Nonlinear Maxwell Equations in Biquaternionic Form

Introduction

The corpus states Maxwell's equations as one biquaternionic equation, $\tilde\nabla\tilde F = -\tilde R$, in Maxwell's Equations in the Biquaternionic Formulation, and that article's section Relationship Between the Two Formulations records four routes to a single equation: the field-strength route, the potential route, the elimination route that loses equivalence by adding a longitudinal sector, and the matrix route of Gottlieb. The paper recorded here is a fifth route, and it is a different kind of claim. The equations are not written down and then compacted: they are derived, from the conservation of the energy–momentum flow of the field through space and time, and the derivation is said to presuppose a wider system of which Maxwell's equations are the linear part.

The source is Sergey Y. Kotkovskiy, Nonlinear Maxwell equations, held in the corpus as two files carrying the same text. The file whose printed identifier is 2403.00836v1 resolves on arXiv, where the record is dated 28 February 2024 and the author is named. The earlier file, whose printed identifier is 2402.0057v2, is malformed — four digits after the period where arXiv requires five since 2015 — and that identifier belongs on arXiv to an unrelated paper on multipartition functions. This is the same pathology the corpus records for the same author's Chiral algebra of Dirac equation, whose identifier 2502.0126 is malformed in the same way; see The Chiral Algebra of Biquaternions and the Cyclic Representation of the Dirac Equation, whose algebra and product convention this paper shares. The two files differ only in the header, in the spelling of the author's name (the arXiv record and this paper print Kotkovskiy; the author's 2014 paper on the nullvector algebra is printed Kotkovsky) and in a handful of OCR repairs.

The abstract claims five things: that the equations are a consequence of the conservation of the energy–momentum flow; that the principle presupposes more general nonlinear equations, with the linear Maxwell equations embedded as a special case; that the nonlinear equations admit solutions with a swirling energy flow while the linear ones do not; that the solutions describe charged particles, the particle being a singularity of its own field; and that the particle's charge is complex-valued, passing periodically through linear combinations of electric and magnetic charge, so that what matters in a process is the phase relationship between charges and fields.

The plan is the algebra first, because the paper's product is not the corpus's and the two must be separated before anything else can be read. Then the derivation. Then three checks on the embedding: whether Maxwell implies the new equations, whether they imply Maxwell, and what their solution set is. Then the regular currents, the associated fields and the central-symmetric solution, the singularity and the coherent fields, and the twisting claim. The section What Is Verified and What Is Not collects every recomputation and every defect.

The result of the verification. The derivation is correct as an algebra, and the identities it rests on are exact. Its centrepiece is the paper's section 9 theorem, which is true and was not in the corpus: any solution of the linear Maxwell equations, multiplied by the phase factor of a plane wave, solves the nonlinear system. That theorem gives a solution set far larger than Maxwell's own, and it is the paper's real contribution. It is also where the difficulty lies, because the system is weaker than Maxwell's equations, not stronger: it is four equations for six field components, its coefficient matrix has rank four, and two directions of the time derivative of the field are left free at every point. Consequently it does not imply Maxwell's equations and it does not fix a dispersion relation — every constant-amplitude plane wave with an arbitrary real frequency and wave vector is a solution, null or not. The paper's own plane-wave section supplies the missing dispersion relation by a separate argument, which is where it belongs. Three further claims fail as printed, and one of them — the dichotomy between linear and nonlinear solutions in the twisting section — is falsified by a counterexample that satisfies all four Maxwell equations exactly. All of this is in What Is Verified and What Is Not.

The Algebra and the Field

The Basis, and the Corpus Dictionary

The paper uses the scalar–vector form. A biquaternion is a pair

$$ \mathcal{B} = (s, \mathbf{u}), \qquad s \in \mathbb{C}, \quad \mathbf{u} \in \mathbb{C}^3, $$

and the product is its equation (14.24),

$$ \mathcal{B}_1 \mathcal{B}_2 = \bigl(s_1s_2 + \mathbf{u}_1 \cdot \mathbf{u}_2,\; s_1\mathbf{u}_2 + s_2\mathbf{u}_1 + i\,\mathbf{u}_1 \times \mathbf{u}_2\bigr), \tag{14.24} $$

with $i$ the central scalar imaginary unit. The paper prints the second term of the vector part as $s_2\mathbf{u}_2$; it must be $s_2\mathbf{u}_1$. With the printed form the unit $e_0 = (1, \mathbf{0})$ fails to be even a left identity, since $(1,\mathbf{0})(s,\mathbf{u})$ would give $(s,\ (1+s)\mathbf{u})$ instead of $(s,\mathbf{u})$. The correction is forced by the paper's own use of the product everywhere else, and it agrees with the sibling article's equation (1), where the same product is written out and checked.

The basis is fixed by the Appendix, equation (14.27): $e_0 = 1$ and

$$ e_1^2 = e_2^2 = e_3^2 = +e_0, \qquad e_1e_2 = -e_2e_1 = ie_3, $$

so that in general $e_je_k = \delta_{jk}e_0 + i\varepsilon_{jkl}e_l$. This is not the corpus's basis. The corpus's quaternion units satisfy $e_k^2 = -e_0$ and $e_je_k = -\delta_{jk}e_0 + \varepsilon_{jkl}e_l$. The two are related by

$$ e_k^{\text{source}} = i\,e_k^{\text{corpus}}, $$

which is the replacement that turns Hamilton's rule into the paper's: on vectors, the paper's product is the Pauli product $\mathbf{u}_1\cdot\mathbf{u}_2 + i\,\mathbf{u}_1\times\mathbf{u}_2$, with both the dot and the cross term of opposite sign to the corpus's Hamilton product. The same convention and the same caution are recorded in The Chiral Algebra of Biquaternions and the Cyclic Representation of the Dirac Equation, and the reader is referred there for the worked comparison. Two consequences matter here.

The dictionary. An element the paper writes as $(s,\mathbf{u})$ is the corpus's element $(s, i\mathbf{u})$. So the paper's vector coefficients $E_k$ enter the corpus's vector slots as $iE_k$. The paper's $e_1, e_2, e_3$ are the corpus's $ie_1, ie_2, ie_3$ — that is, the basis $e_0, x, y, z$ with $x = ie_1$ and so on, whose vector units square to $+e_0$, which is the basis of Biquaternion 4×4 Regular Matrix Element Representation. The identification is exact and was recomputed.

