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arxiv: 2503.05410 · v4 · submitted 2025-03-07 · 🧮 math.AP

Decay of solutions of nonlinear Dirac equations

Sebastian Herr , Christopher Maul\'en , Claudio Mu\~noz This is my paper

Pith reviewed 2026-05-23 01:09 UTC · model grok-4.3

classification 🧮 math.AP
keywords nonlinear Dirac equationsdecay of solutionsvirial identitiesmassless casemassive casesolitary wavesbreatherslight cone
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The pith

Any globally defined solution of the one-dimensional massless nonlinear Dirac equation converges to zero inside a spatial region expanding at rate t over log squared t.

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

The paper proves long-time decay for solutions of nonlinear Dirac equations across several dimensions and mass regimes by building weighted virial identities that exploit the algebraic properties of the Dirac operator. In the one-dimensional massless case these identities imply that every global solution disperses to zero inside a region whose width grows like t divided by the square of the logarithm of t, with no restriction on initial-data size or nonlinearity power. The same approach yields decay on compact sets for small odd solutions in the massive case when the nonlinearity is holomorphic and odd, and extends to three dimensions under an H1 bound without requiring parity. In all dimensions the identities also give L2 decay outside the light cone. The results directly exclude the survival of non-dispersive localized structures such as breathers or standing waves in the regimes where the decay holds.

Core claim

By constructing weighted virial identities adapted to the Dirac algebra the authors show that, in the one-dimensional massless case, every globally defined solution tends to zero inside the region whose radius grows proportionally to t log to the minus two t; the statement requires neither smallness of the initial data nor any restriction on the power of the nonlinearity and therefore rules out standing breather-like or solitary-wave structures. Parallel decay statements are obtained for small odd solutions in the massive case under holomorphic odd nonlinearities, for bounded-H1 solutions in three dimensions without a parity assumption, and for the L2 norm in the exterior of the light cone.

What carries the argument

Weighted virial identities adapted to the Dirac algebra, which integrate localized energy quantities to produce time-decay estimates.

If this is right

  • No breather-like or solitary-wave structures persist in the one-dimensional massless regime.
  • Small odd solutions with holomorphic odd nonlinearities decay to zero on compact sets in the one-dimensional massive case.
  • Decay holds in three dimensions under H1 boundedness without any parity condition on the data.
  • The L2 norm decays in the exterior light-cone region in all dimensions, confirming that no solutions propagate faster than light.
  • The method supplies decay proofs for an important class of nonlinear Dirac models.

Where Pith is reading between the lines

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

  • If global existence can be established independently, the decay statements would give a complete description of long-time behavior for the massless equation.
  • The same virial identities might be modified to prove asymptotic stability of specific solitary waves in the massive regime.
  • The light-cone decay result indicates that the nonlinear Dirac flow respects relativistic causality even for large data.
  • Analogous identities could be tested on other first-order hyperbolic systems whose principal part carries a similar algebraic structure.

Load-bearing premise

Solutions are assumed to exist globally in time and the virial identities must close without extra smallness or power restrictions on the nonlinearity.

What would settle it

A concrete global solution in the one-dimensional massless case that stays bounded away from zero inside the region |x| less than c t over log squared t for some positive c and for arbitrarily large times would disprove the decay statement.

read the original abstract

We study the long-time behavior of small and large solutions to a broad class of nonlinear Dirac-type equations. Our results are classified in 1D massless and massive cases, 3D general and $n$ dimensional in generality. In the 1D massless case we prove that any globally defined solution converges to zero as time tends to infinity, within a spatial region expanding at a rate proportional to $ t \log^{-2} t$. This result holds without assumptions on the smallness of initial data or specific power of nonlinearity, ruling out the existence of standing breather-like or solitary wave structures in this regime. In the 1D massive case, solitary waves are known to exist. Introducing new virial identities adapted to the Dirac's distinctive algebra, we prove that there are ``holomorphic'' odd nonlinearities under which globally defined small odd solutions decay to zero on spatial compact sets as time tends to infinity. This result is extended to the 3D case under boundedness of the $H^1$ norm but without requiring the parity condition on the data, giving decay proofs for an important class of nonlinear Dirac models, and opening the door to the future use of virial identities to prove asymptotic stability of well-chosen Dirac solitary waves. Finally, in higher dimensions $ n \geq 1$, we prove the $L^2$ decay for global solutions of nonlinear Dirac equations in the ``exterior light-cone'' region. This confirms the non-existence of breathers and other solutions propagating faster than the speed of light. Our proofs rely on carefully constructed weighted virial identities.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript studies the long-time behavior of solutions to a broad class of nonlinear Dirac equations. In the 1D massless case, it proves that any globally defined solution decays to zero in a spatial region expanding at rate proportional to t (log t)^{-2}, without smallness assumptions on initial data or restrictions on the power of the nonlinearity. In the 1D massive case, new virial identities are introduced to show decay to zero on compact sets for globally defined small odd solutions when the nonlinearity is holomorphic and odd. This is extended to the 3D case under an a-priori H^1 bound (without parity). In n dimensions, L^2 decay is obtained in the exterior light-cone region. All proofs rely on carefully constructed weighted virial identities adapted to the Dirac algebra.

