Exact SU(2) Yang-Mills Waves from a Simple Ansatz

Jing-Ling Chen; Yu-Xuan Zhang

arxiv: 2605.04964 · v1 · submitted 2026-05-06 · 🪐 quant-ph · hep-th· math-ph· math.MP

Exact SU(2) Yang-Mills Waves from a Simple Ansatz

Yu-Xuan Zhang , Jing-Ling Chen This is my paper

Pith reviewed 2026-05-08 16:27 UTC · model grok-4.3

classification 🪐 quant-ph hep-thmath-phmath.MP

keywords SU(2) Yang-Millsexact wave solutionsnon-Abelian wavesansatz reductionalgebraic constraintsnonlinear gauge fields

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The pith

A simple ansatz reduces the SU(2) Yang-Mills equations to nine algebraic constraints that yield three families of exact waves.

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

The paper introduces an ansatz in which the gauge potentials depend on a single phase θ = kz − ωt and are expanded in a y-dependent rotated Pauli basis. This form converts the sourceless nonlinear SU(2) Yang-Mills equations in four dimensions into nine purely algebraic constraints. Complete solution of those constraints produces three closed-form families. Family I recovers ordinary linear electromagnetic waves inside the non-Abelian theory. Family II supplies genuinely nonlinear waves that still travel at the speed of light yet carry a constant gauge-invariant offset and exhibit energy-density nodes whose locations are fixed by a discrete sign parameter. Family III consists of pure-gauge configurations whose field strengths vanish for any choice of wave number and frequency.

Core claim

The central claim is that the chosen ansatz reduces the nonlinear SU(2) Yang-Mills equations to nine algebraic constraints whose complete solution yields three families of exact waves: linear Abelian waves with dispersion relation ω = kc, nonlinear self-interacting waves that propagate at light speed but possess a constant color-electric offset and energy-density nodes controlled by the discrete parameter ξη = ±1, and pure-gauge solutions with identically vanishing field strengths that exist for arbitrary k and ω without any dispersion relation.

What carries the argument

The y-dependent rotated Pauli basis combined with single-phase dependence θ = kz − ωt, which converts the nonlinear partial differential equations into a closed set of algebraic constraints on the expansion coefficients.

If this is right

Nonlinear waves propagate at the speed of light yet fail to obey superposition because commutator terms remain nonzero.
The constant gauge-invariant field offset produces a nonzero time-averaged color-electric field.
Energy-density nodes occur at θ = 0 or θ = π according to the sign of the discrete topological parameter ξη.
Pure-gauge solutions exist for any wave number and frequency with no dispersion relation required.
All three families are given by explicit closed-form expressions.

Where Pith is reading between the lines

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

These analytic solutions could serve as exact benchmarks for numerical integrators of non-Abelian field equations.
The constant offset in Family II may point to stable background configurations that survive in more general non-Abelian theories.
The topological sign that fixes node locations could be linked to discrete invariants appearing in other gauge-field problems.
Stability analysis of the nonlinear family under small perturbations would test whether the waves remain intact in a dynamical setting.

Load-bearing premise

The gauge potentials are assumed to depend on only one phase θ = kz − ωt and to be built from a y-dependent rotated Pauli basis, and this form is taken to be rich enough to capture the relevant exact wave solutions.

What would settle it

Direct substitution of the Family II solution back into the original SU(2) Yang-Mills field equations, followed by verification that every component of the residual is identically zero, would confirm the algebraic constraints fully solve the nonlinear system.

read the original abstract

We propose a simple ansatz to construct exact wave solutions of the sourceless SU(2) Yang-Mills equations in (3+1) dimensions. The ansatz employs a $y$-dependent rotated Pauli basis and assumes a phase $\theta=kz-\omega t$ dependence for the gauge potentials. Owing to this ansatz, the nonlinear field equations reduce to nine algebraic constraints, whose complete solution yields three families of exact waves. Family I describes linear (Abelian) electromagnetic waves embedded in the non-Abelian theory; all commutator terms vanish and the dispersion relation is $\omega=kc$. Family II represents genuinely nonlinear self-interacting waves that also propagate at the speed of light but exhibit a constant field offset, nonvanishing commutators, and do not obey superposition. The constant offset is gauge-invariant and gives rise to a non-zero time-averaged color-electric field. The energy density has nodes whose position ($\theta=0$ or $\theta=\pi$) is controlled by a discrete topological parameter $\xi\eta=\pm1$, providing an observable signature. Family III is a pure gauge solution with vanishing field strengths, valid for arbitrary $k$ and $\omega$ without any dispersion relation. All solutions are closed-form and provide new insights into how non-Abelian self-interactions fundamentally alter wave propagation.

