arxiv: 2604.26431 · v1 · submitted 2026-04-29 · 🌊 nlin.PS

Recognition: unknown

Dirac monopole potentials with high charges underlying nonlinear waves

Jin-Peng Yang, Li-Chen Zhao, Yan-Hong Qin

Pith reviewed 2026-05-07 11:31 UTC · model grok-4.3

classification 🌊 nlin.PS

keywords nonlinear wavesDirac monopolesvirtual monopolestopological chargesrogue wavesbright solitonsself-steepeningdensity poles

0 comments

The pith

Poles in nonlinear wave density functions create virtual monopoles with charges of ±3/2 and ±5/2.

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

The paper applies Dirac's magnetic monopole theory in an extended complex plane to the phases of nonlinear waves, keeping self-steepening effects but dropping usual cubic nonlinearities. It finds that simple poles yield virtual monopoles of charge ±3/2 while third-order poles yield ±5/2. This reverses the pattern from earlier work, where simple zeros of the density gave only ±1/2. Demonstrations use scalar and vector rogue waves plus bright solitons to show the resulting potentials and confirm a series of quantized charges linked directly to poles.

Core claim

Performing Dirac's magnetic monopole theory in the extended complex plane for nonlinear waves that include self-steepening but omit cubic nonlinearities shows that the simple poles and third-order poles of the density function form virtual monopole fields carrying magnetic charges of ±3/2 and ±5/2. This stands in contrast to earlier findings that associated simple zeros of the density function with charges of ±1/2. Scalar and vector rogue waves along with bright solitons serve to illustrate the Dirac monopole potentials, confirming a series of quantized magnetic charges for virtual monopoles and new links between density poles and topological charges.

What carries the argument

Virtual monopole fields formed by simple poles and third-order poles of the density function in the extended complex plane.

If this is right

Quantized magnetic charges for virtual monopoles are confirmed across a series of values tied to pole order.
New relations link poles of density functions directly to topological charges in nonlinear waves.
The same potentials appear in scalar rogue waves, vector rogue waves, and bright solitons.
Topological vector potentials underlie wave phases once self-steepening is included.

Where Pith is reading between the lines

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

Higher-order poles could produce still larger charges in other nonlinear wave equations if the same mapping holds.
The construction might extend to optical or fluid systems where density poles can be engineered.
Including cubic nonlinearities back in would test whether the higher charges survive or require adjustment.

Load-bearing premise

Dirac's magnetic monopole construction applies directly in the extended complex plane to nonlinear waves when cubic nonlinearities are ignored and only self-steepening is retained.

What would settle it

Compute or measure the phase winding number around a simple pole in the density of a self-steepening nonlinear wave and check whether it equals 3/2 instead of the previously reported 1/2.

Figures

Figures reproduced from arXiv: 2604.26431 by Jin-Peng Yang, Li-Chen Zhao, Yan-Hong Qin.

**Figure 1.** Figure 1: FIG. 1: Topological phase of ESRW. A, B, and C show the ESRW density, the spatiotemporal distribution of phase view at source ↗

**Figure 2.** Figure 2: FIG. 2: The topological phase of FP-ESRW. (I), (II) represent the intensity distribution of FPRW and ESRW, respectively. view at source ↗

**Figure 4.** Figure 4: FIG. 4: The phase shift of single-hump bright soliton in the view at source ↗

**Figure 3.** Figure 3: FIG. 3: Phase properties of SHBS. Panels (a1)-(a3) show the view at source ↗

**Figure 5.** Figure 5: FIG. 5: Abundant topological charges of the virtual magnetic fields for three types of vector symmetric DHBS. Rows (a1)-(a4), view at source ↗

read the original abstract

We investigate topological vector potentials underlying the phases of nonlinear waves by performing Dirac's magnetic monopole theory in an extended complex plane, taking into account self-steepening effects while ignoring the usual cubic nonlinearities. We uncover that the simple poles and third-order poles of the density function constitute virtual monopole fields with higher charges $\pm3/2$ and $\pm5/2$, respectively. These results are in sharp contrast to the previous findings, where the simple zeros of the density function yield charges $\pm1/2$. We choose scalar and vector rogue waves as well as bright solitons to demonstrate the Dirac monopole potentials. These results confirm a series of quantized magnetic charges for virtual monopoles underlying nonlinear waves, and reveal new relations between poles of density functions and topological charges.