Significance. If the virial identities close without smallness or power restrictions as claimed, the results would be significant for ruling out breathers and superluminal structures in large-data regimes for nonlinear Dirac systems. The adaptation of virial methods to the distinctive matrix algebra of the Dirac operator is a methodological strength, and the explicit conditioning on global existence avoids overclaiming.

minor comments (3)
  1. [Abstract] Abstract: the expansion rate is written as 't log^{-2} t'; replace with the unambiguous form t (log t)^{-2} for clarity.
  2. [Theorem statements (throughout)] The statements of the main theorems should explicitly restate the global-existence hypothesis in the theorem hypotheses rather than only in the surrounding text.
  3. [Sections deriving the virial identities] Ensure that the precise form of the weighted virial multipliers and the resulting error terms are displayed in an equation environment with numbered labels for easy reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading, positive summary, and recommendation of minor revision. The assessment correctly captures the scope of our results on decay via adapted virial identities for nonlinear Dirac equations across dimensions, including the absence of smallness assumptions in the 1D massless case and the use of holomorphic odd nonlinearities in the massive case. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives decay statements from newly adapted weighted virial identities that close for a broad class of nonlinearities without smallness or power restrictions (conditional only on global existence). No derivation step reduces by construction to a fitted parameter, self-citation chain, or renamed input; the identities are presented as constructed for the Dirac algebra and rely on external functional-analysis tools. The central claims therefore remain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on abstract; no free parameters, invented entities, or ad-hoc axioms are mentioned. Relies on standard PDE existence and functional analysis background.

axioms (2)
  • domain assumption Global existence of solutions is given or assumed for the statements to apply
    Decay statements are conditioned on globally defined solutions (abstract).
  • standard math Standard Sobolev embeddings and multiplier techniques from dispersive PDE theory
    Implicit in construction of virial identities.

pith-pipeline@v0.9.0 · 5819 in / 1345 out tokens · 39494 ms · 2026-05-23T01:09:17.490877+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    Finally, in higher dimensions n ≥ 1, we prove the L² decay for global solutions of nonlinear Dirac equations in the 'exterior light-cone' region. This confirms the non-existence of breathers and other solutions propagating faster than the speed of light. Our proofs rely on carefully constructed weighted virial identities.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    In the 1D massless case we prove that any globally defined solution converges to zero as time tends to infinity, within a spatial region expanding at a rate proportional to t log^{-2} t.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

67 extracted references · 67 canonical work pages · 2 internal anchors

  1. [1]

    Aldunate, J

    D. Aldunate, J. Ricaud, E. Stockmeyer, H. Van Den Bosch,Results on the spectral stability of standing wave solutions of the Soler model in 1-D, Comm. Math. Phys., Vol. 401, pp. 227–273, (2023)

    work page 2023

  2. [2]

    M. Á. Alejo, A. J. Corcho,On the nonexistence of NLS breathers, accepted in Phys. D. arXiv:2408.09862

    work page arXiv

  3. [3]

    M. Á. Alejo, F. Cortez, C. Kwak and C. Muñoz,On the dynamics of zero-speed solutions for Camassa-Holm type equations, IMRN, rnz038,https://doi.org/10.1093/imrn/rnz038

    work page doi:10.1093/imrn/rnz038

  4. [4]

    M. Á. Alejo, L. Fanelli and C. Muñoz,Stability and instability of breathers in the U(1) Sasa– Satsuma and nonlinear Schrödinger models, Nonlinearity 34 3429 (2021)

    work page 2021

  5. [5]

    M. Á. Alejo, and C. Maulén,Decay for Skyrme wave maps, Lett. Math. Phys. 112 (2022), no. 5, Paper No. 90, 33 pp

    work page 2022

  6. [6]

