Relativistic k-fields with Massless Soliton Solutions in 3+1 Dimensions

M. Mohammadi; R. Gheisari

REVIEW 3 major objections 3 minor 89 references

Relativistic k-fields with Massless Soliton Solutions in 3+1 Dimensions

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A nonlinear field theory in 3+1 dimensions admits a single zero-energy soliton that is stable against deformation.

desk verdict A checkable zero-energy k-field construction in 3+1D, but the uniqueness claim is false: R=0 with a null ψ1 wave gives an infinite zero-energy family. read the letter →

arxiv 1908.10291 v6 pith:7RCQ2T45 submitted 2019-08-23 nlin.PS hep-th

M. Mohammadi , R. Gheisari This is my paper · ORCID

classification nlin.PShep-th MSC 35Q5135Q75

keywords k-fieldsnon-topologicalsolitonmasslesssolutionzerorest-massextendedKlein-Gordonsystemenergystabilitynon-standardLagrangiansolitarywave

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Every known relativistic scalar soliton carries a nonzero rest mass. This paper asks whether a classical field theory can support a zero-rest-mass soliton in 3+1 dimensions and proposes a construction that answers yes. The construction is a non-standard Lagrangian density, called a k-field model, whose kinetic terms enter nonlinearly; it is built so that a particular localized configuration makes the Lagrangian, the field equations, and the energy density all vanish. The zero-energy solution is $R=1/(1+r^2)$, $\theta=\pm\sqrt{2}\,t$, and $\psi_j=\pm x_j/(1+r^2)$, and the paper argues it is the unique such solution. Because every term in the energy density is nonnegative and any deformation creates a positive contribution, the solution is stable against arbitrary deformation and can be called a massless soliton.

What carries the argument

The load-bearing object is the engineered Lagrangian density $L=B\sum_{i=1}^{12}K_i^3$ in Eq. (19), where the $K_i$ are twelve independent linear combinations of twelve scalar quantities $L_i$ formed from the five fields through the kinetic scalars $S_{ij}=\partial_\mu\varphi_i\partial^\mu\varphi_j$. The cubic power is what makes a simultaneous zero of all $L_i$ a solution: at such a configuration $L$, its derivatives with respect to the fields, and the derived quantities entering the equations of motion all vanish, so the Euler-Lagrange equations and the zero-energy condition are automatic. The energy density then takes the form $B\sum_i K_i^2(3C_i-K_i)$ with the bracket positive at the massless solution, which is what turns the zero-energy configuration into an energy minimum.

What would settle it

Find a localized solution of the twelve equations $L_i=0$ (equivalently $K_i=0$) that is not related to (41) by the space-time symmetries of the theory. If one exists, the massless solution is not single, and the global energy-minimum argument loses its force; local stability could still hold.

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Extended reading notes

Core claim

The paper's central claim is that the extended Klein-Gordon system (19), a sum of cubes of twelve functionals $K_i$ built from five real scalar fields $R$, $\theta$, $\psi_1$, $\psi_2$, $\psi_3$, has a single non-topological massless solitary wave solution (41). The fields are $R=1/(1+r^2)$, $\theta=\pm\sqrt{2}\,t$, and $\psi_j=\pm x_j/(1+r^2)$ with $x_j=x,y,z$ and $r^2=x^2+y^2+z^2$. On this configuration all twelve scalars $L_i$ (equivalently all $K_i$) vanish, so by design the equations of motion and the energy density, $\varepsilon=B\sum_i K_i^2(3C_i-K_i)$, vanish identically; hence the lump has total energy zero. The paper further claims that any arbitrary deformation above this background makes at least one $K_i$ nonzero and therefore makes the total energy positive, so the massless solution is the energetic ground state of the model.

Load-bearing premise

The global stability conclusion rests on the assumption that the field configuration (41) is the only localized zero-energy solution; the paper verifies a list of candidate ansätze but does not prove exhaustiveness.