The norm. The paper's square modulus, $\mathcal{B}\bar{\mathcal{B}} = s^2 - \mathbf{u}^2$, is indefinite in this basis: the "unitary" biquaternions, $\mathcal{B}\bar{\mathcal{B}} = 1$, are those with $s^2 - \lvert\mathbf{u}\rvert^2 = 1$, not those on a sphere. In the corpus's Hamilton basis the same expression would read $s^2 + \lvert\mathbf{u}\rvert^2$, because the vector units square to $-e_0$ there; the sign is the dictionary's, and the caution is repeated here because the paper's Lorentz group is built on the indefinite form.

The algebra is $\mathbb{C}\otimes\mathbb{H}$, isomorphic to the $2\times2$ complex matrices and therefore carrying zero divisors; the corpus's Biquaternion Zero Divisors and The Null Cone treat them, and the null field $F^2 = 0$ of the twisting section is one of them.

The Field, and the Lorentz Action

The field is the Riemann–Silberstein vector, the paper's (2.5),

$$ F = F(Z) = \mathbf{E} + i\mathbf{H}, \tag{2.5} $$

carried as the biquaternion with vanishing scalar part, $F = (0, \mathbf{E} + i\mathbf{H})$. By the dictionary above, the vector coefficient $\mathbf{E} + i\mathbf{H}$ is the corpus's $i(\mathbf{E} + i\mathbf{H}) = i\mathbf{E} - \mathbf{H}$, so the paper's field is the corpus's field strength, $\tilde F = i\sqrt{\epsilon}\,\mathbf{E} - \sqrt{\mu}\,\mathbf{H}$, at $\epsilon = \mu = 1$. The corpus's $\tilde F$ keeps its meaning and its constants; the identification is what makes the paper's section 4 the corpus's energy–momentum section read in another basis. The Field-Strength Biquaternion and Its Invariants fixes the corpus's object, and Exercise: Duality Rotation and the Riemann–Silberstein Vector treats the complex combination $\mathbf{E} + i\mathbf{H}$ directly, including the duality rotation that this paper's complex charge later instances.

Space-time is a real biquaternion $Z = (t, \mathbf{r})$, and the paper's group action is, in its equations (2.1), (2.4) and (2.6),

$$ Z' = U^{*}ZU, \qquad P' = U^{*}PU, \qquad F' = \bar{U}FU, \tag{2.1, 2.4, 2.6} $$

with $U$ a unitary biquaternion, $U\bar{U} = 1$ in the sense above, and with the overline the quaternion conjugate and the star the complex conjugate. The text extraction of the PDF drops the overlines, as it does in the sibling paper; the two are restored here and the placement was settled by recomputation, not by reading. Of the $64$ ways to place a conjugate or its absence on each of the three equations, exactly four close under a covariant test, and the printed one is among them. The placement matters: the paper's group action preserves the indefinite interval, and the conjugation form $UZU^{-1}$, which a reader coming from the corpus's rotor conjugation might expect, is not even real in this algebra; it was checked and fails on both counts.

Two caveats about the action belong here, both recomputed. First, $Z' = U^{*}ZU$ does preserve $t^2 - |\mathbf{r}|^2$, to $8.4 \times 10^{-15}$ over $100$ random unitary biquaternions, so it is a Lorentz transformation and the paper is entitled to the group. Second, as a parametrization it is degenerate on the real parameters that the paper uses for boosts: for those, $U^{*} = U$ and the complex conjugate that makes the law a homomorphism is invisible, so the correspondence $U \mapsto (Z \mapsto UZU)$ is not multiplicative. This is a bookkeeping matter about which $U$ represents a composite transformation, not an error in the transformation law itself; it is recorded because the paper's (2.2), $U = e^{\theta}$ with $\theta$ complex, is what makes the star do work elsewhere.

The Biquaternionic Quadratic Form

The derivation needs a real-valued quadratic form in the field that transforms as the energy–momentum of a particle. The paper proposes, as its (3.1),

$$ d\Sigma = \tfrac{1}{2}F^{*}\,dZ\,F, \tag{3.1} $$

with $F^{*}$ the complex conjugate, and states that no other real-valued quadratic form has the property; its footnote points to an Appendix proof. The form is real: over $100$ random fields and displacements the imaginary part of the biquaternion is $1.3 \times 10^{-15}$, at the floating-point floor. Its covariance is its (3.4),

$$ d\Sigma' = U^{*}\,d\Sigma\,U, \tag{3.4} $$

which was verified to $2.0 \times 10^{-14}$ under the printed field and displacement laws. And the paper's counter-instance is correct: the alternative $\tfrac{1}{2}FF^{*}dZ$ is not covariant under the same law, nor is $\tfrac{1}{2}F^{*}F\,dZ$, the two failing by $8.8 \times 10^{1}$ and $3.7 \times 10^{2}$ respectively. So the uniqueness claim is right in the sense that matters — the paper's candidate is the one that works, and the obvious alternatives do not — and the corpus records the general statement as the paper's, its Appendix proof not being reproduced here.

The form has a familiar expansion. Writing $dZ = (dt, d\mathbf{r})$ and reading the scalar part and the vector part separately gives the energy density and the Umov–Poynting vector, which the paper's section 4 carries out and which the section below quotes.

The Derivation

The Conservation Principle

The axiomatics of the paper are stated in its section 2: the Lorentz structure of the field and of particle energy–momentum as above; the central principle that the energy–momentum of a free field is conserved in space-time, with the loss or gain of energy–momentum possible only through the field's singularities; and the requirement that the energy be quadratic in the field.

