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Optimal convergence rates of the Klein-Gordon-Schr\"odinger system in the nonrelativistic limit

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Pith's one-line read The Klein-Gordon–Schrödinger system converges to two decoupled Schrödinger equations at the optimal O(ε²) rate.

desk verdict New long-time O(ε²) bounds for the Klein-Gordon–Schrödinger system are real; the 'optimal' claim is a heuristic overreach that should be qualified or proven. read the letter →

arxiv 2607.28963 v1 pith:AOLJ5QA7 submitted 2026-07-31 math.AP

Weizhu Bao , Yong Lu , Zhiwei Zheng This is my paper · ORCID
classification math.AP MSC 35B4035Q55
keywords Klein-Gordon-SchrödingersystemnonrelativisticlimitoptimalconvergencerateWKBexpansiongeometricopticsdispersivedecaylongtimeestimatesSchrödingerapproximation
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

The paper proves that, in the nonrelativistic limit ε→0, the coupled Klein-Gordon–Schrödinger system is approximated, on a long time interval of order 1/ε, by two decoupled linear Schrödinger equations, with errors proportional to ε² (growing at most linearly in time). With more regular initial data the validity extends to times of order ε^{-4/3}, and the error in the Schrödinger component becomes (1+εt+ε²t²)ε². These rates are shown to be optimal: the ε² order coincides with the unavoidable initial error, and adding higher-order WKB terms cannot improve the final rate. The result rigorously justifies numerical observations that had suggested exactly these error forms.

What carries the argument

The argument runs through a WKB (geometric-optics) expansion of the symmetric-hyperbolic reformulation of the system, with phase θ=µt/ε². The load-bearing components are the Schrödinger profiles ϕℓ and ψℓ and, at higher order, the explicit correctors ϕc(t)=−it/(8µ³)Δ²ϕℓ(t) and ψc(t)=∫₀ᵗ e^{i(t−s)Δ}(iλ²/µ²)|ψℓ|²ψℓ(s) ds. The proof exploits the dispersive decay estimate ∥f(t)∥_{W^{σ,∞}} ≤ C t^{-d/2}∥f₀∥_{W^{σ,1}} for the linear Schrödinger evolution, which controls the nonlinear differences, and then closes a bootstrap/Grönwall estimate for the perturbation ˙u=(Φ−Φa, ψ−ψa). The WKB approximate solution is constructed so that its residual and initial error are O(ε²) (or O(ε³+ε⁴t) at higher orde

What would settle it

Run a numerical simulation of the Klein-Gordon–Schrödinger system in dimension d≥3 with initial data in H^{σ+4}∩W^{σ,1}, and measure ∥ψ(t)−ψℓ(t)∥_{H^σ} at t=T₁/ε; if the error is not bounded by C ε² or grows faster than (1+t)ε², the theorem's bound is false. Conversely, a simulation in d=1 or d=2 showing the same O(ε²) rate would indicate the dispersive-decay hypothesis is not necessary.

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Extended reading notes

Core claim

The central claim is that the exact solution (ϕ,ψ) of the Klein-Gordon–Schrödinger system with well-prepared initial data can be written, up to an H^σ error of order ε², as the leading WKB profiles e^{iµt/ε²}ϕℓ(t)+c.c. for µϕ and ψℓ(t) for ψ, where ϕℓ and ψℓ solve the free Schrödinger equations in (1.5). Theorem 1.3 establishes that, for d≥3 and initial data in H^{σ+4}∩W^{σ,1}, ∥µϕ(t)−(e^{iµt/ε²}ϕℓ(t)+e^{−iµt/ε²}ϕℓ(t))∥_{H^σ} + ∥ψ(t)−ψℓ(t)∥_{H^σ} ≤ C(1+t)ε² for 0≤t≤T₁/ε. Under higher regularity, Theorem 1.4 gives the refined bounds ∥µϕ(t)−(… )∥_{H^σ} ≤ C(1+t+ε²t²)ε² and ∥ψ(t)−ψℓ(t)∥_{H^σ} ≤ C(1+εt+ε²t²)ε² on the longer interval 0≤t≤T₂/ε^{4/3}, and on the 1/ε interval the ψ error is uniformly

Load-bearing premise

The proof rests on the dispersive-decay bound for the limiting Schrödinger profiles, which requires the initial data to lie in a W^{σ,1}-based space and the spatial dimension to be at least three; if that decay is unavailable, the bootstrap interval and the ε² rate would not follow.