Editorial extensions

If this is right

Every solution of the new system other than the massless lump has a positive total energy, because at least one of the nonnegative squared terms in the energy density is nonzero.
A relativistic boost of the massless lump also has zero energy for any velocity, so the model contains a classical zero-rest-mass object that can be at rest or moving.
Because the lump is non-topological, widely separated copies can be superposed; at early times the combination is approximately a solution with negligible interaction energy.
When copies with different velocities are combined, the phase field $\theta$ must interpolate between the two vacuum phases in the region where $R$ and $\psi_j$ are nearly zero, which the paper argues leaves the energy zero to the same approximation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial extension: the paper's uniqueness claim is checked against a finite list of ansätze rather than proven. A systematic numerical or analytical search over all localized solutions of equations (48)-(50) would provide a decisive test.
Editorial extension: the design recipe can be generalized. Adding further scalar fields and extra independent $L_i$ constraints should produce other models with unique massless solitons of different shapes, turning the present construction into a template rather than a single example.
Editorial extension: if the model is taken as a classical description of a massless particle, its zero rest energy means a non-rigid lump would be slightly deformed by any interaction and would acquire a tiny positive energy; whether such an object can remain stable under self-interaction is a separate dynamical question.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here is my read. The construction is real: (41) is an explicit localized solution of (19) with zero energy, and the energy density is positive semidefinite because each term is K_i^2 times positive coefficients. The local variation argument (52) is fine, and the numerics for the listed ad hoc deformations behave as advertised. Credit where due: the paper is transparent about reverse-engineering the Lagrangian, and it makes no false claim about physical particles. The literature review is broad and mostly standard.

The central problem is the uniqueness claim, and the stress-test note lands. The statement after Eq. (24) that K_i=0 is equivalent to L_i=0 is not true in general: the twelve K_i are linear combinations of the L_i, but the coefficients are not all field-independent, and at R=0 the combination loses L2. Concretely, take R=0, ψ1=f(x−t) with f compactly supported, ψ2=ψ3=0, θ=0. Then L1=0, L4=0 because ψ1 is a null wave, L5=L6=L7=L8=L9=0, L10=L11=L12=0, while L2=−2. Every K_i vanishes because the only term containing L2 is multiplied by R^2. The equations of motion and ε=0 follow. So there is an infinite-dimensional family of localized zero-energy solutions; (41) is not single. The θ=0 can be any function, and R=ψ_j=0 is itself a degenerate zero-energy branch. This is not a nit about proof technique: the paper's own wording 'any arbitrary deformation increases total energy' is false in the strict sense, because deformations within the zero-energy family do not increase energy.

What remains true is modest: E≥0 and (41) has zero energy, so it is a global energy minimum in the weak sense of attaining the lower bound. The overdetermined-system heuristic ('twelve PDEs, five fields, so rare') does not prove uniqueness, and the example shows why. A revised version that explicitly characterizes the zero-energy manifold, or simply drops the uniqueness claim and states that (41) is a localized zero-energy solution with E≥0 for all configurations, would be credible.

Who is this for? Researchers working on k-field or nonstandard-Lagrangian solitons, especially as a cautionary example about reverse-engineered Lagrangians and degenerate vacuum branches. It deserves a serious referee: the algebra is checkable, the flaw is concrete and fixable, and the explicit solution is a legitimate construction even if the interpretation must be scaled back. I would not publish as is; with the claims weakened or the zero-energy family characterized, it could be a short paper. Send it to review rather than desk-reject, but expect major revision.

Referee Report

3 major / 3 minor

Summary. The manuscript constructs a k-field Lagrangian L = B Σ_{i=1}^{12} K_i^3 for five real scalar fields R, θ, ψ1, ψ2, ψ3, and proposes the explicit configuration (41) as a single massless non-topological solitary wave whose energy density vanishes everywhere. The authors argue that because the energy density is a sum of positive terms quadratic in the Ki's, and because the zero-energy solution is claimed to be unique, the configuration is energetically stable in the strong sense that any arbitrary deformation above its background increases the total energy.

Significance. The explicit solution (41) does satisfy the equations of motion by construction, and the local second-order positivity of the energy variation is straightforward and nicely demonstrated. If the uniqueness claim were correct, the paper would provide a curious example of a massless classical soliton in 3+1 dimensions. However, the uniqueness claim is not only unproven but false: there exists an infinite-dimensional family of exact zero-energy solutions that the paper overlooks. Consequently the central physical claim of a single massless soliton is invalid, and the global stability conclusion drawn from it does not follow.

major comments (3)