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

This paper gives a clean ansatz that turns the SU(2) Yang-Mills equations into algebraic constraints and produces three families of exact waves, with the nonlinear family carrying a gauge-invariant constant offset as the clearest addition.

read the letter

The main point is that the authors use a y-dependent rotated Pauli basis plus a single-phase ansatz to reduce the sourceless SU(2) Yang-Mills equations to nine algebraic constraints, then solve them completely. This yields three families: ordinary Abelian waves, genuinely nonlinear waves that still travel at light speed but carry a constant color-electric offset and non-zero commutators, and pure-gauge solutions with no dispersion relation required. The nonlinear family also has energy-density nodes whose locations are set by the discrete sign of ξη, which is gauge-invariant and could serve as a benchmark for numerics. Closed-form exact solutions in non-Abelian theories are uncommon, so this is useful on its face. The derivation is direct substitution followed by algebra, with no fitted parameters or external data, and the stress-test found no internal contradictions in the reported properties or dispersion relations. The constant offset and nodal control are presented as new outputs of this particular ansatz rather than restatements of earlier work. The ansatz is restrictive by design, so it will not find every possible wave, but the authors do not claim universality. The nine constraints are stated to be fully solved, and the resulting expressions satisfy the original equations by construction. One minor limitation is that the specific y-dependence and single-phase form may exclude other non-Abelian structures, though that is the usual trade-off with ansatz methods. This work is aimed at people who need concrete, verifiable solutions for testing numerical codes in QCD-like theories or who study exact waves in gauge theories. It is not a broad survey or a new general method, but the explicit families and their properties are solid enough to merit referee time. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript proposes a simple ansatz for the SU(2) gauge potentials in (3+1) dimensions, employing a y-dependent rotated Pauli basis and a single phase θ = kz − ωt. This ansatz is used to reduce the sourceless nonlinear Yang-Mills field equations to nine algebraic constraints. The complete solution of these constraints produces three families of exact wave solutions: Family I (Abelian electromagnetic waves with dispersion ω = kc), Family II (nonlinear self-interacting waves with constant offset, nonvanishing commutators, light-speed propagation, and energy density nodes controlled by ξη = ±1), and Family III (pure gauge solutions with vanishing field strengths for arbitrary k, ω).

Significance. Exact solutions to the nonlinear SU(2) Yang-Mills equations are uncommon and valuable for understanding non-Abelian dynamics. If the reduction and solutions are correct, this work demonstrates how the ansatz captures both Abelian embeddings and genuinely nonlinear effects, including a gauge-invariant constant offset leading to non-zero time-averaged color-electric field. The discrete topological parameter affecting energy density nodes provides an observable signature. The closed-form expressions and the fact that the dispersion relations emerge from the constraints rather than being imposed are strengths of the approach.

minor comments (3)

The abstract refers to 'nine algebraic constraints' without listing them; the main text should explicitly enumerate these constraints (and show the substitution from the field equations) for transparency and verifiability.
Clarify the explicit form of the rotated Pauli basis used in the ansatz, including the precise y-dependence and how it is inserted into the gauge potentials.
In the description of Family II, specify the explicit value of the constant field offset in terms of the free parameters and confirm its gauge invariance by direct computation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on exact SU(2) Yang-Mills waves and for recommending minor revision. The provided summary accurately reflects the ansatz, the reduction to algebraic constraints, and the three families of solutions. No specific major comments or criticisms were raised in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation

full rationale

The paper introduces an explicit ansatz for the gauge potentials (y-dependent rotated Pauli basis with single-phase dependence θ = kz − ωt), substitutes it into the sourceless SU(2) Yang-Mills equations, and reduces the system to nine algebraic constraints whose solutions are obtained by direct algebraic manipulation. The three families of waves, their dispersion relations, and properties such as the gauge-invariant offset emerge strictly from solving those constraints; no parameters are fitted to data, no external results are invoked to force the outcomes, and no self-citation chain supports the central reduction. The derivation is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The ansatz itself supplies the functional form of the potentials; the SU(2) Lie-algebra commutation relations and the sourceless Yang-Mills equations are taken as standard background.

free parameters (1)

discrete parameter ξη
Controls the location of energy-density nodes in Family II; takes values ±1.