    Balabane, T

    M. Balabane, T. Cazenave, A. Douady and F. Merle,Existence of excited states for a nonlinear Dirac field. Comm. Math. Phys. 119, 153-176 (1988)

    work page 1988

  7. [7]

    Bejenaru and S

    I. Bejenaru and S. Herr,The cubic Dirac equation: Small initial data inH1pR3q, Comm. Math. Phys. (2014), 1–40

    work page 2014

  8. [8]

    Berkolaiko and A

    G. Berkolaiko and A. Comech,On Spectral Stability of Solitary Waves of Nonlinear Dirac Equation in 1D, Math. Model. Nat. Phenom. Vol. 7, No. 2, 2012, pp. 13-31

    work page 2012

  9. [9]

    Borrelli, A

    W. Borrelli, A. Malchiodi, & R. Wu,Ground state Dirac bubbles and Killing spinors. Commun. Math. Phys. 383, 1151–1180 (2021)

    work page 2021

  10. [10]

    Bournaveas and T

    N. Bournaveas and T. Candy.Global well-posedness for the massless cubic Dirac equationIMRN (2016)

    work page 2016

  11. [11]

    Boussaïd and A

    N. Boussaïd and A. Comech,Nonlinear Dirac equations: Spectral stability of solitary waves, Math- ematical Surveys and Monographs, Vol. 244, American Mathematical Society, Providence, RI, 2019, vi+297 pp., ISBN 978-1-4704-4395-5

    work page 2019

  12. [12]

    Boussaïd and A

    N. Boussaïd and A. Comech,Spectral stability of bi-frequency solitary waves in Soler and Dirac– Klein–Gordon, 2018, Vol. 17, Issue 4: 1331-1347.https://doi.org/10.3934/cpaa.2018065

    work page doi:10.3934/cpaa.2018065 2018

  13. [13]

    Boussaïd, A

    N. Boussaïd, A. Comech and N. Kulkarni,On spectral stability of one- and bi-frequency solitary waves in Soler model in (3+1)D, arXiv:2412.21170

    work page arXiv

  14. [14]

    Boussaïd and S

    N. Boussaïd and S. Cuccagna,On Stability of Standing Waves of Nonlinear Dirac Equations, Comm. PDE, 37:6 (2012), 1001–1056,https://doi.org/10.1080/03605302.2012.665973

    work page doi:10.1080/03605302.2012.665973 2012

  15. [15]

    Boussaïd, P

    N. Boussaïd, P. D’Ancona, and L. Fanelli,Virial identity and weak dispersion for the magnetic Dirac equation, J. Math. Pures Appl. 95 (2011) 137–150

    work page 2011

  16. [16]

    M. B. Erdoğan, W. R. Green,On the one dimensional Dirac equation with potential, J. Math. Pures Appl. 151 (2021) 132–170

    work page 2021

  17. [17]

    Cacciafesta,Virial Identity and Dispersive estimates for the n-Dimensional Dirac Equation, J

    F. Cacciafesta,Virial Identity and Dispersive estimates for the n-Dimensional Dirac Equation, J. Math. Sci. Univ. Tokyo 18 (2011), 441–463

    work page 2011

  18. [18]

    Candy,Global existence for anL 2 critical nonlinear Dirac equation in one dimension.Adv

    T. Candy,Global existence for anL 2 critical nonlinear Dirac equation in one dimension.Adv. Diff. Eqns. 16 (7/8) 643 – 666, 2011.https://doi.org/10.57262/ade/1355703201

    work page doi:10.57262/ade/1355703201 2011

  19. [19]

    Candy and S

    T. Candy and S. Herr,The massless and the non-relativistic limit for the cubic Dirac equation, preprint arXiv:2308.12057 (2023)

    work page arXiv 2023

  20. [20]

    Cazenave and L

    T. Cazenave and L. Vazquez,Existence of localized solutions for a Classical Nonlinear Dirac Field. Comm. Math. Phys. 105, 35–47 (1986)

    work page 1986

  21. [21]

    Strichartz estimates for the Dirac equation on spherically symmetric spaces

    F. Cacciafesta, A.-S. de Suzzoni,Strichartz estimates for the dirac equation on spherically sym- metric spacesarXiv:1902.07572

    work page internal anchor Pith review Pith/arXiv arXiv 1902

  22. [22]

    Y. Cho, H. Seokchang, and L. KiyeonScattering and Nonscattering of the Hartree-Type Nonlinear Dirac System at Critical Regularity. SIAM J. Math. Anal. 55 (2023): 3395–3419

    work page 2023

  23. [23]