[§3, Eqs. (20)-(21) and the sentence after Eq. (24)] The statement that the conditions Ki = 0 are equivalent to Li = 0 is false because the linear transformation from the Li to the Ki has a field-dependent determinant that vanishes when R=0. For instance, K1 = R^2 L2, so K1 = 0 does not force L2 = 0 on configurations with R = 0. This invalidates the counting argument used to justify the uniqueness of the zero-energy solution.
[§3, around Eqs. (41)-(47)] The family R = 0, θ = 0 (or any constant), ψ1 = f(x−t), ψ2 = ψ3 = 0, with f any compactly supported function, satisfies all twelve conditions Ki = 0. Each term in the equations of motion (22)-(24) contains a factor Ki, so these equations are satisfied, and the energy density (25) is identically zero. This exact zero-energy solution is distinct from (41) and from all configurations listed in (42)-(47) and (53)-(63), so the 'single massless solution' assertion is false.
[§3, Eq. (52) and the concluding paragraph] The local positivity argument only establishes that the second variation of the energy is positive for infinitesimal perturbations around (41). It does not establish that 'any arbitrary deformation above its background' increases the energy, and the existence of the zero-energy family above shows that the claimed uniqueness, which is the premise for the global minimum statement, fails. The paper's own caveat that 'if one succeeds to find another massless solution' further indicates that uniqueness was not proven.

minor comments (3)

[Eq. (31)] Eq. (31) repeats K4 in the prefactor of ε5; it should presumably be K5.
[Eq. (25)] In Eq. (25), the phrase 'the energy-density belongs to the new extended Lagrangian-density (6)' should refer to Eq. (19).
[Fig. 2 and Eqs. (53)-(63)] The numerical plots in Fig. 2 only probe the specific ansatze (43) and (53)-(63), so they do not constitute a general check of stability; the claim of stability for arbitrary deformations requires a proof that is not supplied.

Circularity Check

2 steps flagged · score 4.0 of 10

The zero-energy solution is built into the Lagrangian by construction, and the claimed uniqueness rests on an equivalence that fails at R=0, leaving the paper's central stability claim partly circular and partly unsupported.

self definitional [Section 3, around Eqs. (19), (25), and after Eq. (41)]
"However, we build the new extended KG system (19) in such a way that there is a single massless solution exceptionally, for which Li = 0, as follows: ... these coupled equations are built deliberately in such a way that made Eq. (41) an exceptional static common solution; meaning that, we first consider Eq. (41), and then try to find the proper restrictive conditions Li = 0 (i = 1,··· , 12) to support it as an exceptional massless solution."

Because L = BΣK_i^3 and ε = BΣK_i^2[3C_i−K_i], any field configuration with all K_i = 0 automatically satisfies the equations of motion (22)-(24) and has ε = 0. The L_i (and hence K_i) were chosen after fixing the target profile (41), so the 'result' that (41) is a massless solution is a direct restatement of the construction, not an independent prediction. The nontrivial checks are limited to positivity of C_i and the absence of identically zero constraints; the existence of a zero-energy solution is essentially guaranteed by definition.
other [Section 3, between Eqs. (20)-(24) and after Eq. (41)]
"Note that, since Ki’s are twelve independent linear combination of twelve independent scalars Li’s, it is easy to understand that the conditions Ki = 0 are equivalent to Li = 0 (i = 1,··· , 12). ... In sum, it seems right that the special solution (41) is a single massless solution and we use this name in the rest of the paper."

The asserted equivalence K_i = 0 ⇔ L_i = 0 is false when R = 0. K_1 = R^2L_2, and all K_i containing L_2 are multiplied by R^2 or h_jR^2, so at R = 0 the K_i conditions do not force L_2 = 0. Explicitly, R = 0, ψ_1 = f(x−t), ψ_2 = ψ_3 = 0, θ = 0 gives L_2 = −2 but every K_i = 0, hence all equations of motion (22)-(24) hold and ε = 0. This is a zero-energy solution distinct from (41), so the uniqueness premise used to conclude that (41) is the global energy minimum is not derived; it is an assertion, and in fact it is false as stated. The load-bearing stability conclusion therefore rests on an unproved—and contradicted—uniqueness premise.

full rationale

This paper is an inverse-design model-building exercise: the Lagrangian (19) is deliberately constructed so that a preselected profile (41) makes all twelve K_i vanish, and the paper is transparent about this ('we first consider Eq. (41), and then try to find the proper restrictive conditions'). Consequently, the existence of a zero-energy solution is largely a tautology of the construction, which is the self-definitional component of the circularity. The genuinely independent content lies in the positivity calculation (all C_i > 0) and in verifying that the constraints are not identically zero. However, the paper's central claim that (41) is the single massless solution and hence that any arbitrary deformation increases the total energy depends on the equivalence K_i = 0 ⇔ L_i = 0 and on a finite list of candidate deformations. That equivalence fails at R = 0, where R = 0, ψ_1 = f(x−t), ψ_2 = ψ_3 = 0, θ = 0 gives all K_i = 0 and ε = 0 for a continuum of configurations not listed among the excluded families. Thus the uniqueness premise is unsupported and actually false. The self-citations to [66]-[70] provide background methodology but are not load-bearing enough by themselves to raise the score; the main problem is the construction-as-prediction for existence and the assumed uniqueness for stability. Overall, the paper is partially circular but not a fully tautological derivation, so a score of 4 is appropriate.

