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arxiv: 2605.23364 · v1 · pith:GZCTBHDFnew · submitted 2026-05-22 · ❄️ cond-mat.stat-mech · nlin.CD

Inviscid scaling in the Kuramoto-Sivashinsky equation from functional renormalization group and direct numerical simulations

Liubov Gosteva , Dipankar Roy , Nicol\'as Wschebor , L\'eonie Canet This is my paper

Pith reviewed 2026-05-25 03:17 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech nlin.CD
keywords Kuramoto-Sivashinsky equationdynamical scalingfunctional renormalization groupinviscid-Burgers universalityeffective viscosityKPZ equationdirect numerical simulations
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The pith

The one-dimensional Kuramoto-Sivashinsky equation exhibits an intermediate scaling regime with dynamical exponent z=1 arising from the vanishing of effective viscosity.

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

The paper establishes that the KS equation possesses a scaling regime at intermediate scales where the dynamical exponent takes the value z=1. This regime sits between the large-scale KPZ scaling with z=3/2 and small-scale non-universal behavior. The regime emerges because the effective viscosity starts negative at microscopic scales and becomes positive at macroscopic scales, passing through zero. This zero-crossing imprints universal z=1 scaling on the correlations, placing the regime in the inviscid-Burgers universality class. The result is obtained from both functional renormalization group calculations and direct numerical simulations.

Core claim

The one-dimensional Kuramoto-Sivashinsky equation features a scaling regime characterized by the dynamical exponent z=1 at intermediate scales between the large-scale Kardar-Parisi-Zhang scaling with z=3/2 and the small-scale non-universal behavior. This scaling regime is intrinsic to the KS dynamics since it arises from the vanishing of the effective viscosity when evolving from its microscopic negative KS value to its macroscopic effective positive KPZ value. This vanishing of the viscosity deeply imprints the behavior of correlations at intermediate scales, which exhibit a universal z=1 scaling. This behavior pertains to the inviscid-Burgers universality class, which corresponds to the零 -

What carries the argument

The vanishing of the effective viscosity during its evolution from negative microscopic KS value to positive macroscopic KPZ value, which corresponds to the zero-viscosity fixed point of the KPZ equation and enforces inviscid-Burgers scaling at intermediate scales.

If this is right

  • Correlations at intermediate scales follow the inviscid-Burgers universality class independently of microscopic details.
  • The intermediate regime is intrinsic to the KS equation and appears whenever the effective viscosity changes sign.
  • Both functional renormalization group and direct numerical simulations locate and characterize the same z=1 scaling window.
  • The large-scale KPZ regime with z=3/2 and the small-scale non-universal regime remain separated by this intermediate window.

Where Pith is reading between the lines

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

  • Similar sign-changing viscosity flows could produce intermediate scaling regimes in other nonlinear partial differential equations that interpolate between different universality classes.
  • The inviscid-Burgers scaling might be observable in experimental systems whose effective viscosity can be tuned through zero, such as certain fluid interfaces or flame fronts.
  • The existence of this regime suggests that the transition between KPZ and inviscid dynamics is not abrupt but contains an extended scaling window controlled by the viscosity zero-crossing.

Load-bearing premise

The vanishing of the effective viscosity imprints the correlations at intermediate scales to produce universal z=1 scaling rather than resulting from other dynamical mechanisms or from artifacts of the approximation scheme.

What would settle it

A direct numerical simulation or functional renormalization group flow that finds no z=1 regime or a different exponent at the intermediate scales where viscosity crosses zero would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.23364 by Dipankar Roy, L\'eonie Canet, Liubov Gosteva, Nicol\'as Wschebor.

Figure 1
Figure 1. Figure 1: FIG. 1. Evolution of the functions (a) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Inverse of the decorrelation frequencies [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Correlation function [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

We show that the one-dimensional Kuramoto-Sivashinsky (KS) equation features a scaling regime characterized by the dynamical exponent $z=1$ at intermediate scales between the large-scale Kardar-Parisi-Zhang (KPZ) scaling with $z=3/2$ and the small-scale non-universal behavior. This scaling regime is intrinsic to the KS dynamics since it arises from the vanishing of the effective viscosity when evolving from its microscopic negative KS value, to its macroscopic effective positive KPZ value. This vanishing of the viscosity deeply imprints the behavior of correlations at intermediate scales, which exhibit a universal $z=1$ scaling. This behavior pertains to the inviscid-Burgers universality class, which corresponds to the zero-viscosity fixed point of the KPZ equation. We evidence and characterize this so-far-overlooked scaling regime using both functional renormalization group and direct numerical simulations.