    Chugunova and D

    M. Chugunova and D. Pelinovsky,Block-Diagonalization of the Symmetric First-Order Coupled- Mode System. SIAM J. Applied Dyn. Syst. Vol. 5, Iss. 1 (2006)https://doi.org/10.1137/ 050629781

    work page 2006

  24. [24]

    Comech, T

    A. Comech, T. Van Phan and A. Stefanov,Asymptotic stability of solitary waves in generalized Gross-Neveu models, Ann. I. H. Poincaré Analyse Nonlinéaire 34 (2017) 157–196. 50 S. HERR, C. MAULÉN, AND C. MUÑOZ

    work page 2017

  25. [25]

    Standing Waves and Global Well-Posedness for the 2d Hartree Equa- tion with a Point Interaction

    A. Contreras, D. E. Pelinovsky and Y. Shimabukuro,L2 orbital stability of Dirac solitons in the massive Thirring model, Comm. PDE, 41:2 (2016), 227–255,https://doi.org/10.1080/03605302. 2015.1123272

    work page doi:10.1080/03605302 2016

  26. [26]

    Cossetti, L

    L. Cossetti, L. Fanelli, and D. Krejčiřík,Absence of eigenvalues of Dirac and Pauli Hamiltonians via the method of multipliers. Comm. Math. Phys. 379 (2020), no. 2, 633–691

    work page 2020

  27. [27]

    Cuevas-Maraver, N

    J. Cuevas-Maraver, N. Boussaïd, A. Comech, R. Lan, P.G. Kevrekidis, and A. Saxena (2018). Solitary Waves in the Nonlinear Dirac Equation. In: Carmona, V., Cuevas-Maraver, J., Fernández- Sánchez, F., García- Medina, E. (eds) Nonlinear Systems, Vol. 1. Understanding Complex Systems. Springer, Cham.https://doi.org/10.1007/978-3-319-66766-9_4

    work page doi:10.1007/978-3-319-66766-9_4 2018

  28. [28]

    D’Ancona, and L

    P. D’Ancona, and L. Fanelli,Decay estimates for the wave and Dirac equations with a magnetic potential. Comm. Pure Appl. Math. 60 (2007), no. 3, 357–392

    work page 2007

  29. [29]

    D’Ancona, and L

    P. D’Ancona, and L. Fanelli,Strichartz and smoothing estimates of dispersive equations with magnetic potentials. Comm. Partial Differential Equations 33 (2008), no. 4-6, 1082–1112

    work page 2008

  30. [30]

    D’Ancona, and L

    P. D’Ancona, and L. Fanelli, and N. M. Schiavone,Eigenvalue bounds for non-selfadjoint Dirac operators. Math. Ann. 383 (2022), no. 1-2, 621–644

    work page 2022

  31. [31]

    R. F. Dashen, B. Hasslacher, and A. Neveu,Semiclassical bound states in an asymptotically free theory, Phys. Rev. D 12, 2443 – 15 October, 1975,https://doi.org/10.1103/PhysRevD.12.2443

    work page doi:10.1103/physrevd.12.2443 1975

  32. [32]

    P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford University Press, USA, 1982

    work page 1982

  33. [33]

    D’Ancona, and O

    P. D’Ancona, and O. M. Okamoto.Blowup and ill-posedness results for a Dirac equation without gauge invariance, Evolution Equations and Control Theory 5 (2016): 225–234

    work page 2016

  34. [34]

    T. V. Dudnikova,Virial Identities and Energy–Momentum Relation for Solitary Waves of Non- linear Dirac Equations. Lobachevskii J Math 41, 956–981 (2020).https://doi.org/10.1134/ S1995080220060074

    work page 2020

  35. [35]

    Escobedo and L

    M. Escobedo and L. Vega,A semilinear Dirac equation inH spR3qforsą1. SIAM J. Math. Anal. 28(1997), no.2, 338–362

    work page 1997

  36. [36]

    M. J. Esteban and É. Séré,Stationary states of the nonlinear Dirac equation: a variational approach. Comm. Math. Phys. 171 (1995), no.2, 323–350

    work page 1995

  37. [37]

    Fefferman, J.P

    C.L. Fefferman, J.P. Lee-Thorp, and M.I. Weinstein,Edge States in Honeycomb Structures. Ann. PDE 2, 12 (2016)

    work page 2016

  38. [38]