Assumptions & free parameters 1 free parameters · 3 assumptions · 1 invented entities

The model's existence result is essentially built in: the Lagrangian is constructed so that (41) solves the constraint system K_i=0. The only substantive mathematical content is the verification of that solution and the positivity of the energy functional. No new physical entities are predicted; the psi fields are bookkeeping devices. The arbitrary constant B scales the stability response but is not fitted. The uniqueness of the zero-energy solution is an unproven assumption that carries the global stability claim.

free parameters (1)

B = arbitrary positive number
Overall positive constant in the Lagrangian (19). It scales the energy of deformed configurations but does not affect the zero-energy solution. Chosen by hand, not fitted to data.

assumptions (3)

standard math The Euler-Lagrange equations from a Lagrangian density are the correct equations of motion.
Equations (22)-(24) are derived from the least-action principle applied to the Lagrangian (19).
domain assumption Positive definiteness of the energy density and a global minimum of total energy are the correct criteria for soliton stability.
This is the 'energetic stability' criterion adopted from the authors' previous work and refs. [65-70]. It is an assumption about how to judge stability, not a derived theorem.
ad hoc to paper The massless solution (41) is the unique solution of the system of conditions K_i=0 (i=1,...,12).
The global stability claim requires this uniqueness. The paper does not prove it; it only checks a finite list of candidate families and argues that twelve equations for five fields make common solutions rare.

invented entities (1)

Auxiliary scalar fields psi_1, psi_2, psi_3
purpose: Remove the degeneracy of zero-energy solutions that appears with only R and theta in 3+1 dimensions, allowing the desired configuration (41) to be the unique massless solution.
The fields have no physical origin, no couplings to standard matter, and no predicted observable. They are introduced solely to support a chosen ansatz and to make the energy density positive definite.

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Pith. "Pith review of Relativistic k-fields with Massless Soliton Solutions in 3+1 Dimensions." pith.science (2026). https://pith.science/paper/7RCQ2T45

@misc{pith2026190810291,
  author       = {Pith},
  title        = {Pith review of: Relativistic k-fields with Massless Soliton Solutions in 3+1 Dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7RCQ2T45}},
  note         = {Machine review of arXiv:1908.10291}
}

read the original abstract

In this work, the relativistic non-standard Lagrangian densities (k-fields) with massless solutions are generally introduced. Such solutions are not necessarily energetically stable. However, in 3+1 dimensions, we introduce a new k-field model that results in a single non-topological massless solitary wave solution. This special solution is energetically stable; that is, any arbitrary deformation above its background leads to an increase in the total energy. In other words, its energy is zero which is the least energy in all solutions. Hence, it can be called a massless soliton solution.

Figures

Figures reproduced from arXiv: 1908.10291 by the authors.

**Figure 2.** Plots. a-l are representing variations of the total energy [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

89 extracted references · 80 canonical work pages

[1]

Rajaraman, R. (1982). Solitons and instantons. North Holland, Elsevier, Amsterdam

1982
[2]

Das, A. (1989). Integrable models (Vol. 30). World scientiﬁc

1989
[3]

Lamb Jr, G. L. (1980). Elements of soliton theory . New York, Wiley-Interscience

1980
[4]

Manton, N., & Sutcliﬀe, P. (2004). Topological solitons. Cambridge University Press

2004
[5]

K., Peyrard, M., & Sodano, P

Campbell, D. K., Peyrard, M., & Sodano, P. (1986). Kink-antikink interactions in the double sine-Gordon equation . Physica D: Nonlinear Phenomena, 19(2), 165-205

1986
[6]

H., & Haberman, R

Goodman, R. H., & Haberman, R. (2005). Kink-Antikink Collisions in the φ4 Equa- tion: The n-Bounce Resonance and the Separatrix Map . SIAM Journal on Applied Dynamical Systems, 4(4), 1195-1228

2005
[7]

Riazi, N. (1993). Particle aspects of kink interactions . International journal of theo- retical physics, 32(11), 2155-2160