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 that the one-dimensional Kuramoto-Sivashinsky equation possesses an intrinsic intermediate scaling regime with dynamical exponent z=1, lying between large-scale KPZ scaling (z=3/2) and small-scale non-universal behavior. This regime is attributed to the zero-crossing of the running effective viscosity as it evolves from its microscopic negative value to the macroscopic positive KPZ value, placing the intermediate dynamics in the inviscid-Burgers universality class. The claim is supported by both functional renormalization group analysis of the effective action and direct numerical simulations of the two-point correlation spectra.

Significance. If substantiated, the result identifies a previously overlooked universal scaling window intrinsic to the KS dynamics, arising directly from the viscosity sign change rather than from external tuning. The dual use of FRG flow equations and DNS provides complementary evidence that strengthens the case for an inviscid-Burgers imprint at intermediate wave-numbers. This could influence the analysis of crossover phenomena in dissipative nonlinear PDEs and related turbulence models.

major comments (2)
  1. [FRG analysis and flow equations] The central claim that the viscosity zero-crossing enforces a regulator-independent z=1 regime (rather than an artifact of the chosen truncation or finite integration interval) lacks an explicit analytic demonstration that the inviscid-Burgers fixed point fixes the dynamical exponent independently of the cutoff function; the observed scaling in the FRG flow could arise from the specific local-potential or derivative-expansion scheme employed.
  2. [Direct numerical simulations section] DNS spectra are presented as confirming the z=1 window, but the manuscript does not report quantitative error bars, convergence tests with respect to system size or integration time, or explicit criteria for selecting the intermediate wave-number range; without these, it is difficult to rule out that the apparent z=1 is influenced by the crossover regions or finite-resolution effects.
minor comments (2)
  1. Notation for the running viscosity and the precise definition of the dynamical exponent extraction from the two-point function should be clarified with an explicit equation reference.
  2. The abstract states the scaling is 'universal' but the manuscript should add a brief comparison to known results in the inviscid-Burgers class to strengthen the universality claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment below, indicating where revisions will be incorporated.

read point-by-point responses

  1. Referee: [FRG analysis and flow equations] The central claim that the viscosity zero-crossing enforces a regulator-independent z=1 regime (rather than an artifact of the chosen truncation or finite integration interval) lacks an explicit analytic demonstration that the inviscid-Burgers fixed point fixes the dynamical exponent independently of the cutoff function; the observed scaling in the FRG flow could arise from the specific local-potential or derivative-expansion scheme employed.

    Authors: We acknowledge that our FRG results are obtained within a derivative-expansion truncation and that a fully analytic, regulator-independent proof of the z=1 scaling at the viscosity zero-crossing is not provided. The central physical mechanism remains the robust sign change of the running viscosity from its negative microscopic value to the positive IR value fixed by the KPZ fixed point; when the viscosity vanishes, the dynamics reduce to the inviscid-Burgers class whose scaling z=1 follows from dimensional analysis in the absence of the linear viscous term. We will revise the manuscript to add an explicit discussion of this point, including a brief comparison of the flow behavior near the zero-crossing for two different cutoff functions within the same truncation, to clarify that the scaling window is tied to the viscosity sign change rather than to scheme-specific artifacts. revision: partial

  2. Referee: [Direct numerical simulations section] DNS spectra are presented as confirming the z=1 window, but the manuscript does not report quantitative error bars, convergence tests with respect to system size or integration time, or explicit criteria for selecting the intermediate wave-number range; without these, it is difficult to rule out that the apparent z=1 is influenced by the crossover regions or finite-resolution effects.

    Authors: We agree that the DNS section would benefit from additional quantitative controls. In the revised manuscript we will add error bars obtained from ensemble averaging over independent realizations, report convergence checks with respect to system size and total integration time, and state explicit selection criteria for the intermediate wave-number window (the range in which the running effective viscosity remains near zero). These additions will make it possible to assess the separation from the KPZ and small-scale regimes more rigorously. revision: yes

Circularity Check

0 steps flagged

No significant circularity; scaling emerges from dynamics via FRG and DNS

full rationale

The paper derives the z=1 regime from the vanishing effective viscosity in the KS equation, evidenced by FRG flows and DNS spectra. No quoted step reduces a prediction to a fitted input by construction, nor does any load-bearing claim collapse to a self-citation chain or ansatz smuggled via prior work. The central result is presented as numerically observed and intrinsic to the flow from negative microscopic viscosity to positive KPZ value, remaining self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, invented entities, or ad-hoc axioms; the work relies on standard FRG and DNS techniques whose background assumptions are not enumerated here.

axioms (1)
  • standard math Standard assumptions underlying the functional renormalization group in statistical mechanics
    The FRG analysis presupposes the validity of the renormalization-group flow equations and regulator choices typical in the field.

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