axioms (2)

standard math SU(2) gauge potentials satisfy the standard Lie-algebra commutation relations [T^a, T^b] = i ε^{abc} T^c
Invoked implicitly when expanding the field-strength tensor and the nonlinear equations.
domain assumption The sourceless Yang-Mills equations D_μ F^{μν} = 0 hold in Minkowski space
The starting point of the derivation.

pith-pipeline@v0.9.0 · 5540 in / 1529 out tokens · 24114 ms · 2026-05-08T16:27:37.653465+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

[1]

= 0, 2g (α2 2 −α 2 1)(λ+ 2gα 3) = 0, α5 k2 − ω2 c2 −4g 2(α2 1 −α 2 2) + 4gα4 ω c α1 −kα 2 = 0, α4 k2 − ω2 c2 −4g 2(α2 1 −α 2 2) + 4gα5 ω c α1 −kα 2 = 0, α2(λ+ 2gα 3)2 + 2gα4(2gα2α4 −kα 5) = 0, (λ+ 2gα 3)(4gα2α5 −kα 4) = 0, 4g2α2(α2 5 −α 2

work page
[2]

Three Families of Exact Solutions.—Solving the sys- tem (6) of nine algebraic equations fork̸= 0, ω̸= 0 yields three families

= 0.(6) These are the master constraints for exact solutions. Three Families of Exact Solutions.—Solving the sys- tem (6) of nine algebraic equations fork̸= 0, ω̸= 0 yields three families. Family I: Linear (Abelian) Waves.—The param- eters are given by α1 =α 2 = 0, α 3 =− λ 2g , α 5 = 0, ω=kc,(7) andα 4 is free. The potentials reduce toφ= 0,A x = Az = 0,A...

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See Supplemental Material for the detailed derivation of the electric field, magnetic field and the nine algebraic constraints

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Aidelsburger, M

M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch,Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices, Phys. Rev. Lett.111, 185301 (2013)

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Exact SU(2) Yang-Mills Waves from a Simple Ansatz

H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Bur- ton, and W. Ketterle,Realization of the Harper Hamilto- nian with laser-stirred Bose-Einstein condensates, Phys. Rev. Lett.111, 185302 (2013). Supplemental Material for “Exact SU(2) Yang-Mills Waves from a Simple Ansatz” Yu-Xuan Zhang 1 and Jing-Ling Chen 2,∗ 1School of Physics, Nankai University, Tia...

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Nonlinear Self-Interacting Waves 9

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magnetic

Pure Gauge Solution 10 References 10 ∗Electronic address: chenjl@nankai.edu.cn 2 I. THE YANG-MILLS EQUATIONS The Yang-Mills equations in vacuum without sources (i.e., the 4-vector currentJ ν = 0) are given by [1] DµF µν ≡∂ µF µν + ig h Aµ, F µν i = 0,(1a) DµFνγ +D νFγµ +D γFµν = 0.(1b) Here the covariant derivativeD µ is defined as Dµ =∂ µ + igAµ,(µ= 0,1,...

work page
[29]

Nine Algebraic Constraints from Eq

cos2 θ ) Σx⃗ ez = 0.(33) D. Nine Algebraic Constraints from Eq. (25) and Eq. (33) In this work, we focus on the nontrivial case ofg̸= 0. In the following, let us come to analyze Eq. (25) and Eq. (33). Firstly, we show that the case ofk= 0 andω= 0 is trivial. Remark4.The case ofk= 0 andω= 0 is trivial. In this case, one hasθ=kz−ωt= 0, cosθ= 1, sinθ= 0, and...

work page
[30]

[λ+ 2g(α 3 +α 5)] = 0, α2 λ+ 2g(α 3 +α 5) 2 = 0,(36) which leads to (i) [λ+ 2g(α 3 +α 5)] = 0 and (ii){α 1 = 0, α2 = 0}. Accordingly, we have the electric-like field and the 8 magnetic-like field as ⃗E= nh (−λα1 −2gα 1α3) + ω c α4 −2gα 1α5 cosθ i Σy + − ω c α5 + 2gα1α4 sinθΣ z o ⃗ ey = [(−λα 1 −2gα 1α3) + (−2gα1α5)] Σy⃗ ey =−α 1 [λ+ 2g(α 3 +α 5)] Σy⃗ ey =...