    C. L. Fefferman, and M. I. Weinstein,Waves in honeycomb structures, Journees Equations aux derivees partialles, Biarretz, 3-7 juin 2012, GDR 243 4 (CNRS)

    work page 2012

  39. [39]

    Feng, F.F

    C. Fitzner and M. Thies,Breathers and their interaction in the massless Gross-Neveu model. Phys. Rev. D 87, 025001 – Published 2 January, 2013, DOI:https://doi.org/10.1103/PhysRevD. 87.025001

    work page doi:10.1103/physrevd 2013

  40. [40]

    Georgiev, and B

    V. Georgiev, and B. Shakarov,Global Large Data Solutions for 2D Dirac Equation with Hartree Type Interaction, IMRN, Volume 2022, Issue 17, August 2022, Pages 12803–12820,https://doi. org/10.1093/imrn/rnab082

    work page doi:10.1093/imrn/rnab082 2022

  41. [41]

    Hong,Scattering of cubic Dirac equations with a general class of Hartree-type nonlinearity for the critical Sobolev data, Preprint arXiv:2209.0023

    S. Hong,Scattering of cubic Dirac equations with a general class of Hartree-type nonlinearity for the critical Sobolev data, Preprint arXiv:2209.0023

    work page arXiv

  42. [42]

    H.Huh, andD.E.Pelinovsky,Nonexistence of self-similar blowup for the nonlinear Dirac equations in (1+1) dimensions, Appl. Math. Letters 92 (2019) 176-183

    work page 2019

  43. [43]

    E. A. Kopylova,Weighted energy decay for 1D Dirac equation.Dyn. Partial Diff. Eqns. 8 (2011), no. 2, 113–125

    work page 2011

  44. [44]

    Kowalczyk, Y

    M. Kowalczyk, Y. Martel, Y. & C. Muñoz,Nonexistence of small, odd breathers for a class of nonlinear wave equations. Lett. Math. Phys. 107, 921–931 (2017).https://doi.org/10.1007/ s11005-016-0930-y

    work page 2017

  45. [45]

    C. Kwak, C. Muñoz,Extended Decay Properties for Generalized BBM Equations, Nonlinear dis- persive partial differential equations and inverse scattering, 397–411, Fields Inst. Commun., 83, Springer, New York, 2019

    work page 2019

  46. [46]

    Machihara, K

    S. Machihara, K. Nakanishi; K. Tsugawa,Well-posedness for nonlinear Dirac equations in one dimension, Kyoto J. Math. 50 (2010), no. 2, 403–451

    work page 2010

  47. [47]

    Martel,Asymptotic stability of solitary waves for the 1D cubic-quintic Schrödinger equation with no internal mode

    Y. Martel,Asymptotic stability of solitary waves for the 1D cubic-quintic Schrödinger equation with no internal mode. Probability and Mathematical Physics Vol. 3 (2022), No. 4, 839–867 DOI: 10.2140/pmp.2022.3.839

    work page doi:10.2140/pmp.2022.3.839 2022

  48. [48]

    M. E. Martinez,Decay of small odd solutions for long range Schrödinger and Hartree equations in one dimension. Nonlinearity 33, 1156–1182 (2020). DECAY OF SOLUTIONS OF NONLINEAR DIRAC EQUATIONS 51

    work page 2020

  49. [49]

    Morales and C

    M. Morales and C. Muñoz,On local decay of inflaton and axion fields. Partial Differ. Equ. Appl. 5 (2024), no. 3, Paper No. 19, 30 pp

    work page 2024

  50. [50]

    Maulén and C

    C. Maulén and C. Muñoz,Decay in the one dimensional generalized Improved Boussinesq equation. Partial Diff. Eqns. Appl. 1, 1 (2020).https://doi.org/10.1007/s42985-019-0002-0

    work page doi:10.1007/s42985-019-0002-0 2020

  51. [51]

    Merle,Existence of stationary states for nonlinear Dirac equations

    F. Merle,Existence of stationary states for nonlinear Dirac equations. Journal of Differential Equations 74 (1988), 50–68. doi:10.1016/0022-0396(88)90018-6

    work page doi:10.1016/0022-0396(88)90018-6 1988

  52. [52]

    Muñoz, G

    C. Muñoz, G. Ponce, J.-C. Saut,On the long time behavior of solutions to the intermediate long wave equation, SIAM J. Math. Anal. Vol. 53, Iss. 1 (2021) 10.1137/19M1293181

    work page doi:10.1137/19m1293181 2021

  53. [53]