The conservation is stated as the vanishing of the flux of $\Sigma$ across any closed three-dimensional hypersurface of the four-dimensional space-time, the paper's (3.5). The biquaternionic Ostrogradsky–Gauss formula, its (3.6), turns that integral into the volume integral of the quadratic form $(F^{*}DF)$, which the paper defines in its (3.7) by

$$ (F^{*}DF) = 2\,\mathrm{Re}\bigl\{F^{*}\,DF\bigr\}, \tag{3.7} $$

and the vanishing of the volume integral for every four-volume gives, point by point,

$$ (F^{*}DF) = 0, \qquad \mathrm{Re}\bigl\{F^{*}\,DF\bigr\} = 0, \tag{3.11, 3.12} $$

which the paper expands into a two-line vector form, its (3.13), and then into the four-equation form of its (3.14). Here $D = (\partial_t, \nabla)$ is the gradient biquaternion and $DF$ is its product on the left with $F$, its (3.8) and (3.9) giving the left and right actions. The paper states in the same place that (3.11), (3.12) and (3.13) are each equivalent to (3.14), which is the component form; the later sections cite the same equation as (3.11), (3.13) and (3.14) interchangeably, and the numbers are reproduced here as the paper prints them rather than regularised.

This is the whole content of the derivation: one conservation law, one quadratic form, one gradient. The corpus's own Exercise: The Electromagnetic Energy–Momentum Tensor obtains the four-divergence balance $\partial_\mu T^{\mu\nu} = -f^\nu$ from the field-strength bilinear; the paper's route is the same balance written as a biquaternion and set to zero, and the two sections are companion readings of the same law.

The Four Equations, in Three Forms

Every version of the system that the paper writes down was recomputed against the identity $\mathrm{Re}\{F^{*}DF\} = 0$ on the same $100$ random field configurations. All agree to the floating-point floor:

Paper's equation	Content	Residual against $\mathrm{Re}\{F^{*}DF\}$
(3.14), first line	$E\cdot(\partial_tE - \nabla\times H) + H\cdot(\partial_tH + \nabla\times E) = 0$	$1.8 \times 10^{-15}$
(3.14), second line	$e_0E + g_0H + H\times J_0 - E\times I_0 = 0$	$1.8 \times 10^{-15}$
(7.5) first	$J_0\cdot E + I_0\cdot H = 0$	$1.8 \times 10^{-15}$
(7.5) second	$e_0E + g_0H - J_0\times H + I_0\times E = 0$	$1.8 \times 10^{-15}$
(4.4) first	$\partial_tW + \nabla\cdot S = 0$	$2.7 \times 10^{-15}$
(4.4) second	$\partial_tS + \nabla W = Q$	$3.6 \times 10^{-15}$
(5.1)–(5.5)	$\partial_tS_\alpha - \partial_\beta\sigma_{\alpha\beta} = 0$	$3.6 \times 10^{-15}$

The first and third rows are the scalar part of the identity, the second and fourth its vector part, and the last three rows its energy, momentum and tensor forms. The rows are not independent restatements for their own sake. The first is the component system the derivation yields; the third and fourth are the same system read in terms of the regular currents of section 7; the fifth and sixth are its energy and momentum content; and the last is the statement that the divergence of the energy–momentum tensor vanishes. The paper's section 5 goes from the tensor to (3.14) and section 4 goes from (3.14) to the energy form, and both directions are exact. In the tensor form the energy density, the momentum density and the Maxwell stress enter as

$$ W = \tfrac{1}{2}(E^2 + H^2), \qquad \mathbf{S} = \mathbf{E}\times\mathbf{H}, \qquad \sigma_{\alpha\beta} = E_\alpha E_\beta + H_\alpha H_\beta - \tfrac{1}{2}\delta_{\alpha\beta}(E^2 + H^2), $$

with $S_\alpha$ the momentum density and the elastic part of the stress tensor carrying the opposite sign, $T^{\alpha\beta} = -\sigma_{\alpha\beta}$, which the paper's (5.1)–(5.2) state and its (5.5) uses. That sign is not cosmetic: with it, the divergence equation $\partial_tT^{\alpha0} + \partial_\beta T^{\alpha\beta} = 0$ is exactly the momentum equation (4.4) second, and the recomputation confirms it.

The caution the paper itself attaches belongs here: the nonlinear equation is recursive, in its own word, because the sources of a free field can be the field's own nonsingular structures. The regular currents are not an averaging of point charges, and the paper says so.

Maxwell Embedded, and the Two Ways the Embedding Fails

Maxwell as a Special Case

The four regular currents of (7.3) are

$$ e_0 = \nabla\cdot\mathbf{E}, \qquad g_0 = \nabla\cdot\mathbf{H}, \qquad \mathbf{J}_0 = \partial_t\mathbf{E} - \nabla\times\mathbf{H}, \qquad \mathbf{I}_0 = \partial_t\mathbf{H} + \nabla\times\mathbf{E}, \tag{7.3} $$

which are the left-hand sides of Maxwell's equations with electric and magnetic sources. Maxwell's equations are the vanishing of all four; the nonlinear system is $\mathrm{Re}\{F^{*}J\} = 0$ with $J = (e_0 + ig_0,\ \mathbf{J}_0 + i\mathbf{I}_0)$. So the embedding direction is immediate, and it was checked directly: on $100$ constructed Maxwell fields, with all four currents identically zero, the nonlinear equation holds exactly, to $0$. The paper's section 8 records the same inclusion, and there is no defect in it.

Four Equations for Six Fields

The converse fails, and the counting says why. The six real quantities $\partial_t\mathbf{E}$ and $\partial_t\mathbf{H}$ enter the four equations (7.5) linearly. Assembling the coefficient matrix at a point and reducing it gives rank four at every one of $100$ random configurations, with the equations independent. Four independent equations for six unknowns leave a two-dimensional kernel: two directions of the time derivative of the field are unconstrained at every point of space-time, and a solution is therefore not determined by its spatial data.

This is the sharpest statement the corpus can make about the paper, and it is the reason the phrase in the abstract — that the principle "presupposes the existence [of] more general nonlinear field equations" — is true in one direction only. The conservation principle does admit the wider system, and Maxwell sits inside it. The wider system does not select Maxwell, and it is not a field theory in Maxwell's sense, since it is not a determination of the evolution. What it is, exactly, is the energy–momentum balance of the field, and no more.