Editorial extensions

If this is right

  • For initial data in the stated H^σ∩W^{σ,1} spaces in dimension d≥3, the nonrelativistic limit is rigorously justified on the natural 1/ε time scale, with the exact solution staying within O(ε²) of the two decoupled Schrödinger profiles.
  • Over the longer ε^{-4/3} time scale allowed by higher regularity, the Schrödinger component ψ attains a uniform O(ε²) bound on the 1/ε subinterval, matching the numerical observation that ψ is better approximated than the Klein-Gordon component ϕ.
  • Because the ε² order is fixed by the initial error and by the unavoidable presence of ε²-order WKB terms, no higher-order expansion can beat the O(ε²) rate; the result is optimal.
  • The specific error forms (1+t)ε² for µϕ and ε² for ψ explain the numerically observed asymmetry between the two components on the long time scale.
  • The reformulation as stability of WKB approximate solutions in nonlinear geometric optics provides a template for proving optimal nonrelativistic-limit rates in other singularly perturbed dispersive systems.

Reading between the lines

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

  • The proof's dependence on the L¹-based dispersive-decay estimate ties the optimal-rate statement to dimension d≥3; in d=1 or d=2 the same bootstrap would not close, so the optimality should not be assumed to carry over without additional hypotheses or a different mechanism.
  • A natural testable extension is to replace the free Schrödinger limit profiles by nonlinear Schrödinger profiles (adding the cubic term to ψℓ); the WKB construction suggests the same ε² barrier would appear, meaning the optimality argument would have to be revisited for a nonlinear limit system.
  • The explicit correctors ϕc and ψc identify the leading sub-ε² corrections; numerical experiments could directly check that the dominant error after subtracting the leading profiles is described by these correctors, providing a sharper verification than the H^σ norm bound alone.
  • The same WKB-plus-bootstrap framework, with a suitable phase and corrected profiles, likely yields analogous optimal rates for related systems such as Klein-Gordon-Zakharov or Maxwell-Klein-Gordon in their nonrelativistic limits, though each system would need its own phase analysis.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 6 minor

Summary. The paper studies the nonrelativistic limit ε→0 of the Klein-Gordon–Schrödinger system (1.1)–(1.2) in dimensions d≥3. It constructs WKB approximate solutions whose leading profiles solve two free Schrödinger equations (1.5), and proves by Duhamel/bootstrap estimates that the exact solution stays within C(1+t)ε² in H^σ on time scales of order ε^{-1} (Theorem 1.3). Under additional regularity it proves the longer time scale ε^{-4/3} with bounds C(1+εt+ε²t²)ε², yielding a uniform O(ε²) bound for the Schrödinger component on the ε^{-1} interval (Theorem 1.4 and Corollary 1.5). The title and abstract further claim that the O(ε²) rates are optimal.

Significance. If read as an upper-bound paper, this is a solid contribution. The WKB construction in §2 and the bootstrap argument in §3 are coherent and the stated bounds follow from them; the proof is self-contained modulo standard dispersive estimates and the Kato–Ponce inequality, with no fitted parameters and no circular dependence on the numerical results of [6], which serve only as motivation. The results rigorously justify the numerically observed rates and improve the existing long-time convergence literature for this system. The central weakness is the advertised optimality: the paper proves only upper bounds, and the argument offered in §1.3 shows only that the WKB approximate solution is not closer to the leading profile; it does not show that the exact solution cannot be closer. This is a load-bearing overstatement in the title and abstract, but it is local and fixable by either proving a lower bound or qualifying the claim.

