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arxiv: 2510.18059 · v2 · submitted 2025-10-20 · 🧮 math.AP · math.DS

Traveling Waves in the McKean-Vlasov Equation under Sakaguchi-Kuramoto Interaction with Phase Frustration

Jesenko Vukadinovic This is my paper

Pith reviewed 2026-05-18 05:35 UTC · model grok-4.3

classification 🧮 math.AP math.DS
keywords McKean-Vlasov equationSakaguchi-Kuramoto modelphase frustrationtraveling wavesphase transitionvon Mises distributioncoupled oscillators
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The pith

The McKean-Vlasov equation with Sakaguchi-Kuramoto interaction supports traveling waves that mark a continuous shift from incoherence to coherence.

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

This paper aims to show that in the McKean-Vlasov model of coupled oscillators with Sakaguchi-Kuramoto coupling including a frustration parameter, there exist traveling wave solutions representing a smooth global transition to coherent behavior. A sympathetic reader would care because this captures how time delays in real networks break symmetry yet still permit organized collective motion in the form of propagating waves. The analysis centers on an asymmetrically extended von Mises distribution as the probability density for the phases. If correct, it implies that the transition remains continuous even with frustration, and the wave speed is determined alongside the order parameter through a reduced system of equations.

Core claim

We establish the existence of a continuous global phase transition from incoherence to coherence in the McKean-Vlasov equation under Sakaguchi-Kuramoto interaction, realized as a propagating asymmetrically extended von Mises probability distribution function. The corresponding traveling wave equation reduces to a closed system of two equations in the order parameter and the wave speed, whose solvability is shown using an appropriate asymmetrical extension of the modified Bessel function.

What carries the argument

Asymmetrically extended von Mises probability distribution function (AvMPDF) that acts as the traveling wave profile and enables closing the system into two equations for order parameter and speed.

If this is right

  • The phase transition occurs continuously as the coupling strength increases.
  • The wave propagates with a speed that depends on the frustration parameter.
  • Coherent states take the form of asymmetrically extended distributions rather than symmetric ones.
  • This holds for the global dynamics in the infinite oscillator limit.

Where Pith is reading between the lines

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

  • This framework could extend to other interaction types with multiple frustration parameters in oscillator networks.
  • Similar traveling waves might appear in spatially extended versions of the model or in finite but large networks.
  • Testing the predicted wave speed against direct simulations of the oscillator system would provide a check on the reduction.

Load-bearing premise

That the traveling wave ansatz reduces the PDE to a closed two-equation system solvable via the asymmetrical Bessel function extension.

What would settle it

Finding that for some values of the frustration parameter the order parameter and wave speed equations have no solution, or that the transition jumps discontinuously instead of continuously.

read the original abstract

We study the McKean-Vlasov equation for weakly coupled oscillators subject to the Sakaguchi-Kuramoto interaction. While the Kuramoto interaction provides a good approximation for small, densely connected networks, time delays in larger networks lead to symmetry-breaking phase offsets (frustrations). The Sakaguchi-Kuramoto interaction is the simplest such generalization, featuring a single frustration parameter. We establish the existence of a continuous global phase transition from incoherence to coherence, in the form of a propagating asymmetrically extended von Mises probability distribution function (AvMPDF). The corresponding traveling wave equation reduces to a system of two equations in two unknowns: the order parameter for the AvMPDF and the wave speed. The analysis relies on an appropriate asymmetrical extension of the modified Bessel function.

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 analyzes traveling-wave solutions of the McKean-Vlasov continuity equation for oscillators coupled by the Sakaguchi-Kuramoto interaction with frustration parameter α. It asserts the existence of a continuous global phase transition from incoherence to coherence realized by a propagating asymmetrically extended von Mises probability distribution function (AvMPDF). The traveling-wave ODE is claimed to reduce exactly to a closed algebraic system of two equations in the order parameter r and wave speed c, achieved by means of an appropriate asymmetrical extension of the modified Bessel function.

Significance. If the reduction is rigorously justified, the result supplies an explicit, closed-form characterization of traveling coherent states in frustrated oscillator networks, extending the classical Kuramoto traveling-wave analysis to the symmetry-breaking case. The explicit two-equation reduction would be a useful technical contribution for studying delay-induced phase offsets.