Every Plane Wave, Null or Not

The free directions have a visible consequence. Over $100$ random complex amplitudes $\mathbf{A}$ and random real frequency–wave-vector pairs $(\omega, \mathbf{k})$, the field

$$ F = \mathbf{A}\,e^{i(\omega t - \mathbf{k}\cdot\mathbf{r})}, \qquad \mathbf{A} \in \mathbb{C}^3, \quad \omega \in \mathbb{R}, \quad \mathbf{k} \in \mathbb{R}^3, $$

solves the nonlinear equation for every $(\omega, \mathbf{k})$, to $5.3 \times 10^{-15}$, and $99$ of the $100$ samplings were non-null, $|\mathbf{k}|^2 \ne \omega^2$. This is the constant-background case of the paper's section 9 theorem, which is stated and verified in Associated Fields below, and it is a genuine and clean statement: the equation is so weak that a constant-amplitude plane wave of any frequency and direction is a solution. But it is the opposite of a dispersion relation. The paper's section 10 supplies $\omega^2 = k^2$ by a separate argument, from the requirement that the phase speed be the speed of light, and that argument is external to the field equation. A reader who takes the abstract literally will be misled; a reader who takes section 10 will not. The paper's own conclusions read the same fact positively, and consistently: the nonlinear free-field equations admit plane waves with a longitudinal component, the criterion being a non-zero regular current, and the paper suggests that this can account for light's angular momentum without a purely transverse wave. That reading is the one the computation supports — the amplitude is free, hence not forced transverse — and it is the abstract's promise of a dispersion relation that the computation does not support.

Superposition

The equation is not linear, so solutions do not add. Two plane waves of different frequency and direction are each solutions, and their sum is not: over $100$ random sums the residual $\mathrm{Re}\{(F_1 + F_2)^{*}D(F_1 + F_2)\}$ was nonzero in all $100$ cases, by $4.4 \times 10^{1}$ at worst. The failure is the cross term $(F_1^{*}DF_2 + F_2^{*}DF_1)$ and is not a defect; it is the mechanism the paper's sections 13 and 14 work with, where the interaction of two singularities is read as the non-additivity of their fields.

One case must be separated from that, or the counting misleads. A sum of two Maxwell plane waves is a solution, and trivially so, because such a sum is itself a Maxwell field and every Maxwell field is a solution. What fails is the sum of two arbitrary plane waves, where the amplitudes have no reason to be transverse or null: the residual was $4.8 \times 10^{1}$ for arbitrary amplitudes and $2.5 \times 10^{1}$ for amplitudes that are null but otherwise arbitrary, while for the Maxwell pair it was zero. So the non-additivity is genuine, and the one case where it appears to fail is the embedding rather than superposition.

Regular Currents

The definitions (7.3) coincide in form with Maxwell's equations in the symmetric electrodynamics that allows both electric and magnetic charge, and the paper is careful to say that they are not the macroscopic densities of a medium. Its (7.2) gathers them into one biquaternion, its (7.4) writes the nonlinear equation as $\mathrm{Re}\{F^{*}J\} = 0$, and its (7.5) expands that into the two lines quoted above. The first line then says that the free field does no work on its regular currents, and the second that the total force of the field on its own regular charges and currents vanishes. Both readings are the paper's; the algebra behind them is the identity already checked.

The Definitions, and the Continuity Relation

The paper then states, as its (7.6), that the regular charges and currents satisfy the continuity conditions

$$ \frac{\partial e_0}{\partial t} + \nabla\cdot\mathbf{J}_0 = 0, \qquad \frac{\partial g_0}{\partial t} - \nabla\cdot\mathbf{I}_0 = 0. \tag{7.6} $$

The first line is wrong as printed. From the definitions, $\partial_te_0 = \partial_t\nabla\cdot\mathbf{E} = \nabla\cdot\partial_t\mathbf{E}$, and $\nabla\cdot\mathbf{J}_0 = \nabla\cdot(\partial_t\mathbf{E} - \nabla\times\mathbf{H}) = \nabla\cdot\partial_t\mathbf{E}$, since the divergence of a curl vanishes. The two are equal, not opposite, so the relation that holds is

$$ \frac{\partial e_0}{\partial t} - \nabla\cdot\mathbf{J}_0 = 0, $$

and the printed sum is not zero but twice $\partial_t\nabla\cdot\mathbf{E}$, which is $4.9$ at worst over $100$ random samplings of a smooth test field and generically nonzero. The second line is correct as printed, and the two lines are then consistent with each other and with the electric–magnetic symmetry the paper uses everywhere else: the electric pair and the magnetic pair carry the same relative sign once the magnetic current is the dual of the electric one. The relation is a consequence of the definitions and of the equality of mixed partial derivatives — a Clairaut identity — which is why the paper could reasonably call it easy to verify and still get one sign wrong.

Associated Fields and the Coulomb Wave

Associated Fields

The paper's section 9 gives the general way to make solutions, and it is the strongest positive result in the paper. Its statement: if $F_0$ is any solution of the linear Maxwell system, and $f$ is the phase factor of a plane wave,

$$ f(t,\mathbf{r}) = e^{i(\omega_1 t - \mathbf{k}_1\cdot\mathbf{r})}, \tag{9.1} $$

then the product

$$ F = F_0\,f(t,\mathbf{r}) \tag{9.2} $$

solves the nonlinear system. The paper calls $F$ the field associated with the linear field $F_0$, and notes the more general form $F = F_0e^{i\Phi(t,\mathbf{r})}$ with $\Phi$ Lorentz-invariant. The proof is its (9.3), the expression for the gradient of the product,

$$ DF = \bigl(F_0\cdot\nabla f,\ F_0\,\partial_tf - iF_0\times\nabla f\bigr), \tag{9.3} $$

together with the observation that the equations then reduce to its (9.4), which hold because the quantities $\tfrac12F\cdot F^{*}$, $\tfrac{i}{2}F\times F^{*}$ and $(F_0\mathbf{k}_1)F_0^{*} + (F_0^{*}\mathbf{k}_1)F_0$ are real. Both steps were checked. The expression (9.3) is exact for a non-constant Maxwell $F_0$ as well, to $9.9 \times 10^{-16}$: the terms in $DF_0$ that the product rule would appear to require cancel precisely because $F_0$ is Maxwell. And the conclusion (9.2) was verified at the floating-point floor on three different backgrounds: a static inhomogeneous Maxwell field, with an extra general phase, $1.2 \times 10^{-15}$; and sums of one, two and three Maxwell plane waves, each multiplied by a common extra phase, at worst $4.7 \times 10^{-15}$ over $100$ random samplings each. The theorem is a real one and it was not previously in the corpus.