major comments (1)
  1. [Abstract; §1.3, final paragraph; Remark 2.1] The paper claims the O(ε²) rates are 'optimal,' but Theorems 1.3 and 1.4 establish upper bounds only. The argument in the final paragraph of §1.3 — that the WKB expansion contains O(ε²) components and hence higher-order WKB cannot improve the rate — concerns the distance between the WKB approximate solution and the leading profile, not the distance between the exact solution and the leading profile. The exact solution could in principle have a different O(ε²) correction that cancels these terms. No lower bound such as ‖error‖ ≥ c ε² for some t is proved. Moreover, Remark 2.1 explicitly states that the initial error can be reduced to O(ε³), so the abstract's inference 'coincide with the order of initial error, and thus are optimal' is not valid. I recommend either proving a lower bound for a natural class of data or replacing 'optimal' by a qualified statement such as 'optimal among WKB l
minor comments (6)
  1. [§1.2, notation] The symbol T1 is used both for the dimensionless constant T1 = 1/(2M1) and for the time scale T1/ε. This makes the bootstrap definition and the statement of Theorem 1.3 confusing. Suggest using, e.g., τ1 for the constant and T1/ε for the time horizon.
  2. [§2.3, initial error formula] The displayed formula for rΦ reads rΦ = −ε²(0, µ∂tϕℓ, λ/µ |ψℓ|²)^T. This appears dimensionally inconsistent: from the ansatz (2.27), the ε² term in the second component is ε²/µ(∂tϕℓ e^{iθ} + c.c.), so the formula should contain (∂tϕℓ + ∂tϕℓ)/µ (or the real part) rather than µ∂tϕℓ. The estimate ‖rΦ‖ ≤ Cε² is unaffected, but the displayed formula should be corrected.
  3. [§2.2, order n=2] The notation [F(ψa,ψa)]_2^{(2)} is ambiguous: the displayed computation contains e^{±iθ} terms but then writes '= 2λ Re(ψℓψc)', silently projecting onto the zero mode. Please clarify that the p=0 mode is being selected here and that the nonzero modes are treated separately.
  4. [§3.3, bootstrap exponent] In the display following (3.22), the term '1/2 T2 ε^{4/3}' should read '1/2 T2^2 ε^{4/3}'. The subsequent bound is unaffected, but the typo makes the verification harder.
  5. [§3.3, first line] The sentence 'The proof is based on the energy inequality energy-expan' contains a broken cross-reference; it should refer to (3.6). Also, 'bootstarp' in the same section is a typo.
  6. [General] Minor typographical issues: 'the the approximate system' in §2.4, 'compoents' in §2.3, and in the bibliography [35] 'Ann. Mat. Pura Appl.4198' should be 'Ann. Mat. Pura Appl. 198'. These do not affect the mathematics.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity in the convergence proof; the 'optimality' claim is heuristic but not circular.

full rationale

The proof of Theorems 1.3 and 1.4 is self-contained and does not reduce any prediction to its inputs. The WKB profiles are constructed in Section 2 by imposing vanishing-residual conditions (2.1) and solving for the profiles from the leading Schrödinger equations; the residuals and initial errors are then estimated directly in Propositions 1.6 and 1.7. The stability analysis in Section 3 is a standard bootstrap/Duhamel argument using the external dispersive-decay lemma (Lemma 1.1, from [9]) and the Kato–Ponce inequality, closing via inequalities (3.12) and (3.22). No parameter is fitted to numerical data, and no predicted quantity is used to define an input. The numerical results [6] are cited only as motivation and comparison, not as evidence in the proof. Author-overlapping citations ([21,22] for the symmetric hyperbolic reformulation, [5,23] in a remark, [6] for numerics) are methodological or contextual and are not load-bearing for the main error estimates. The paper's claim that the O(ε²) rates are 'optimal' is not established: the final paragraph of §1.3 only shows that the WKB approximate solution contains O(ε²) components, which does not rule out cancellations making the exact error smaller; no lower bound is proved. This is a rigor/correctness limitation, not a circularity, and is flagged here as such.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted and no new entities are postulated. The result is a theorem under explicit regularity and integrability assumptions; all constants depend only on initial-data norms and (μ,λ). The only non-standard inputs are the WKB phase ansatz and the W^{σ,1} decay assumptions, which are standard tools in this literature.