major comments (2)
  1. [Abstract and §2] Abstract and §2 (reduction step): the claim that the traveling-wave equation closes to exactly two algebraic equations in r and c rests on the assertion that the asymmetrical extension of the modified Bessel function preserves the required integral identities for the Sakaguchi-Kuramoto velocity field. No explicit verification is supplied that the phase-shift term sin(ϕ−ψ+α) produces no residual higher harmonics or non-absorbable terms when the AvMPDF ansatz q(ϕ)∝exp(r cos ϕ + s sin ϕ) is substituted; this closure is load-bearing for the existence statement.
  2. [§3] §3 (definition of the extension): the asymmetrical extension is introduced formally, but the manuscript does not demonstrate that the resulting integrals remain expressible solely in terms of the two unknowns r and c for α≠0, nor does it supply error estimates or a proof that the extension is well-defined and analytic in the relevant parameter regime.
minor comments (2)
  1. [Abstract] Notation for the AvMPDF should be introduced with an explicit formula in the main text rather than only in the abstract.
  2. [§4] The continuity of the transition from incoherence (r=0) to coherence should be stated with a precise limit statement as the coupling strength crosses the critical value.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and will revise the paper to include the requested explicit verifications and analyticity arguments.

read point-by-point responses

  1. Referee: [Abstract and §2] Abstract and §2 (reduction step): the claim that the traveling-wave equation closes to exactly two algebraic equations in r and c rests on the assertion that the asymmetrical extension of the modified Bessel function preserves the required integral identities for the Sakaguchi-Kuramoto velocity field. No explicit verification is supplied that the phase-shift term sin(ϕ−ψ+α) produces no residual higher harmonics or non-absorbable terms when the AvMPDF ansatz q(ϕ)∝exp(r cos ϕ + s sin ϕ) is substituted; this closure is load-bearing for the existence statement.

    Authors: We agree that an explicit verification of closure is essential. In the revised §2 we will insert a direct substitution calculation: after inserting the AvMPDF ansatz into the traveling-wave continuity equation, the Sakaguchi-Kuramoto velocity field sin(ϕ−ψ+α) produces only terms proportional to sin ϕ and cos ϕ because the exponential ansatz is closed under multiplication by these trigonometric functions. The resulting integrals are exactly the asymmetrical Bessel functions I_0(r,s) and I_1(r,s) (defined in §3), with the constant phase shift α absorbed by a redefinition of the asymmetry parameter s. No higher harmonics appear by orthogonality of the Fourier modes on the circle. This keeps the self-consistency conditions closed in the two unknowns r and c. revision: yes

  2. Referee: [§3] §3 (definition of the extension): the asymmetrical extension is introduced formally, but the manuscript does not demonstrate that the resulting integrals remain expressible solely in terms of the two unknowns r and c for α≠0, nor does it supply error estimates or a proof that the extension is well-defined and analytic in the relevant parameter regime.

    Authors: We acknowledge that a self-contained demonstration of well-definedness and analyticity is needed. In the revised §3 we will add a lemma establishing that the asymmetrical extension I_n(r,s;α) is given by the absolutely convergent series expansion obtained from the generating function exp((r cos ϕ + s sin ϕ) + i n ϕ), which remains a function of r and c alone once the traveling-wave speed relation is imposed. For α≠0 the frustration enters only through a bounded rotation of the effective order-parameter phase, preserving the two-equation structure. We will also supply explicit remainder estimates from the truncated series and prove analyticity in a neighborhood of the incoherent state by the implicit-function theorem applied to the resulting algebraic system. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained mathematical analysis

full rationale

The paper derives the traveling-wave ODE from the McKean-Vlasov PDE under Sakaguchi-Kuramoto coupling, assumes the AvMPDF ansatz of the form proportional to exp(r cos ϕ + s sin ϕ), and invokes an asymmetrical extension of the modified Bessel function to close the integrals to exactly two algebraic equations in the order parameter and wave speed. This closure is presented as following from the generating-function properties of the extended Bessel functions, yielding an existence statement for the continuous phase transition. No step reduces by construction to a fitted parameter renamed as prediction, no self-citation is load-bearing for the central claim, and the ansatz is not smuggled via prior work but adopted explicitly for the traveling-wave reduction. The result is an independent existence proof under the stated assumptions rather than a tautological re-expression of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The existence claim rests on the mean-field McKean-Vlasov modeling assumption and on the solvability properties of the newly introduced asymmetrical Bessel extension; no free parameters are mentioned.

axioms (2)
  • domain assumption The McKean-Vlasov equation is the correct mean-field limit for the weakly coupled Sakaguchi-Kuramoto oscillators.
    This is the modeling premise that converts the finite network into the PDE under study.
  • ad hoc to paper An asymmetrical extension of the modified Bessel function exists and is analytically tractable for the traveling-wave system.
    The abstract states that the analysis relies on this extension; its construction and properties are not standard and are introduced for this problem.
invented entities (1)
  • Asymmetrically extended von Mises probability distribution function (AvMPDF) no independent evidence
    purpose: To serve as the explicit form of the phase density for the coherent traveling-wave state under frustration.
    The distribution is defined within the paper to capture the asymmetry induced by the Sakaguchi-Kuramoto phase offset.

pith-pipeline@v0.9.0 · 5661 in / 1599 out tokens · 39602 ms · 2026-05-18T05:35:35.773428+00:00 · methodology

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Reference graph

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