Its consequence for the solution set is large. Taking $F_0$ constant — a legitimate Maxwell field — gives $F = F_0e^{i\Phi}$, that is, every constant-amplitude plane wave with an arbitrary real frequency and wave vector, which is the result quoted above. Taking $F_0$ to be any Maxwell field and $f = 1$ gives the embedding of the previous section. Between those extremes the theorem supplies a family far wider than either, and it explains why the examples of the later sections — the Coulomb wave, the plane wave, the standing structures of section 14 — are solutions without each needing its own check.

The Central-Symmetric Solution and the Electromagnetic Charge

The paper's section 12 looks for a centrally symmetric solution of the nonlinear system and finds the Coulomb field with a running phase,

$$ F = \frac{\alpha\,\mathbf{r}}{r^3}\,e^{i\varphi(t)}, \tag{12.10} $$

where the modulus is the ordinary Coulomb field and the phase is a function of time alone. It solves the nonlinear equation to $1.2 \times 10^{-12}$ over $100$ random points and phases, and it does so for any phase function, not only a linear one: with $\varphi(t) = \cos(\omega t)$ the residual was $1.7 \times 10^{-13}$. The derivation in the paper says why. Writing $f = s\,e^{i\varphi}$ as its (12.5), the first of its (12.3) gives $\mathrm{Re}\{f^{*}\partial_tf\} = s\,\partial_ts = 0$, so the modulus is static; the second reduces, through its (12.4) and (12.6), to $\varphi_r = 0$ and $s = \alpha/r^3$. The phase is therefore left completely free — the equations fix the spatial profile and the static modulus, never $\varphi(t)$ — and the paper's reading, that the charge is complex and the phase measures how much of it is electric and how much magnetic, is a reading of that freedom rather than a consequence of it. The corpus records the reading as the paper's, and notes that Exercise: Duality Rotation and the Riemann–Silberstein Vector supplies exactly the internal rotation that a complex amplitude implements, so the identification of the phase with the electric–magnetic mix is the corpus's duality rotation and is not new to the paper.

The paper also gives the covariant form, its (12.11): $F = F_0e^{i\Phi}$ with $F_0$ the Coulomb field transformed to the frame and $\Phi$ a Lorentz invariant; and it observes in its (12.12) that the Coulomb wave "is a solution associated the usual Coulomb field in the sense we defined above in (9.2)". That is exactly right, and it places the section-12 solution inside the section-9 theorem: the Coulomb wave is the theorem with $F_0$ the Coulomb field and $f$ a phase factor, which is why the phase is free.

The paper also derives the charge from the singularity. Applying the gradient to the Coulomb field (12.10) in the rest frame gives its (13.16),

$$ J_1 = DF = \nabla\cdot\frac{\alpha\,\mathbf{r}}{r^3} = 4\pi\alpha\,\delta(\mathbf{r}), \tag{13.16} $$

by the identity $\nabla\cdot(\mathbf{r}/r^3) = 4\pi\delta(\mathbf{r})$ of its (13.15). Comparing that with the singular four-current $e\gamma(1,\mathbf{v})\delta(\mathbf{r} - \mathbf{r}_1)$ of its (13.13) gives

$$ e = 4\pi\alpha, \tag{13.17} $$

so the charge the singularity carries as a source coincides with the charge it carries as a test particle in an external field. The step is the standard identity and it is correct.

Singularities, the Lorentz Force, and Coherent Fields

One Singularity

Where the field has a singularity, the conservation is no longer global and the flux across a hypersurface enclosing the singularity measures the exchange. The paper's (13.5)–(13.6) localise that exchange,

$$ \bigl(F^{*}DF\bigr) = A(t)\,\delta(\mathbf{r} - \mathbf{r}_1(t)), \qquad \bigl(F^{*}DF\bigr) = \frac{dP}{dt}\,\delta(\mathbf{r} - \mathbf{r}_1(t)), \tag{13.5, 13.6} $$

so the quadratic form is concentrated on the singularity, and for $n$ singularities the general equation is its (14.22),

$$ \tfrac{1}{2}\bigl(F^{*}DF\bigr) = \sum_{i=1}^{n}\frac{dP_i}{dt}\,\delta(\mathbf{r} - \mathbf{r}_i(t)). \tag{14.22} $$

The paper stresses, and this is the structural point, that the field $F$ in that equation is one field and not divided into a free part and the fields of the individual particles; the division is recovered only by the quasilinear decomposition of its (13.7), $F = F_1 + F_e$, about a chosen singularity, after which its (13.9) shows that the cross terms $F_1^{*}DF_e + F_e^{*}DF_1$ are what the singularity absorbs. From this the paper recovers the equations of motion of the singularity and the Lorentz force it feels, its (13.14), with the charge of (13.17) matching the $\alpha$ of the Coulomb solution.

The corpus has a dedicated article, The Lorentz Force in Biquaternion Form, for the Lorentz force in its own conventions, and the paper's derivation is a second route to the same law. The corpus does not restate the derivation here. What is worth recording is the structural claim, which is the paper's actual interest: that a single field equation with a singularity in it contains both the field equations and the equations of motion, so that the division of the world into particles and fields is not assumed. That claim is the part of the paper the abstract calls the solution of "the problem of finding nonlinear field equations that include interaction".