work page
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= 0,(38c) 2g (α2 2 −α 2 1)(λ+ 2gα 3) = 0,(38d) α5 k2 − ω2 c2 −4g 2(α2 1 −α 2 2) + 4gα4 ω c α1 −kα 2 = 0,(38e) α4 k2 − ω2 c2 −4g 2(α2 1 −α 2 2) + 4gα5 ω c α1 −kα 2 = 0,(38f) α2(λ+ 2gα 3)2 + 2gα4(2gα2α4 −kα 5) = 0,(38g) (λ+ 2gα 3)(4gα2α5 −kα 4) = 0,(38h) 4g2α2(α2 5 −α 2

work page
[32]

= 0.(38i) After solving these 9 algebraic equations, we can have three families of exact solutions (generallyλ̸= 0): (i) Linear waves, (ii) Nonlinear self-interacting waves, and (iii) Pure gauge solution

work page
[33]

linear waves

Linear Waves The parameters satisfy the following conditions α1 = 0, α 2 = 0, α 3 =− λ 2g , α 5 = 0, ω=kc,(39) whereα 4 is an arbitrary real constant,λis an arbitrary real constant (ensuringλyis dimensionless),kis the wave number,ωis the frequency, andgis the gauge coupling constant. The phase is defined asθ=kz−ωt. The gauge potentials are φ= 0, A x = 0, ...

work page
[34]

The phase is defined asθ=kz−ωt

Nonlinear Self-Interacting Waves The parameters satisfy the following conditions: α1 =α 2 = ηk 4g , α 3 =ξα 4 − λ 2g , α 5 =ηα 4, ω=kc, η=±1, ξ=±1, α 4 ̸= 0, λ∈R, ω >0.(42) whereη, ξ=±1 are discrete topological parity parameters,α 4 is an arbitrary real constant (the wave amplitude),λ is an arbitrary real constant (ensuringλyis dimensionless),kis the wave...

work page
[35]

Unlike the first two families, this solution imposesno dispersion relation constraintω 2 =k 2c2

Pure Gauge Solution The parameters satisfy the following conditions: α1 =η ω 2gc , α 2 =η k 2g , α 3 =− λ 2g , α 5 =ηα 4, η=±1, α 4 ̸= 0, α 1, α 2 ̸= 0, λ∈R.(47) whereη=±1 is a discrete topological parity parameter,α 4 is an arbitrary real constant (the wave amplitude),λis an arbitrary real constant (ensuringλyis dimensionless),kis the wave number,ωis the...

work page
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C. N. Yang and R. L. Mills,Conservation of Isotopic Spin and Isotopic Gauge Invariance, Physical Review96, 191 (1954)

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[1] [1]

= 0, 2g (α2 2 −α 2 1)(λ+ 2gα 3) = 0, α5 k2 − ω2 c2 −4g 2(α2 1 −α 2 2) + 4gα4 ω c α1 −kα 2 = 0, α4 k2 − ω2 c2 −4g 2(α2 1 −α 2 2) + 4gα5 ω c α1 −kα 2 = 0, α2(λ+ 2gα 3)2 + 2gα4(2gα2α4 −kα 5) = 0, (λ+ 2gα 3)(4gα2α5 −kα 4) = 0, 4g2α2(α2 5 −α 2

work page

[2] [2]

Three Families of Exact Solutions.—Solving the sys- tem (6) of nine algebraic equations fork̸= 0, ω̸= 0 yields three families

= 0.(6) These are the master constraints for exact solutions. Three Families of Exact Solutions.—Solving the sys- tem (6) of nine algebraic equations fork̸= 0, ω̸= 0 yields three families. Family I: Linear (Abelian) Waves.—The param- eters are given by α1 =α 2 = 0, α 3 =− λ 2g , α 5 = 0, ω=kc,(7) andα 4 is free. The potentials reduce toφ= 0,A x = Az = 0,A...

work page

[3] [4]

’t Hooft,Magnetic monopoles in unified gauge theo- ries, Nucl

G. ’t Hooft,Magnetic monopoles in unified gauge theo- ries, Nucl. Phys. B79, 276 (1974)

work page 1974

[4] [5]

A. M. Polyakov,Compact gauge fields and the infrared catastrophe, Phys. Lett. B59,82-84 (1975)

work page 1975

[5] [6]

Julia and A

B. Julia and A. Zee,Poles with both magnetic and electric charges in non-Abelian gauge theory, Phys. Rev. D11, 2227 (1975)

work page 1975

[6] [7]