assumptions (6)
  • standard math Sobolev embedding and algebra property H^σ ⊂ L∞ for σ>d/2
    Used throughout §2–3 to control nonlinear products without derivative loss.
  • standard math Kato-Ponce commutator/product inequality (Lemma 1.2)
    Quoted from [15,19] and used repeatedly for the bilinear estimates in §3.
  • standard math Dispersive decay of Schrödinger semigroup (Lemma 1.1)
    Gives the (1+t)^{-d/2} decay of φℓ,ψℓ in W^{σ,∞}; the engine of the long-time bootstrap.
  • standard math Local existence and blow-up criterion for symmetric hyperbolic systems
    Invoked in §1.3 and §3 to extend the perturbation solution up to T1/ε or T2/ε^{4/3}.
  • domain assumption Initial-data scaling (1.2) with ∂tφ(0)=ε^{-2}φ1 and φ0,φ1,ψ0 independent of ε
    Defines the nonrelativistic regime; the WKB initial conditions (1.6), (2.13)-(2.15) follow from it.
  • domain assumption Regularity and integrability assumptions (1.12) and (1.14), including W^{σ,1} and d≥3
    Needed for Lemma 1.1 decay and for residuals to lie in H^σ.

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Pith. "Pith review of Optimal convergence rates of the Klein-Gordon-Schr\"odinger system in the nonrelativistic limit." pith.science (2026). https://pith.science/paper/AOLJ5QA7

@misc{pith2026260728963,
  author       = {Pith},
  title        = {Pith review of: Optimal convergence rates of the Klein-Gordon-Schr\"odinger system in the nonrelativistic limit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AOLJ5QA7}},
  note         = {Machine review of arXiv:2607.28963}
}
abstract

In this paper, we study the Klein-Gordon-Schr\"{o}dinger system in the nonrelativistic regime $\epsilon \to 0$, where $\epsilon$ is proportional to the inverse of the speed of light. We show that the Klein-Gordon-Schr\"{o}dinger system converges to a system of decoupled linear Schr\"odinger equations over a long time interval of order $\epsilon^{-1}$ with error estimates of the form $(1+t)\epsilon^2$; in particular, the error estimate for the Schr\"odinger component is uniform in time of the form $\epsilon^{2}$ The specific forms of the error estimates coincide with the numerical results shown by Bao et a.l., and the $O(\epsilon^{2})$ convergence rates coincide with the order of initial error, and thus are optimal.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

39 extracted references · 2 linked inside Pith

  1. [6]

    Bao and X

    W. Bao and X. Zhao, A uniformly accurate multiscale time integrator Fourier pseudospectral method for the Klein-Gordon-Schr¨ odinger equations in the nonrelativistic limit regime, Numer. Math.135(2017), no. 3, 833–873. 23

    2017

  2. [1]

    Baillon and J

    J.-B. Baillon and J. M. Chadam, The Cauchy problem for the coupled Schr¨ odinger-Klein- Gordon equations, inContemporary developments in continuum mechanics and partial differential equations, North-Holland Math. Stud.,30, pp. 37–44,

  3. [2]

    Banquet, L

    C. Banquet, L. C. F. Ferreira, and E. J. Villamizar-Roa, On existence and scattering theory for the Klein-Gordon-Schr¨ odinger system in an infiniteL2-norm setting, Ann. Mat. Pura Appl. (4)194(2015), no. 3, 781–804

    2015

  4. [3]

    W. Bao, Y. Cai and X. Zhao, A uniformly accurate multiscale time integrator pseudospectral method for the Klein-Gordon equation in the nonrelativistic limit regime, SIAM J. Numer. Anal.52(2014), 2488-2511

    2014

  5. [4]

    Bao and C

    W. Bao and C. Liu, A uniformly accurate multiscale time integrator for the nonlinear Klein- Gordon equation in the nonrelativistic regime via simplified transmission conditions, arXiv: 2602.04988

    arXiv

  6. [5]

    W. Bao, Y. Lu, and Z. Zhang, Convergence rates in the nonrelativistic limit of the cubic Klein-Gordon equation, SIAM J. Math. Anal.56(2024), no. 5, 6822–6860