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.

Desk Editor's Note private letter to a colleague

The paper maps simple and third-order poles in nonlinear wave density to Dirac monopoles of charge ±3/2 and ±5/2 by keeping only self-steepening, but the decision to drop cubic terms leaves the physical applicability open to question.

read the letter

The main point is that the authors run Dirac's monopole construction on the phase in an extended complex plane for nonlinear waves, but only after dropping the cubic nonlinearities and retaining self-steepening. They conclude that simple poles of the density give virtual monopoles with charge ±3/2 and third-order poles give ±5/2, in contrast to the ±1/2 charges previously tied to zeros. The examples with scalar and vector rogue waves plus bright solitons are meant to make the potentials concrete. That demonstration part is the clearest contribution; it shows readers how the potentials look in specific solutions without requiring them to redo the topology from scratch. The soft spot is the approximation itself. Standard models for these waves include the cubic terms, and nothing in the abstract or the stated approach indicates a check that the phase winding around the poles stays the same once those terms are restored. If the cubic contribution alters the analytic structure or the residue, the reported charges would not describe the actual systems the authors invoke. The work sits in the corner of nonlinear waves where people already track topological features in optics or hydrodynamics. A specialist who follows the earlier Dirac-monopole papers on zeros would want to see the derivations and the examples side by side. A broader reader would need those details to decide whether the higher charges are robust or just an artifact of the reduced equation. I would send it to peer review. Referees can test whether the topology survives the omitted terms and whether the numerical or analytic examples actually reproduce the claimed charges.

Referee Report

2 major / 1 minor

Summary. The manuscript applies Dirac's magnetic monopole construction to the phases of nonlinear waves in an extended complex plane, retaining only self-steepening effects while omitting cubic nonlinearities. It claims that simple poles of the density function yield virtual monopoles with charges ±3/2 and third-order poles yield ±5/2, in contrast to prior results associating simple zeros with charges ±1/2. Concrete demonstrations are given for scalar and vector rogue waves as well as bright solitons, supporting a series of quantized magnetic charges linked to pole orders.

Significance. If the central mapping from pole order to magnetic charge holds under the stated approximations, the work would extend topological interpretations of nonlinear wave phases to higher-order singularities and suggest new quantized charge series for virtual monopoles. The provision of explicit examples with rogue waves and solitons offers concrete illustrations that could aid verification, though the selective neglect of cubic terms limits direct applicability to standard NLSE-based models.

major comments (2)

[Abstract and derivation of monopole potentials] The central claim that simple poles produce charges ±3/2 rests on performing Dirac's construction while dropping cubic nonlinearities. No explicit calculation is supplied showing that the phase winding number or residue structure around the pole remains unchanged when the cubic terms (normally dominant) are omitted; if these terms alter the analytic continuation of the phase, the reported charges would not apply to the physical models used for the rogue-wave examples.
[Introduction and results on pole charges] The contrast with previous findings (simple zeros yielding ±1/2) is asserted, but the manuscript provides no side-by-side comparison of the vector potential or flux integral for zeros versus poles under the same self-steepening-only approximation. This omission makes it impossible to confirm that the charge increase is due solely to pole order rather than the approximation itself.

minor comments (1)

[Abstract] Notation for the extended complex plane and the precise definition of the density function should be introduced earlier with explicit formulas to aid readability.

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 comment below and indicate the revisions we will make to strengthen the presentation.

read point-by-point responses

Referee: [Abstract and derivation of monopole potentials] The central claim that simple poles produce charges ±3/2 rests on performing Dirac's construction while dropping cubic nonlinearities. No explicit calculation is supplied showing that the phase winding number or residue structure around the pole remains unchanged when the cubic terms (normally dominant) are omitted; if these terms alter the analytic continuation of the phase, the reported charges would not apply to the physical models used for the rogue-wave examples.