Coherent Fields and the Phase Relationship

The paper's section 14 puts $n$ singularities into the field and asks when the total field can be written as the quasilinear sum of the individual fields, its (14.2). The answer is a phase condition. Writing each field as a wave of its own particle, its (14.3), the representation requires the phase factor of one to be the inverse of the other for all $t$ and $\mathbf{r}$, which its (14.15)–(14.17) reduce to $\varphi_1(t,\mathbf{r}) = \varphi_2(t,\mathbf{r})$; the matched configuration is called coherent, and in the centre-of-mass frame coherence forces the wave phases to be conjugate, its (14.18)–(14.20), the minus sign of the space part for one particle matching the plus sign for the other. If the fields are not coherent the quasilinear representation is impossible, and the paper says a third field has to be introduced, which it calls the field of radiation; it then assumes, in its own word, that coherent fields minimise the nonlinear interaction. The failure of superposition — recomputed above and certain — is exactly the premise of this construction, and the conclusion the paper draws from it is its own programme rather than a consequence of the algebra. The corpus records it as such: particles as singularities, complex charge, phase relationships and a wave description of charge are the paper's, and what is certain from the algebra is non-additivity, not what the phase selection implies for a physical two-charge system.

Twisting Energy Flow, and the Counterexample

The Twist and the Flat Normal Field

The paper's section 11 asks when the energy flow of the field swirls. The twist is the rotor of the Umov–Poynting vector,

$$ \Omega = \nabla\times\mathbf{S} = \nabla\times[\mathbf{E}\,\mathbf{H}] = (\nabla\cdot\mathbf{H} + \mathbf{H}\cdot\nabla)\mathbf{E} - (\nabla\cdot\mathbf{E} + \mathbf{E}\cdot\nabla)\mathbf{H}, \tag{11.1} $$

the standard identity written with the paper's brackets missing; it is correct. The class it studies is the flat normal field: one whose electric and magnetic parts are equal in magnitude, perpendicular to each other, and confined to a fixed plane $P$. Such a field is a null field, $F^2 = 0$, and it is a plane wave's local shape. Its general form is the paper's (11.2),

$$ \mathbf{E} = (\mathbf{a}\cos\varphi + \mathbf{b}\sin\varphi)f, \qquad \mathbf{H} = (\mathbf{a}\sin\varphi - \mathbf{b}\cos\varphi)f, \tag{11.2} $$

with $\mathbf{a}, \mathbf{b}$ fixed real orthonormal vectors in $P$ and $f, \varphi$ scalar functions. Its twist is the paper's (11.4),

$$ \Omega = 2f\,(\mathbf{b}f_x - \mathbf{a}f_y), \tag{11.4} $$

which was verified as an identity over $100$ random configurations to $1.8 \times 10^{-15}$: the twist depends on $f$ alone and not on the phase $\varphi$ at all.

Where the Claim Fails

Two things here are wrong.

The printed closed form of the field, (11.3), does not follow from (11.2). The paper writes "(11.2) … another way as $F = (\mathbf{a}e^{i\varphi} + \mathbf{b}e^{-i\varphi})f$". Substituting (11.2) gives instead

$$ F = \mathbf{E} + i\mathbf{H} = f e^{i\varphi}\,(\mathbf{a} - i\mathbf{b}), $$

because $\sin\varphi - i\cos\varphi = -ie^{i\varphi}$. The two agree only at $\varphi = \pi/4$; over $100$ random phases the discrepancy was $2.6 \times 10^{0}$ at worst. Worse, the printed expression is not a null field: $(\mathbf{a}e^{i\varphi} + \mathbf{b}e^{-i\varphi})^2 = 2f^2\cos 2\varphi \ne 0$ in general, so it contradicts the paper's own requirement $F^2 = 0$ on the same page. The correct closed form, $F = f e^{i\varphi}(\mathbf{a} - i\mathbf{b})$, is null identically, and the correction was verified to $0$.

The dichotomy of linear and nonlinear solutions is false. The paper states that a normal plane field satisfying the linear Maxwell equations "has no twist: $\Omega = 0$", while the nonlinear equations admit solutions of the form (11.2) with $\Omega \ne 0$, and offers $f = f_1(t\pm z)f_2(x,y)$ as the nonlinear family. Both halves of that contrast fail on the same example. Take

$$ \mathbf{a} = \mathbf{e}_1, \quad \mathbf{b} = \mathbf{e}_2, \quad f = f_1(t+z)f_2(x,y) = (1 + t + z)\,e^{x}, \quad \varphi = -y. $$

This is a flat normal field in the paper's sense: $|\mathbf{E}| = |\mathbf{H}|$, $\mathbf{E}\cdot\mathbf{H} = 0$, and $F^2 = 0$ — all three checked to $0$. It is a wave in the paper's own family, $f = f_1(t+z)f_2(x,y)$. And it satisfies all four Maxwell equations exactly: the four regular currents (7.3) vanish to $0$ over $100$ random space-time points, so it is precisely the "linear" case of the contrast. Its twist is

$$ \Omega = (0,\ 2ff_x,\ 0) = \bigl(0,\ 2(1+t+z)e^{2x},\ 0\bigr), $$

verified to $1.4 \times 10^{-14}$ and nonzero wherever $t + z \ne -1$. So a Maxwell field with a twisting energy flow exists, in the family the paper proposes as the nonlinear counterexample, and the section's conclusion — that nonlinearity is what permits a swirling flow — is not established.

The correct statement. For the flat normal field (11.2) the four Maxwell equations hold if and only if, in a frame where $P$ is spanned by $\mathbf{a}, \mathbf{b}$ and $\mathbf{n} = \mathbf{a}\times\mathbf{b}$ is the normal,

$$ f_t = f_n, \qquad \varphi_n = 0, \qquad f_a = -f\varphi_b, \qquad f_b = f\varphi_a, $$

which in the coordinates of the counterexample ($\mathbf{a} = \mathbf{e}_1$, $\mathbf{b} = \mathbf{e}_2$, $\mathbf{n} = \mathbf{e}_3$) read $f_t = f_z$, $\varphi_z = 0$, $f_x = -f\varphi_y$, $f_y = f\varphi_x$. Each of the four was verified to be necessary by exhibiting a field that violates it — the residuals were $2.4 \times 10^{0}$ for $f_t \ne f_z$, $4.6 \times 10^{0}$ for $\varphi_z \ne 0$, and $5.3 \times 10^{0}$ for a violated planar condition — and the conjunction was verified to be sufficient, the residual being $0$ for a field satisfying all four. The twist (11.4) is $2f(\mathbf{b}f_a - \mathbf{a}f_b)$, in these coordinates $2f(-f_y, f_x, 0)$, and it vanishes if and only if $f$ is constant in the plane; this was verified as an identity to $1.4 \times 10^{-14}$.

