Non-relativistic Limit from Complex Klein-Gordon Equation to a System of Coupled Nonlinear Schrödinger Equations in 2D and 3D

Abstract

In this paper, the authors consider the non-relativistic limit of solutions to the defocusing cubic nonlinear Klein-Gordon equations. Inspired by the work of Lei and Wu (2023), which considered the real-valued case, the authors show that the solutions to the complex Klein-Gordon equation can be described by using a system of two coupled nonlinear Schrödinger equations as the speed of light tends to infinity both in two and three dimensions. On the one hand, if the initial data belong to a lower-regularity Sobolev space H^α, α ∈ [1, 4], the error is proportionate to the α order of the reciprocal of the speed of light, with a constant that grows linearly in time. On the other hand, if initial data are smoother (at least in H⁴), the error is proportionate to the square of the reciprocal of the speed of light, with a constant that grows linearly in time. The error of two different regularity requirements holds globally over time.

Keywords

Nonlinear Klein-Gordon equations / Nonlinear Schrödinger equations / Non-relativistic limit / Convergence rate / 35L70 / 35Q55 / 35B25 / 35B40

References

[1] Bao W, Cai Y. Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator. SIAM J. Numer. Anal., 2012, 50(2): 492-521

[2] Bao W, Lu Y, Zhang Z. Convergence rates in the nonrelativistic limit of the cubic Klein-Gordon equation. SIAM J. Math. Anal., 2024, 56(5): 6822-6860

[3] Bourgain J, Li D. On an endpoint Kato-Ponce inequality. Differential Integral Equations, 2014, 27(11–12): 1037-1072

[4] Dodson B. Global well-posedness and scattering for the defocusing, L²-critical, nonlinear Schrödinger equation when d = 2. Duke Math. J., 2016, 165(18): 3435-3516

[5] Kato T, Ponce G. Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math., 1988, 41(7): 891-907

[6] Keel M, Tao T. Endpoint Strichartz estimates. Amer. J. Math., 1998, 120(5): 955-980

[7] Lei, Z. and Wu, Y., Non-relativistic limit for the cubic nonlinear Klein-Gordon equations, 2023, arXiv: 2309.10235.

[8] Li D. On Kato-Ponce and fractional Leibniz. Rev. Mat. Iberoam., 2019, 35(1): 23-100

[9] Machihara S. The nonrelativistic limit of the nonlinear Klein-Gordon equation. Funkcial. Ekvac., 2001, 44(2): 243-252

[10] Machihara S, Nakanishi K, Ozawa T. Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations. Math. Ann., 2002, 322(3): 603-621

[11] Masmoudi N, Nakanishi K. From nonlinear Klein-Gordon equation to a system of coupled nonlinear Schrödinger equations. Math. Ann., 2002, 324(2): 359-389

[12] Morawetz C S, Strauss W A. Decay and scattering of solutions of a nonlinear relativistic wave equation. Comm. Pure Appl. Math., 1972, 25: 1-31

[13] Najman B. The nonrelativistic limit of the nonlinear Klein-Gordon equation. Nonlinear Anal., 1990, 15(3): 217-228

[14] Nakanishi K. Nonrelativistic limit of scattering theory for nonlinear Klein-Gordon equations. J. Differential Equations, 2002, 180(2): 453-470

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