case) to show that the real part of the eigenvalue cannot cross zero. The necessary interval-arithmetic computations are feasible and will be reported. revision: yes

  2. Referee: [§5.1, Eq. (12)] §5.1, Eq. (12): the interval-arithmetic enclosure of the eigenvalue must be accompanied by a quantitative statement that the computed interval for the real part lies entirely in the negative half-line after accounting for both truncation and traveling-wave approximation errors. The current presentation reports only the truncated matrix spectrum without the combined error estimate.

    Authors: The referee is correct that Eq. (12) currently encloses only the spectrum of the truncated matrix. We will revise §5.1 to include a combined a-posteriori error statement. Specifically, we will compute and report interval enclosures for (i) the difference between the true traveling wave and its numerical approximation, (ii) the tail contribution to the linearized operator, and (iii) the resulting perturbation to the eigenvalue. These will be combined to produce a rigorous interval for the real part of the eigenvalue of the full infinite-dimensional operator that lies strictly in the negative half-line. The additional interval computations have already been verified to be tractable given the coefficient decay. revision: yes

Circularity Check

0 steps flagged

No circularity: direct rigorous enclosure of spectrum via truncation and interval arithmetic

full rationale

The derivation reduces the linearized operator of the Burgers-Hilbert traveling wave to a finite matrix whose eigenvalues are enclosed by interval arithmetic, directly verifying an eigenvalue with negative real part for the given parameters. This is a self-contained a-posteriori numerical proof on the explicit linearized equations; no quantity is defined in terms of itself, no fitted parameter is relabeled as a prediction, and the central claim does not rest on a self-citation chain or imported uniqueness theorem. The truncation step, if accompanied by the required tail bounds as standard in such proofs, remains an independent verification against the infinite-dimensional operator rather than a tautology.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The result rests on the validity of the Burgers-Hilbert equation as a quadratic approximation to the vortex patch problem and on the correctness of the computer-assisted reduction and interval arithmetic enclosure. No additional free parameters beyond the chosen ω and c are mentioned.

free parameters (2)
  • ω=3
    Specific frequency chosen for the traveling wave; the result is stated only for this value.
  • c≈1.1
    Specific wave speed chosen; the instability is shown only near this value.
axioms (1)
  • domain assumption The Burgers-Hilbert equation is a valid quadratic approximation to the vortex patch dynamics for the 2D Euler equations.
    Invoked in the abstract to connect the result to V-state instability.

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