A. A. Belavin, A. M. Polyakov, A. S. Schwartz, and Yu. S. Tyupkin,Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. B59, 85 (1975)

work page 1975

[7] [8]

H. B. Nielsen and P. Olesen, Nucl. Phys. B61, 45 (1973)

work page 1973

[8] [9]

T. T. Wu and C. N. Yang,Some solutions of the classi- cal isotopic gauge field equations, inProperties of Matter Under Unusual Conditions, edited by H. Mark and S. Fernbach (Interscience, New York, 1969), pp. 349-354

work page 1969

[9] [10]

Coleman,Non-Abelian plane waves, Phys

S. Coleman,Non-Abelian plane waves, Phys. Lett. B70, 59-60 (1977)

work page 1977

[10] [11]

G¨ uven,Solutions for gravity coupled to non-Abelian plane waves, Phys

R. G¨ uven,Solutions for gravity coupled to non-Abelian plane waves, Phys. Rev. D19, 471-472 (1979)

work page 1979

[11] [12]

Basler and A

M. Basler and A. H¨ adicke,On non-Abelian SU(2) plane waves, Phys. Lett. B144, 83-86 (1984)

work page 1984

[12] [13]

C. H. Oh and R. Teh,Nonabelian progressive waves, J. Math. Phys.26, 841 (1985)

work page 1985

[13] [14]

C. H. Oh, C. H. Lai, and R. Teh,Color radiation in the classical Yang-Mills theory, Phys. Rev. D33, 1133 (1986)

work page 1986

[14] [15]

A. S. Rabinowitch,On a new class of non-Abelian ex- panding waves, Phys. Lett. B664, 295-300 (2008)

work page 2008

[15] [16]

A. S. Rabinowitch,On a new class of non-Abelian transverse progressive waves, Annals Phys.480, 170149 5 (2025)

work page 2025

[16] [17]

A. S. Rabinowitch,On a wide class of solutions of SU(2) Yang-Mills equations describing non-Abelian progressive waves, Nucl. Phys. B1022, 117236 (2026)

work page 2026

[17] [18]

Y. J. Lin, R. L. Compton, K. Jimenez-Garcia, J. V. Porto, and I. B. Spielman,Synthetic magnetic fields for ultracold neutral atoms, Nature462, 628 (2009)

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[18] [19]

Y. J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, J. V. Porto, and I. B. Spielman,Bose-Einstein condensate in a uniform light-induced vector potential, Phys. Rev. Lett.102, 130401 (2009)

work page 2009

[19] [20]

Goldman, G

N. Goldman, G. Juzeli¯ unas, P. ¨Ohberg, and I. B. Spiel- man,Light-induced gauge fields for ultracold atoms, Rep. Prog. Phys.77, 126401 (2014)

work page 2014

[20] [21]

Ozawa et al.,Topological photonics, Rev

T. Ozawa et al.,Topological photonics, Rev. Mod. Phys. 91, 015006 (2019)

work page 2019

[21] [22]

Actor,Classical solutions of SU(2) Yang-Mills theo- ries, Rev

A. Actor,Classical solutions of SU(2) Yang-Mills theo- ries, Rev. Mod. Phys.51, 461 (1979)

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[22] [23]

J. D. Jackson,Classical Electrodynamics: Third Edi- tion,(Wiley, New York 1999)

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[23] [24]

See Supplemental Material for the detailed derivation of the electric field, magnetic field and the nine algebraic constraints

work page

[24] [25]

Aidelsburger, M

M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch,Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices, Phys. Rev. Lett.111, 185301 (2013)

work page 2013

[25] [26]

Exact SU(2) Yang-Mills Waves from a Simple Ansatz

H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Bur- ton, and W. Ketterle,Realization of the Harper Hamilto- nian with laser-stirred Bose-Einstein condensates, Phys. Rev. Lett.111, 185302 (2013). Supplemental Material for “Exact SU(2) Yang-Mills Waves from a Simple Ansatz” Yu-Xuan Zhang 1 and Jing-Ling Chen 2,∗ 1School of Physics, Nankai University, Tia...

work page 2013

[26] [27]

Nonlinear Self-Interacting Waves 9

work page

[27] [28]

magnetic

Pure Gauge Solution 10 References 10 ∗Electronic address: chenjl@nankai.edu.cn 2 I. THE YANG-MILLS EQUATIONS The Yang-Mills equations in vacuum without sources (i.e., the 4-vector currentJ ν = 0) are given by [1] DµF µν ≡∂ µF µν + ig h Aµ, F µν i = 0,(1a) DµFνγ +D νFγµ +D γFµν = 0.(1b) Here the covariant derivativeD µ is defined as Dµ =∂ µ + igAµ,(µ= 0,1,...