1993
[8]

R., Riazi, N., & Mohebbi, A

Gharaati, A. R., Riazi, N., & Mohebbi, A. F. (2006). Internal modes of relativistic solitons. International Journal of Theoretical Physics, 45(1), 53-63

2006

Show all 89 references

[9]

Mohammadi, M., & Riazi, N. (2011). Approaching integrability in bi-dimensional nonlinear ﬁeld equations. Progress of Theoretical Physics, 126(2), 237-248

2011
[10]

Mohammadi, M., & Riazi, N. (2019). The aﬀective factors on the uncertainty in the collisions of the soliton solutions of the double ﬁeld sine-Gordon system . Communi- cations in Nonlinear Science and Numerical Simulation, 72, 176-193

2019
[11]

Mohammadi, M., & Riazi, N. (2014). Bi-dimensional soliton-like solutions of the non- linear complex sine-Gordon system. Progress of Theoretical and experimental Physics, 2014(2), 023A03

2014
[12]

Hassanabadi, H., Lu, L., Maghsoodi, E., Liu, G., & Zarrinkamar, S. (2014). Scattering of Klein-Gordon particles by a Kink-like potential . Annals of Physics, 342, 264-269

2014
[13]

Alonso-Izquierdo, A., & Guilarte, J. M. (2012). On a family of ( 1 + 1)-dimensional scalar ﬁeld theory models: kinks, stability, one-loop mass shifts . Annals of Physics, 327(9), 2251-2274

2012
[14]

A., & Kudryavtsev, A

Gani, V. A., & Kudryavtsev, A. E. (1999). Kink–antikink interactions in the double sine-Gordon equation and the problem of resonance frequencies . Physical Review E, 60(3), 3305. 12

1999
[15]

Popov, C. A. (2005). Perturbation theory for the double sine-Gordon equation . Wave Motion, 42(4), 309-316

2005
[16]

Riazi, N., & Gharaati, A. R. (1998). Dynamics of sine-Gordon solitons . International journal of theoretical physics, 37(3), 1081-1120

1998
[17]

Mohammadi, M., Riazi, N., & Azizi, A. (2012). Radiative Properties of Kinks in the sin4(ϕ) System. Progress of theoretical physics, 128(4), 615-627

2012
[18]

Dorey, P., Mersh, K., Romanczukiewicz, T., & Shnir, Y. (2011). Kink-antikink colli- sions in the φ6 model. Physical review letters, 107(9), 091602

2011
[19]

A., Kudryavtsev, A

Gani, V. A., Kudryavtsev, A. E., & Lizunova, M. A. (2014). Kink interactions in the (1 + 1)-dimensional ϕ6 model. Physical Review D, 89(12), 125009

2014
[20]

C., & Saxena, A

Khare, A., Christov, I. C., & Saxena, A. (2014). Successive phase transitions and kink solutions in φ8, φ10, and φ12 ﬁeld theories. Physical Review E, 90(2), 023208

2014
[21]

M., Gani, V

Marjaneh, A. M., Gani, V. A., Saadatmand, D., Dmitriev, S. V., & Javidan, K. (2017). Multi-kink collisions in the φ6 model. Journal of High Energy Physics, 2017(7), 28

2017
[22]

A., Marjaneh, A

Gani, V. A., Marjaneh, A. M., & Saadatmand, D. (2019). Multi-kink scattering in the double sine-Gordon model. The European Physical Journal C, 79(7), 620

2019
[23]

Bazeia, D., Belendryasova, E., & Gani, V. A. (2018). Scattering of kinks of the sinh- deformedϕ4 model. The European Physical Journal C, 78(4), 340

2018
[24]

A., Marjaneh, A

Gani, V. A., Marjaneh, A. M., Askari, A., Belendryasova, E., & Saadatmand, D. (2018). Scattering of the double sine-Gordon kinks . The European Physical Journal C, 78(4), 345

2018
[25]

(2018).Resonant kink-antikink scattering through quasinormal modes

Dorey, P., & Roma´ nczukiewicz, T. (2018).Resonant kink-antikink scattering through quasinormal modes. Physics Letters B, 779, 117-123

2018
[26]

A., Lensky, V., & Lizunova, M

Gani, V. A., Lensky, V., & Lizunova, M. A. (2015). Kink excitation spectra in the (1+ 1)-dimensional ϕ8 model. Journal of High Energy Physics, 2015(8), 147

2015
[27]