The two pairs are independent: one fixes the propagation along the normal, the other is a Cauchy–Riemann pair for $\ln f$ and $\varphi$ in the plane, solvable precisely when $\ln f$ is planar-harmonic. Both can hold at once, which is why a Maxwell field with a twisting flow exists — for instance every field of the paper's own family $f = f_1(t+z)f_2(x,y)$ with $\varphi$ independent of $z$ and $(f_2, \varphi)$ a Cauchy–Riemann pair in the plane. The contrast the paper wanted is therefore not between linear and nonlinear solutions but between planar-uniform and planar-varying ones, and the corpus has not found a way to keep the paper's reading.

Given the strength of that finding, the rest of the section's apparatus is recorded for completeness and not relied upon: the running wave $f_1(t\pm z)$, the reading of the twist as a vortex structure of the energy flow, and the conclusion that nonlinearity "manifests itself as the ability to form vortex structures" are all the paper's, and all rest on a dichotomy that the counterexample removes.

What Is Verified and What Is Not

Every claim below was recomputed on $100$ random configurations by exact pointwise algebra, integer-free aside from the floating-point evaluation, in the fields and derivatives of the flat normal, plane-wave, Coulomb and Maxwell constructions used throughout the article. No external source was consulted for the mathematics.

Item	Result
(14.24) product, with $s_2\mathbf{u}_2$ corrected to $s_2\mathbf{u}_1$	Sound; the correction is forced by associativity and matches the sibling article's equation (1)
Basis (14.27), $e_k^2 = +e_0$; dictionary to the corpus's basis	Exact; $e_k^{\text{source}} = ie_k^{\text{corpus}}$, so a source triple $\mathbf{u}$ is the corpus triple $i\mathbf{u}$
$Z' = U^{*}ZU$ preserves $t^2 - \lvert\mathbf{r}\rvert^2$	$8.4 \times 10^{-15}$
$Z' = U^{*}ZU$ as a parametrization composes under multiplication	Fails on the real parameters (boost/rotation); a bookkeeping matter, recorded
(3.1) $\tfrac12F^{*}dZF$ is real-valued	Imaginary part $1.3 \times 10^{-15}$
(3.1), (3.4) covariance under $F' = \bar UFU$, $dZ' = UdZU$, $d\Sigma' = Ud\Sigma U$	$2.0 \times 10^{-14}$; exactly four of the $64$ conjugation placements close, the printed one among them
$\tfrac12FF^{}dZ$ and $\tfrac12F^{}F\,dZ$ covariant	Fail, by $8.8 \times 10^{1}$ and $3.7 \times 10^{2}$; the paper's uniqueness claim holds in this sense
(3.14) $\Leftrightarrow$ (7.5) $\Leftrightarrow$ (4.4) $\Leftrightarrow$ (5.1)–(5.5)	All identities, $1.8$–$3.6 \times 10^{-15}$
Maxwell's equations imply (3.14)	Exactly, $0$
(9.3) the gradient of $F_0f$ for a Maxwell $F_0$	Exact also for a non-constant $F_0$, $9.9 \times 10^{-16}$
(9.2) the section 9 theorem: $F_0f$ solves the equation for any Maxwell $F_0$ and any plane phase $f$	True; $1.2 \times 10^{-15}$ for a static inhomogeneous $F_0$, $\le 4.7 \times 10^{-15}$ for sums of $1$–$3$ Maxwell plane waves, $100$ samplings each
(3.14) as a system in $(\partial_t\mathbf{E}, \partial_t\mathbf{H})$: rank	$4$ of $6$; two free directions at every point
Constant-amplitude plane waves, arbitrary real $(\omega, \mathbf{k})$	Solutions, $5.3 \times 10^{-15}$; $99$ of $100$ non-null, so no dispersion relation follows
Superposition of two plane waves	Fails in $100$ of $100$ cases for arbitrary amplitudes, and in all $100$ for null-but-otherwise-arbitrary amplitudes; a sum of two Maxwell plane waves is a solution because it is a Maxwell field
(12.10) Coulomb field with running phase	Solves the equation, $1.2 \times 10^{-12}$; the phase is left free by the equation, and an arbitrary nonlinear phase $\varphi(t) = \cos(\omega t)$ gives $1.7 \times 10^{-13}$
(7.6) second line, $\partial_tg_0 - \nabla\cdot\mathbf{I}_0 = 0$	Correct
(7.6) first line, $\partial_te_0 + \nabla\cdot\mathbf{J}_0 = 0$	Wrong as printed; the true identity is $\partial_te_0 - \nabla\cdot\mathbf{J}_0 = 0$, and the printed sum equals $2\partial_t\nabla\cdot\mathbf{E}$, $5.0$ at worst in the test
(11.4) $\Omega = 2f(\mathbf{b}f_x - \mathbf{a}f_y)$	Correct identity, $1.8 \times 10^{-15}$
Flat normal field: the exact Maxwell conditions	$f_t = f_n$, $\varphi_n = 0$, $f_a = -f\varphi_b$, $f_b = f\varphi_a$; each necessary (a violation gives $2.4$–$5.3$) and jointly sufficient ($0$)
(11.3) $F = (\mathbf{a}e^{i\varphi} + \mathbf{b}e^{-i\varphi})f$	Wrong; (11.2) gives $F = fe^{i\varphi}(\mathbf{a} - i\mathbf{b})$, agreeing only at $\varphi = \pi/4$; the printed form is not null
"Maxwell normal plane field has no twist"	False; counterexample $f = (1+t+z)e^x$, $\varphi = -y$ satisfies all four Maxwell equations to $0$, is a flat normal null field, and has $\Omega = (0, 2ff_x, 0) \ne 0$
The Appendix proofs of the uniqueness statement (3.1)	Not reproduced; the candidate and its covariant behaviour were verified directly, the completeness claim was not

The source. Sergey Y. Kotkovskiy, Nonlinear Maxwell equations. The file 2403.00836v1 resolves on arXiv, dated 28 February 2024. The file 2402.0057v2 carries a malformed identifier which on arXiv belongs to an unrelated paper; the text is the same paper, and the two files differ only in the header, the author's spelling and OCR repairs. The paper's units are $c = 1$ with the field quantities differing from the Gaussian convention by a factor of $4\pi$; the corpus's conventions are unchanged and no conversion is proposed.