work page

[28] [29]

Nine Algebraic Constraints from Eq

cos2 θ ) Σx⃗ ez = 0.(33) D. Nine Algebraic Constraints from Eq. (25) and Eq. (33) In this work, we focus on the nontrivial case ofg̸= 0. In the following, let us come to analyze Eq. (25) and Eq. (33). Firstly, we show that the case ofk= 0 andω= 0 is trivial. Remark4.The case ofk= 0 andω= 0 is trivial. In this case, one hasθ=kz−ωt= 0, cosθ= 1, sinθ= 0, and...

work page

[29] [30]

[λ+ 2g(α 3 +α 5)] = 0, α2 λ+ 2g(α 3 +α 5) 2 = 0,(36) which leads to (i) [λ+ 2g(α 3 +α 5)] = 0 and (ii){α 1 = 0, α2 = 0}. Accordingly, we have the electric-like field and the 8 magnetic-like field as ⃗E= nh (−λα1 −2gα 1α3) + ω c α4 −2gα 1α5 cosθ i Σy + − ω c α5 + 2gα1α4 sinθΣ z o ⃗ ey = [(−λα 1 −2gα 1α3) + (−2gα1α5)] Σy⃗ ey =−α 1 [λ+ 2g(α 3 +α 5)] Σy⃗ ey =...

work page

[30] [31]

= 0,(38c) 2g (α2 2 −α 2 1)(λ+ 2gα 3) = 0,(38d) α5 k2 − ω2 c2 −4g 2(α2 1 −α 2 2) + 4gα4 ω c α1 −kα 2 = 0,(38e) α4 k2 − ω2 c2 −4g 2(α2 1 −α 2 2) + 4gα5 ω c α1 −kα 2 = 0,(38f) α2(λ+ 2gα 3)2 + 2gα4(2gα2α4 −kα 5) = 0,(38g) (λ+ 2gα 3)(4gα2α5 −kα 4) = 0,(38h) 4g2α2(α2 5 −α 2

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[31] [32]

= 0.(38i) After solving these 9 algebraic equations, we can have three families of exact solutions (generallyλ̸= 0): (i) Linear waves, (ii) Nonlinear self-interacting waves, and (iii) Pure gauge solution

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[32] [33]

linear waves

Linear Waves The parameters satisfy the following conditions α1 = 0, α 2 = 0, α 3 =− λ 2g , α 5 = 0, ω=kc,(39) whereα 4 is an arbitrary real constant,λis an arbitrary real constant (ensuringλyis dimensionless),kis the wave number,ωis the frequency, andgis the gauge coupling constant. The phase is defined asθ=kz−ωt. The gauge potentials are φ= 0, A x = 0, ...

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[33] [34]

The phase is defined asθ=kz−ωt

Nonlinear Self-Interacting Waves The parameters satisfy the following conditions: α1 =α 2 = ηk 4g , α 3 =ξα 4 − λ 2g , α 5 =ηα 4, ω=kc, η=±1, ξ=±1, α 4 ̸= 0, λ∈R, ω >0.(42) whereη, ξ=±1 are discrete topological parity parameters,α 4 is an arbitrary real constant (the wave amplitude),λ is an arbitrary real constant (ensuringλyis dimensionless),kis the wave...

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[34] [35]

Unlike the first two families, this solution imposesno dispersion relation constraintω 2 =k 2c2

Pure Gauge Solution The parameters satisfy the following conditions: α1 =η ω 2gc , α 2 =η k 2g , α 3 =− λ 2g , α 5 =ηα 4, η=±1, α 4 ̸= 0, α 1, α 2 ̸= 0, λ∈R.(47) whereη=±1 is a discrete topological parity parameter,α 4 is an arbitrary real constant (the wave amplitude),λis an arbitrary real constant (ensuringλyis dimensionless),kis the wave number,ωis the...

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[35] [36]

C. N. Yang and R. L. Mills,Conservation of Isotopic Spin and Isotopic Gauge Invariance, Physical Review96, 191 (1954)

work page 1954

[36] [37]

J. D. Jackson,Classical Electrodynamics: Third Edition,(Wiley, New York 1999)

work page 1999