V., & Javidan, K

Askari, A., Saadatmand, D., Dmitriev, S. V., & Javidan, K. (2018). High energy den- sity spots and production of kink–antikink pairs in particle collisions . Wave Motion, 78, 54-59

2018
[28]

Morris, J. R. (2018). Small deformations of kinks and walls . Annals of Physics, 393, 122-131

2018
[29]

Morris, J. R. (2019). Interacting kinks and meson mixing . Annals of Physics, 400, 346-365

2019
[30]

Skyrme, T. H. R. (1961). A non-linear ﬁeld theory . Proceedings of the Royal Society A. 260, 127–138

1961
[31]

Skyrme, T. H. R. (1962). A uniﬁed ﬁeld theory of mesons and baryons . Nuclear Physics, 31, 556-569

1962
[32]

S., Schroers, B

Manton, N. S., Schroers, B. J., & Singer, M. A. (2004). The interaction energy of well-separated Skyrme solitons . Communications in mathematical physics, 245(1), 123-147

2004
[33]

Manton, N. S. (1987). Geometry of skyrmions . Communications in Mathematical Physics, 111(3), 469-478. 13

1987
[34]

Li, T. Z. (1988). Are there other types of skyrmions? . International journal of theo- retical physics, 27(3), 307-313

1988
[35]

Li, T. Z. (1991). Skyrmions of diﬀerent types . International journal of theoretical physics, 30(6), 829-836

1991
[36]

’t Hooft, G. (1974). Magnetic monopoles in uniﬁed theories . Nucl. Phys. B, 79 (CERN-TH-1876), 276-284

1974
[37]

Polyakov, A. M. (1974). Spectrum of particles in quantum ﬁeld theory . JETP Lett, 20, 430-433

1974
[38]

Prasad, M. K. (1980). Instantons and monopoles in Yang-Mills gauge ﬁeld theories . Physica D: Nonlinear Phenomena, 1(2), 167-191

1980
[39]

Nishino, S., Matsudo, R., Warschinke, M., & Kondo, K. I. (2018).Magnetic monopoles in pure Yang-Mills theory with a gauge-invariant mass . Progress of Theoretical and Experimental Physics, 2018(10), 103B04

2018
[40]

Eto, M., Hirono, Y., Nitta, M., & Yasui, S. (2014). Vortices and other topological solitons in dense quark matter . Progress of Theoretical and Experimental Physics, 2014(1), 012D01

2014
[41]

Wazwaz, A. M. (2004). The tanh method for traveling wave solutions of nonlinear equations. Applied Mathematics and Computation, 154(3), 713-723

2004
[42]

G., & Smolyakov, M

Panin, A. G., & Smolyakov, M. N. (2017). Problem with classical stability of U(1) gauged Q-balls. Physical Review D, 95(6), 065006

2017
[43]

Kovtun, A., Nugaev, E., & Shkerin, A. (2018). Vibrational modes of Q-balls. Physical Review D, 98(9), 096016

2018
[44]

Smolyakov, M. N. (2018). Perturbations against a Q-ball: Charge, energy, and addi- tivity property. Physical Review D, 97(4), 045011

2018
[45]

I., Copeland, E

Tsumagari, M. I., Copeland, E. J., & Saﬃn, P. M. (2008). Some stationary properties of a Q-ball in arbitrary space dimensions . Physical Review D, 78(6), 065021

2008
[46]

D., & Pang, Y

Lee, T. D., & Pang, Y. (1992). Nontopological solitons. Physics Reports, 221(5-6), 251-350

1992
[47]

Coleman, S. (1985). Q-balls. Nuclear Physics B, 262(2), 263-283

1985
[48]

A., & Menezes, R

Bazeia, D., Marques, M. A., & Menezes, R. (2016). Exact solutions, energy, and charge of stable Q-balls . The European Physical Journal C, 76(5), 241

2016
[49]

A., & Menezes, R

Bazeia, D., Losano, L., Marques, M. A., & Menezes, R. (2017). Split Q-balls. Physics Letters B, 765, 359-364

2017
[50]

N., Axenides, M., Floratos, E

Anagnostopoulos, K. N., Axenides, M., Floratos, E. G., & Tetradis, N. (2001). Large gauged Q balls . Physical Review D, 64(12), 125006

2001
[51]

Axenides, M., Komineas, S., Perivolaropoulos, L., & Floratos, M. (2000). Dynamics of nontopological solitons: Q balls . Physical Review D, 61(8), 085006

2000
[52]