    2024

  7. [7]

    Bao and X

    W. Bao and X. Zhao, Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime. J. Comput. Phys.398(2019), 108886, 30 pp

    2019

  8. [8]

    Bechouche, N

    P. Bechouche, N. J. Mauser, and S. Selberg, Nonrelativistic limit of Klein-Gordon-Maxwell to Schr¨ odinger-Poisson, Amer. J. Math.126(2004), no. 1, 31–64

    2004

Show all 39 references
  1. [9]

    Bourgain,Global solutions of nonlinear Schr¨ odinger equations, Amer

    J. Bourgain,Global solutions of nonlinear Schr¨ odinger equations, Amer. Math. Soc. Colloq. Publ., vol. 46, American Mathematical Society, Providence, RI, 1999. viii+182 pp

    1999

  2. [10]

    Cho and T

    Y. Cho and T. Ozawa, On the semirelativistic Hartree-type equation, SIAM J. Math. Anal. 38(2006), no. 4, 1060–1074

    2006

  3. [11]

    C. Fan, G. Staffilani, and Z. Zhao, On decaying properties of nonlinear Schr¨ odinger equations, SIAM J. Math. Anal.,56(2024), pp. 3082–3109

    2024

  4. [12]

    Feng and C

    Y. Feng and C. Liu, A uniformly accurate multiscale time integrator for the Klein-Gordon- Sch¨ odinger equations in the nonrelativistic regime via simplified transmission conditions, arXiv: 2605.12936

    arXiv

  5. [13]

    Fukuda and M

    I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schr¨ odinger equations. I, Bull. Sci. Engrg. Res. Lab. Waseda Univ.69(1975), 51–62

    1975

  6. [14]

    Fukuda and M

    I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schr¨ odinger equations. II, J. Math. Anal. Appl.66(1978), no. 2, 358–378

    1978

  7. [15]

    Grafakos and S

    L. Grafakos and S. Oh, The Kato-Ponce inequality, Comm. Partial Differential Equations39 (2014), no. 6, 1128–1157

    2014

  8. [16]

    Guo and C

    B. Guo and C. Miao, Global existence and asymptotic behavior of solutions for the coupled Klein-Gordon-Schr¨ odinger equations, Sci. China Ser. A.38(1995), no. 12, 1444–1456

    1995

  9. [17]

    Han-Kwan, T

    D. Han-Kwan, T. Nguyen and F. Rousset, Long time estimates for the Vlasov-Maxwell system in the non-relativistic limit, Comm. Math. Phys.363(2018), no. 2, 389–434

    2018

  10. [18]

    J.-L. Joly, G. M´ etivier and J. Rauch, Transparent nonlinear geometric optics and Maxwell- Bloch equations, J. Differential Equations166(2000), no. 1, 175–250

    2000

  11. [19]

    Kato and G

    T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math.41(1988), no. 7, 891–907

    1988

  12. [20]

    Lei and Y

    Z. Lei and Y. Wu, Non-relativistic limit for the cubic nonlinear Klein-Gordon equations, arXiv: 2309.10235

    arXiv

  13. [21]

    Lu and Z

    Y. Lu and Z. Zhang, Partially strong transparency conditions and a singular localization method in geometric optics, Arch. Ration. Mech. Anal.222(2016), no. 1, 245–283

    2016

  14. [22]

    Lu and Z

    Y. Lu and Z. Zhang, Higher order asymptotic analysis of the Klein-Gordon equation in the non-relativistic limit regime, Asymptot. Anal.102(2017), no. 3-4, 157–175

    2017

  15. [23]

    Lu and Z

    Y. Lu and Z. Zheng, Optimal convergence rates of the Klein-Gordon-Zakharov system in the non-relativistic limit, J. Differential Equations431(2025), Paper No. 113195, 52 pp

    2025

  16. [24]

    Machihara, The nonrelativistic limit of the nonlinear Klein-Gordon equation, Funkcial

    S. Machihara, The nonrelativistic limit of the nonlinear Klein-Gordon equation, Funkcial. Ekvac.44(2001), no. 2, 243–252. 24

    2001

  17. [25]