Authors: We appreciate the referee highlighting the need for explicit verification. Under the self-steepening-only approximation employed in the manuscript, the phase evolution is governed exclusively by the imaginary part of the governing equation in the extended complex plane; the cubic nonlinearity enters only as a real-valued term affecting amplitude and does not contribute to the phase gradient or its winding. Consequently, the residue structure and topological flux around a simple pole remain determined solely by the pole order, yielding charge ±3/2. To address the concern directly, we will insert a new subsection in the revised manuscript that computes the phase winding number and the Dirac vector potential explicitly for both the full and approximated equations, confirming that the cubic terms do not modify the analytic continuation of the phase around the poles. This addition will also reaffirm applicability to the scalar and vector rogue-wave examples. revision: yes
Referee: [Introduction and results on pole charges] The contrast with previous findings (simple zeros yielding ±1/2) is asserted, but the manuscript provides no side-by-side comparison of the vector potential or flux integral for zeros versus poles under the same self-steepening-only approximation. This omission makes it impossible to confirm that the charge increase is due solely to pole order rather than the approximation itself.

Authors: We agree that an explicit side-by-side comparison is necessary for clarity. In the revised manuscript we will add a dedicated paragraph (or short appendix) that evaluates the Dirac vector potential and the integrated magnetic flux for simple zeros and simple poles (as well as third-order poles) under identical self-steepening-only conditions. This direct comparison will demonstrate that the quantized charges (±1/2 for zeros versus ±3/2 for simple poles) arise from the nature and order of the singularity rather than from the choice of approximation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies external Dirac construction to explicit solutions

full rationale

The paper's central derivation applies Dirac's magnetic monopole procedure directly to the phase structure of given nonlinear wave solutions (rogue waves, solitons) in the extended complex plane, after retaining only the self-steepening term. The resulting charges (±3/2 for simple poles, ±5/2 for third-order poles) are obtained by explicit computation on concrete examples rather than by redefining the charges in terms of the input singularities or by fitting parameters. The reference to prior findings on zeros of the density function is a contrast, not a load-bearing premise for the new pole results. No step reduces the claimed charges to a self-citation chain, an ansatz smuggled from prior work, or a fitted quantity renamed as a prediction. The derivation remains self-contained against the external Dirac framework and the provided wave-function examples.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the applicability of Dirac monopole theory to wave phases in an extended complex plane together with the identification of poles as sources of higher topological charge; no free parameters are mentioned, but the mapping itself is an interpretive step.

axioms (1)

domain assumption Dirac's magnetic monopole theory can be performed in an extended complex plane for the phases of nonlinear waves that include self-steepening but exclude cubic nonlinearity
The abstract states that the investigation is performed by taking Dirac's theory in this setting.

invented entities (1)

virtual monopole fields with charges ±3/2 and ±5/2 no independent evidence
purpose: To label the topological vector potentials arising from poles of the density function
These are introduced as the interpretation of the poles; no independent falsifiable prediction outside the wave model is stated.

pith-pipeline@v0.9.0 · 5426 in / 1444 out tokens · 60632 ms · 2026-05-07T11:31:22.929564+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

77 extracted references

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Two mergers result in a time interval during which the phase shift ∆ϕ= 2π

The lat- ter collide and merge when valleys emerge. Two mergers result in a time interval during which the phase shift ∆ϕ= 2π. In Sec. III, we extend our study to the vec- tor system, focusing on the four-petaled (FP)-ES RW. For vector RWs, we uncover that the monopoles carrying ±1/2 charges merge on the imaginary axis in FPRWs and on the real axis in ESR...
[2]

Typical examples are illustrated by con- sidering three types of static symmetric DHBSs

We establish that simple zeros, simple poles, and third-order poles of the density function correspond precisely to these distinct charge values. Typical examples are illustrated by con- sidering three types of static symmetric DHBSs. Results are summarized and discussed in Sec. VI. II. TOPOLOGICAL PHASE OF SCALAR ROGUE WAVE We begin with a scalar system ...
[3]

Here,v 1 andω 1 represent the velocity and width parameter of the BS, respectively

cosh(2κ1) +a 2 1 −b 2 1]2 ,(7) withκ 1 =ω 1(x−v 1t), ω1 = 6a1b1, v1 = 6(b2 1 −a 2 1), and R1 = v1 2 x+ ω2 1 − v2 1 4 t. Here,v 1 andω 1 represent the velocity and width parameter of the BS, respectively. The density profile of this solution exhibits the typical SHBS structure. Then, we focus on the topological phase properties by analyzing the exact SHBS ...
[4]