Bowcock, P., Foster, D., & Sutcliﬀe, P. (2009). Q-balls, integrability and duality . Journal of Physics A: Mathematical and Theoretical, 42(8), 085403

2009
[53]

Shiromizu, T., Uesugi, T., & Aoki, M. (1999). Perturbation analysis of deformed Q-balls and primordial magnetic ﬁeld . Physical Review D, 59(12), 125010. 14

1999
[54]

Shiromizu, T. (1998). Generation of a magnetic ﬁeld due to excited Q-balls . Physical Review D, 58(10), 107301

1998
[55]

Ishihara, H., & Ogawa, T. (2019). Charge-screened nontopological solitons in a sponta- neously brokenU(1) gauge theory. Progress of Theoretical and Experimental Physics, 2019(2), 021B01

2019
[56]

A., Menezes, R., & da Rocha, R

Bazeia, D., Losano, L., Marques, M. A., Menezes, R., & da Rocha, R. (2016).Compact Q-balls. Physics Letters B, 758, 146-151

2016
[57]

P., & Rybakov, Y

Kumar, A., Nisichenko, V. P., & Rybakov, Y. P. (1979). Stability of charged solitons . International Journal of Theoretical Physics, 18(6), 425-432

1979
[58]

Grillakis, M. (1988). Linearized instability for nonlinear schr¨ odinger and klein-gordon equations. Communications on pure and applied mathematics, 41(6), 747-774

1988
[59]

Grillakis, M., Shatah, J., & Strauss, W. (1987). Stability theory of solitary waves in the presence of symmetry, I . Journal of functional analysis, 74(1), 160-197

1987
[60]

D., & Sirlin, A

Friedberg, R., Lee, T. D., & Sirlin, A. (1976). Class of scalar-ﬁeld soliton solutions in three space dimensions. Physical Review D, 13(10), 2739

1976
[61]

Smolyakov, M. N. (2019). Perturbations against a Q-ball. II. Contribution of nonoscil- lation modes. Physical Review D, 100(4), 045002

2019
[62]

G., & Kolokolov, A

Vakhitov, N. G., & Kolokolov, A. A. (1973). Stationary solutions of the wave equation in the medium with nonlinearity saturation . Radiophysics and Quantum Electronics, 16(7), 783-789

1973
[63]

Kolokolov, A. A. (1973). Stability of the dominant mode of the nonlinear wave equation in a cubic medium . Journal of Applied Mechanics and Technical Physics, 14(3), 426- 428

1973
[64]

Cordero, R., & Mota, R. D. (2004). Soliton stability in a generalized sine-Gordon potential. International Journal of Theoretical Physics, 43(11), 2215-2222

2004
[65]

Derrick, G. H. (1964). Comments on nonlinear wave equations as models for elemen- tary particles. Journal of Mathematical Physics, 5(9), 1252-1254

1964
[66]

Mohammadi, M., & Gheisari, R. (2019). Zero rest mass soliton solutions . Physica Scripta, 95(1), 015301

2019
[67]

Mohammadi, M. (2019). The role of the massless phantom term in the stability of a non-topological soliton solution. Iranian Journal of Science and Technology, Transac- tions A: Science, 43(5), 2627-2634

2019
[68]

Mohammadi, M. (2020). Introducing a relativistic nonlinear ﬁeld system with a single stable non-topological soliton solution in 1 + 1 dimensions. Annals of Physics, 414, 168099

2020
[69]

Mohammadi, M. (2020). An energetically stable Q-ball solution in 3+ 1 dimensions . Physica Scripta, 95(4), 045302

2020
[70]

Mohammadi, M. (2020). Stability catalyzer for a relativistic non-topological soliton solution. Annals of Physics, 168304

2020
[71]

Saha, A., & Talukdar, B. (2014). Inverse variational problem for nonstandard La- grangians. Reports on Mathematical Physics, 73(3), 299-309. 15

2014
[72]

Musielak, Z. E. (2008). Standard and non-standard Lagrangians for dissipative dy- namical systems with variable coeﬃcients . Journal of Physics A: Mathematical and Theoretical, 41(5), 055205

2008
[73]

El-Nabulsi, A. R. (2013). Non-linear dynamics with non-standard Lagrangians. Qual- itative theory of dynamical systems, 12(2), 273-291

2013
[74]

El-Nabulsi, R. A. (2015). Classical string ﬁeld mechanics with non-standard La- grangians. Mathematical Sciences, 9(3), 173-179