    Ozawa and K

    T. Ozawa and K. Yamauchi.Structure of Dirac matrices and invariants for nonlinear Dirac equa- tions. Diff. Int. Eqns. 17 (9-10) 971 - 982, 2004.https://doi.org/10.57262/die/1356060310

    work page doi:10.57262/die/1356060310 2004

  54. [54]

    D. E. Pelinovsky and A. Saalmann, Inverse scattering for the massive Thirring model, inNonlinear dispersive partial differential equations and inverse scattering, 497–528, Fields Inst. Commun., 83, Springer, New York, MR3931843

    work page

  55. [55]

    Pelinovsky, Y

    D. Pelinovsky, Y. Shimabukuro,Transverse Instability of Line Solitary Waves in Massive Dirac Equations. J Nonlinear Sci 26, 365–403 (2016)

    work page 2016

  56. [56]

    Pelinovsky and Y

    D.E. Pelinovsky and Y. Shimabukuro,Orbital Stability of Dirac Solitons. Lett. Math. Phys. 104, 21–41 (2014).https://doi.org/10.1007/s11005-013-0650-5

    work page doi:10.1007/s11005-013-0650-5 2014

  57. [57]

    D. E. Pelinovsky; A. Stefanov,Asymptotic stability of small gap solitons in nonlinear Dirac equa- tions, J. Math. Phys. 53, 073705 (2012)https://doi.org/10.1063/1.4731477

    work page doi:10.1063/1.4731477 2012

  58. [58]

    Pelinovsky,Survey on Global Existence in the Nonlinear Dirac Equations in One Spatial Dimension

    D.E. Pelinovsky,Survey on Global Existence in the Nonlinear Dirac Equations in One Spatial Dimension. Harmonic Analysis and Nonlinear Partial Differential Equations, pp. 37–50. RIMS Kkyroku Bessatsu, B26, Kyoto (2011)

    work page 2011

  59. [59]

    Long-time asymptotics for the Massive Thirring model

    A. Saalmann,Long-time asymptotics for the massive Thirring model, arXiv:1807.00623

    work page internal anchor Pith review Pith/arXiv arXiv

  60. [60]

    Selberg and A

    S. Selberg and A. Tesfahun,Low regularity well-posedness for some nonlinear Dirac equations in one space dimension, Diff. Int. Eqns. 23 (2010), no. 3-4, 265–278

    work page 2010

  61. [61]

    Soffer, and M

    A. Soffer, and M. Weinstein,Resonances, radiation damping and instability in Hamiltonian non- linear wave equations, Invent. Math. 136, 9–74 (1999).https://doi.org/10.1007/s002220050303

    work page doi:10.1007/s002220050303 1999

  62. [62]

    Soler,Classical, stable, nonlinear, spinor field with positive rest energy, Phys

    M. Soler,Classical, stable, nonlinear, spinor field with positive rest energy, Phys. Rev. D1(1970), pp. 2766–2769

    work page 1970

  63. [63]

    Strauss and L

    W. Strauss and L. Vázquez.Stability under dilations of nonlinear spinor fields. Phys. Rev. D (3), 34 (2):641–643, 1986

    work page 1986

  64. [64]

    Thaller,The Dirac Equation, Springer-Verlag Berlin Heidelberg 1992

    B. Thaller,The Dirac Equation, Springer-Verlag Berlin Heidelberg 1992

    work page 1992

  65. [65]

    W. E. Thirring,A soluble relativistic field theory, Ann. Phys., 3 (1958), pp. 91–112

    work page 1958

  66. [66]

    Tzvetkov.Existence of global solutions to nonlinear massless Dirac system and wave equation with small data.Tsukuba J

    N. Tzvetkov.Existence of global solutions to nonlinear massless Dirac system and wave equation with small data.Tsukuba J. Math. 22 (1) 193 - 211, June 1998.https://doi.org/10.21099/tkbjm/ 1496163480

    work page doi:10.21099/tkbjm/ 1998

  67. [67]

    35, Issue 6, June 1966, Pages 1117–1141,https://doi.org/10.1143/ PTP.35.1117

    M.Wakano,Intensely Localized Solutions of the Classical Dirac-Maxwell Field Equations, Progress of Theoretical Physics, Vol. 35, Issue 6, June 1966, Pages 1117–1141,https://doi.org/10.1143/ PTP.35.1117. F akultat für Mathematik, Universität Bielefeld, Postf ach 10 01 31, 33501 Bielefeld, Germany. Email address:herr@math.uni-bielefeld.de Email address:cmau...

    work page 1966