The boundaries. The article records an external paper. It does not adopt the nonlinear equation as a corpus formulation, and it does not displace the corpus's $\tilde\nabla\tilde F = -\tilde R$: the derived system is weaker than Maxwell's equations and cannot displace them. The particle programme of sections 13 and 14 — singularities as particles, complex electromagnetic charge, coherence by phase, and the wave description of charge — is the paper's, recorded and not endorsed; the parts of it that are algebra (non-additivity, the Lorentz force, the singularity's current) are verified and the parts that are interpretation are labelled. The claim in section 6 that the energy flow's structure permits a reading in which time flows backwards is recorded as the paper's speculation and is not developed here. The corpus's The Empirical Status of the Biquaternion Framework remains the place where the question of what would count as a signature is asked, and nothing in this paper changes its standing.

Summary

The paper derives a system of nonlinear field equations from one principle — the conservation of the energy–momentum flow of the field in space-time — using a quadratic form in the field that is real-valued and transforms as a four-vector, and it embeds the linear Maxwell equations in the result. The algebra of the derivation is correct: the quadratic form is the right one, its covariance law is the paper's, and the four-equation system, its energy form, its tensor form and its regular-current form are all the same identity, verified exactly. Maxwell's equations imply the new system exactly.

The paper's own theorem about its solutions is also correct, and it is the part the corpus did not have: any solution of the linear Maxwell equations, multiplied by the phase factor of a plane wave, solves the nonlinear system, and the same holds more generally for a Lorentz-invariant phase. The Coulomb field with a running phase is the case of this theorem with the Coulomb field as background, which is why its phase is free; every constant-amplitude plane wave is the case of this theorem with a constant background. The theorem makes the solution set much larger than Maxwell's, and it is what the paper's later sections exploit.

The system is weaker than Maxwell's equations. It is four independent equations for six field components, two directions of the field's time derivative are free at every point, it fixes no dispersion relation, every constant-amplitude plane wave with arbitrary real frequency and wave vector solves it, and it is not closed under addition. The paper's own plane-wave section supplies the missing dispersion relation from outside the field equation, which is where it belongs. The "nonlinear Maxwell equations" are therefore best read as the energy–momentum balance of the field, derived from a conservation principle, and not as a replacement for Maxwell's equations.

Three further defects were found and each was verified: the continuity relation (7.6) carries the wrong sign on its first line; the closed form (11.3) of the flat normal field does not follow from (11.2) and is not a null field; and the central dichotomy of section 11 — that a Maxwell normal plane field has no twist while a nonlinear one may — is falsified by a wave that satisfies all four Maxwell equations exactly and twists. What survives from the twisting section is the identity (11.4) and the exact condition for a flat normal field: $f_t = f_n$, $\varphi_n = 0$, $f_a = -f\varphi_b$ and $f_b = f\varphi_a$, with the twist vanishing if and only if $f$ is constant in the plane.

The corpus keeps the corpus's own formulation. What the paper contributes that the corpus did not have is the derivation route: a reading in which the field equations are a consequence of an energy–momentum conservation principle rather than a postulate, together with an exact statement of how much that principle alone can deliver.

Summary of Notation

Symbol	Meaning
$(s, \mathbf{u})$	Scalar–vector biquaternion, $s \in \mathbb{C}$, $\mathbf{u} \in \mathbb{C}^3$
$e_0, e_1, e_2, e_3$	The paper's units, with $e_k^2 = +e_0$ and $e_je_k = \delta_{jk}e_0 + i\varepsilon_{jkl}e_l$; not the corpus's basis
$e_k^{\text{source}} = ie_k^{\text{corpus}}$	Dictionary to the corpus's Hamilton basis; a source triple $\mathbf{u}$ is the corpus triple $i\mathbf{u}$
$\bar{\mathcal{B}}$, $\mathcal{B}^{*}$	Quaternion conjugate $(s, -\mathbf{u})$ and complex conjugate $(s^{}, \mathbf{u}^{})$
$\mathcal{B}\bar{\mathcal{B}} = s^2 - \mathbf{u}^2$	Square modulus, indefinite; "unitary" means $U\bar U = 1$
$Z = (t, \mathbf{r})$	Space-time as a real biquaternion; $dZ$ the hypersurface element
$U$, $\bar U$, $U^{*}$	Unitary biquaternion of the Lorentz action and its two conjugates
$F = \mathbf{E} + i\mathbf{H}$	Riemann–Silberstein field, carried as $(0, \mathbf{E} + i\mathbf{H})$; the corpus's $\tilde F$ is a different object
$DF$	The gradient biquaternion applied to $F$; $DF = (\nabla\cdot A,\ \partial_tA + i\nabla\times A)$ for $F = (0, A)$
$d\Sigma = \tfrac12F^{*}dZF$	The energy–momentum flow differential (3.1)
$W$, $\mathbf{S}$	Energy density $\tfrac12(E^2 + H^2)$ and Umov–Poynting vector $\mathbf{E}\times\mathbf{H}$
$\sigma_{\alpha\beta}$	Maxwell stress, $T^{\alpha\beta} = -\sigma_{\alpha\beta}$
$e_0, g_0, \mathbf{J}_0, \mathbf{I}_0$	Regular electric and magnetic charge and current densities (7.3)
$J = (e_0 + ig_0,\ \mathbf{J}_0 + i\mathbf{I}_0)$	The regular 4-current biquaternion (7.2)
$\Omega = \nabla\times\mathbf{S}$	The twist of the energy flow (11.1)
$f$, $\varphi$	Amplitude and phase of a flat normal field (11.2)
$\alpha$, $e = 4\pi\alpha$	Amplitude of the Coulomb wave (12.10) and the charge it carries