2015
[75]

El-Nabulsi, R. A. (2014). Nonlinear integro-diﬀerential Einstein’s ﬁeld equations from nonstandard Lagrangians. Canadian Journal of Physics, 92(10), 1149-1153

2014
[76]

El-Nabulsi, R. A. (2013). Generalizations of the Klein-Gordon and the Dirac equations from non-standard Lagrangians . Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 83(4), 383-387

2013
[77]

Bazeia, D., Losano, L., Menezes, R., & Oliveira, J. C. R. E. (2007). Generalized global defect solutions. The European Physical Journal C, 51(4), 953-962

2007
[78]

(2007).k-defects as compactons

Adam, C., Sanchez-Guillen, J., & Wereszczy´ nski, A. (2007).k-defects as compactons. Journal of Physics A: Mathematical and Theoretical, 40(45), 13625

2007
[79]

Babichev, E. (2006). Global topological k-defects. Physical Review D, 74(8), 085004

2006
[80]

Vasheghani, A., & Riazi, N. (1996). Isovector solitons and Maxwell’s equations . In- ternational Journal of Theoretical Physics, 35(3), 587-591

1996
[81]

H., & Riazi, N

Mahzoon, M. H., & Riazi, N. (2007). Nonlinear electrodynamics and NED-inspired chiral solitons. International Journal of Theoretical Physics, 46(4), 823-831

2007
[82]

Armend´ariz-Pic´on, C., Damour, T., & Mukhanov, V. I. (1999). k-Inﬂation. Physics Letters B, 458(2-3), 209-218

1999
[83]

Chiba, T., Okabe, T., & Yamaguchi, M. (2000). Kinetically driven quintessence . Physical Review D, 62(2), 023511

2000
[84]

Armendariz-Picon, C., Mukhanov, V., & Steinhardt, P. J. (2000). Dynamical solution to the problem of a small cosmological constant and late-time cosmic acceleration . Physical Review Letters, 85(21), 4438

2000
[85]

Armendariz-Picon, C., & Lim, E. A. (2005). Haloes of k-essence. Journal of Cosmol- ogy and Astroparticle Physics, 2005(08), 007

2005
[86]

D., & Milosevic, M

Dimitrijevic, D. D., & Milosevic, M. (2012, August). About non standard Lagrangians in cosmology. In AIP Conference Proceedings (Vol. 1472, No. 1, pp. 41-46). American Institute of Physics

2012
[87]

Padmanabhan, T., & Choudhury, T. R. (2002). Can the clustered dark matter and the smooth dark energy arise from the same scalar ﬁeld? . Physical Review D, 66(8), 081301

2002
[88]

Renaux-Petel, S., & Tasinato, G. (2009). Nonlinear perturbations of cosmological scalar ﬁelds with non-standard kinetic terms . Journal of Cosmology and Astroparticle Physics, 2009(01), 012

2009
[89]

El-Nabulsi, R. A. (2013). Nonstandard Lagrangian cosmology. Journal of Theoretical and Applied Physics, 7(1), 58. 16

2013

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Rajaraman, R. (1982). Solitons and instantons. North Holland, Elsevier, Amsterdam

1982

[2] [2]

Das, A. (1989). Integrable models (Vol. 30). World scientiﬁc

1989

[3] [3]

Lamb Jr, G. L. (1980). Elements of soliton theory . New York, Wiley-Interscience

1980

[4] [4]

Manton, N., & Sutcliﬀe, P. (2004). Topological solitons. Cambridge University Press

2004

[5] [5]

K., Peyrard, M., & Sodano, P

Campbell, D. K., Peyrard, M., & Sodano, P. (1986). Kink-antikink interactions in the double sine-Gordon equation . Physica D: Nonlinear Phenomena, 19(2), 165-205

1986

[6] [6]

H., & Haberman, R

Goodman, R. H., & Haberman, R. (2005). Kink-Antikink Collisions in the φ4 Equa- tion: The n-Bounce Resonance and the Separatrix Map . SIAM Journal on Applied Dynamical Systems, 4(4), 1195-1228

2005

[7] [7]

Riazi, N. (1993). Particle aspects of kink interactions . International journal of theo- retical physics, 32(11), 2155-2160

1993

[8] [8]

R., Riazi, N., & Mohebbi, A

Gharaati, A. R., Riazi, N., & Mohebbi, A. F. (2006). Internal modes of relativistic solitons. International Journal of Theoretical Physics, 45(1), 53-63

2006