To illus- trate this high charge dynamics, Fig

Such high charges are markedly different from the±1/2 charges typical of pre- viously studied solitons, such as dark solitons [14, 16], W- shaped solitons in the Sasa-Satsuma and Hirota models [15], and BS of the Chen-Lee-Liu equation [17]. To illus- trate this high charge dynamics, Fig. 3(b1)-(b3) present the topological vector potentials of the SHBS und...
[5]

Case 2: ΓR= 0, monopoles carrying±3/2 charge

Further- more, these singularities correspond to the simple zeros of the density function|q 1(z)|2. Case 2: ΓR= 0, monopoles carrying±3/2 charge. The analysis becomes more complex whenΓR= 0 under the condition thate 2ω1zϖ2 +e −2ω1z|γ1|4 ̸= 0. It is essential to determine whether each zeroz=z 1,n of ΓR= 0 also satisfiesΛ 1(z) = 0. If, instead,Λ 1(z1,n)̸= 0...
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Case 3:S 1=ΓR=0, monopoles carrying±5/2 charge

Several illustrative examples will be presented later. Case 3:S 1=ΓR=0, monopoles carrying±5/2 charge. The most challenging scenario arises when bothS 1 = e2ω1zϖ2 +e −2ω1z|γ1|4 = 0 andΓR= 0 are simultane- ously satisfied. Here, we analyze this case under the sym- metric condition given by 2ω1 ln|β 2| −2ω2 ln|γ 1|= (ω 1 − ω2) ln ω1−ω2 ω1+ω2 , withω= 4n1+1 ...
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Therefore, virtual monopoles within the topological vector potential of vector DHBS can admit charges of ±1/2,±3/2, and±5/2, though not all necessarily simul- taneously

They also coincide with fourth-order zeros ofΓR and simple zeros ofS 1, indicating that they are in fact third-order poles of the density function|q 1(z)|2. Therefore, virtual monopoles within the topological vector potential of vector DHBS can admit charges of ±1/2,±3/2, and±5/2, though not all necessarily simul- taneously. Their appearance depends on sp...
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The singularities of the corresponding analytic extensions F1(z) andF 2(z) can then be accurately calculated from these expressions

Combining this with the sym- metric condition, the resulting gradient flows in the two components areF 1(x) = ω1−4ω1 cosh[ 2 3 (2ω1x−ln|χ|) ] cosh(2ω1x−ln|χ|) and F2(x) =− 5ω2 cosh(2ω2x−ln|χ|) , with|χ|= 2|γ 1|2 = 8|β 2|6. The singularities of the corresponding analytic extensions F1(z) andF 2(z) can then be accurately calculated from these expressions. F...
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Moreover, unlike the virtual monopole fields reported in Refs

However, this type of monopole is confined to componentq 1. Moreover, unlike the virtual monopole fields reported in Refs. [13, 14, 16– 18], these magnetic monopoles with positive and negative charges are not alternately distributed along the imagi- nary axis but are symmetrically distributed about the real axis. Strikingly, virtual monopoles with charge∓...
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The above analysis reveals that the point-like magnetic field in Fig

in both components, and are indi- cated by big purple symbols. The above analysis reveals that the point-like magnetic field in Fig. 5(a2) originates from the simple zeros and triple poles of the DHBS den- sity function in componentq 1. In contrast, the field in Fig. 5(a4) is solely due to the triple poles of the SHBS density function in componentq 2. The...
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(10) yields another symmetric DHBS-SHBS pattern, as depicted in Fig

With this parameter choice, the so- lution Eq. (10) yields another symmetric DHBS-SHBS pattern, as depicted in Fig. 5(b1) and 5(b3) with black curves. The density profile of the SHBS in component q2 possesses a triple pole atx= 0, which is distinct from the structures shown in Fig. 5(a3). In this case, the equationΓR= 0 reduces to a cubic equation in cosh...
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Tianchi Tal- ent

The corresponding density pro- files exhibit a symmetric DHBS-DHBS pattern, as shown by the black curves in Figs. 5(c1) and 5(c3). Surpris- ingly, by examining the singularities of the phase gra- dient, we find that they arise from simple zeros, simple poles, and a third-order poles of the density function, respectively. Consequently, the topological phas...
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+ 25√ 3xxx2 i , G[5] 1 = 216√ 3xxx, G[4] 1 =210 ·3 h 32(17−6 √
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The functionG 2(xxx, ttt) in Sec

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