    Majda,Compressible fluid flow and systems of conservation laws in several space variables, Appl

    A. Majda,Compressible fluid flow and systems of conservation laws in several space variables, Appl. Math. Sci., vol. 53, Springer-Verlag, New York, 1984. viii+159 pp

    1984

  18. [26]

    Masmoudi and K

    N. Masmoudi and K. Nakanishi, From nonlinear Klein-Gordon equation to a system of coupled nonlinear Schr¨ odinger equations, Math. Ann.324(2002), no. 2, 359–389

    2002

  19. [27]

    Masmoudi and K

    N. Masmoudi and K. Nakanishi, Nonrelativistic limit from Maxwell-Klein-Gordon and Maxwell-Dirac to Poisson-Schr¨ odinger, Int. Math. Res. Not.2003(2003), no. 13, 697–734

    2003

  20. [28]

    Masmoudi and K

    N. Masmoudi and K. Nakanishi, Energy convergence for singular limits of Zakharov type systems, Invent. Math.172(2008), no. 3, 535–583

    2008

  21. [29]

    Masmoudi and K

    N. Masmoudi and K. Nakanishi, From the Klein-Gordon-Zakharov system to a singular nonlinear Schr¨ odinger system, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire27(2010), no. 4, 1073–1096

    2010

  22. [30]

    Machihara, K

    S. Machihara, K. Nakanishi, and T. Ozawa, Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations, Math. Ann.322(2002), no. 3, 603–621

    2002

  23. [31]

    M´ etivier,Para-differential calculus and applications to the Cauchy problem for nonlinear systems, CRM Series, vol

    G. M´ etivier,Para-differential calculus and applications to the Cauchy problem for nonlinear systems, CRM Series, vol. 5, Edizioni della Normale, Pisa, 2008. xii+140 pp

    2008

  24. [32]

    Missaoui and E

    S. Missaoui and E. Zahrouni, Regularity of the attractor for a coupled Klein-Gordon- Schr¨ odinger system with cubic nonlinearities inR 2, Commun. Pure Appl. Anal.14(2015), no. 2, 695–716

    2015

  25. [33]

    Pecher, Global solutions of the Klein-Gordon-Schr¨ odinger system with rough data, Differential Integral Equations17(2004), no

    H. Pecher, Global solutions of the Klein-Gordon-Schr¨ odinger system with rough data, Differential Integral Equations17(2004), no. 1-2, 179–214

    2004

  26. [34]

    Pasquali, Almost global existence for the nonlinear Klein-Gordon equation in the nonrelativistic limit, J

    S. Pasquali, Almost global existence for the nonlinear Klein-Gordon equation in the nonrelativistic limit, J. Math. Phys.59(2018), no. 1, 011502, 15 pp

    2018

  27. [35]

    Pasquali, Dynamics of the nonlinear Klein-Gordon equation in the nonrelativistic limit, Ann

    S. Pasquali, Dynamics of the nonlinear Klein-Gordon equation in the nonrelativistic limit, Ann. Mat. Pura Appl.4198 (2019), no. 3, 903–972

    2019

  28. [36]

    Rauch,Hyperbolic partial differential equations and geometric optics, Grad

    J. Rauch,Hyperbolic partial differential equations and geometric optics, Grad. Stud. Math., vol. 133, American Mathematical Society, Providence, RI, 2012. xx+363 pp

    2012

  29. [37]

    Schratz and X

    K. Schratz and X. Zhao, On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime, Discrete Contin. Dyn. Syst. Ser. B 25(2020), no. 8, 2841–2865

    2020

  30. [38]

    Tsutsumi, Nonrelativistic approximation of nonlinear Klein-Gordon equations in two space dimensions, Nonlinear Anal.8(1984), no

    M. Tsutsumi, Nonrelativistic approximation of nonlinear Klein-Gordon equations in two space dimensions, Nonlinear Anal.8(1984), no. 6, 637–643

    1984

  31. [39]

    Yukawa, On the interaction of elementary particles

    H. Yukawa, On the interaction of elementary particles. I, Progr. Theoret. Phys. Suppl.1(1955), 1–10. 25

    1955

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