Dynamics of the nonlinear Klein–Gordon equation in the nonrelativistic limit

Pasquali, Stefano

doi:10.1007/s10231-018-0805-1

Cited by 11 publications

(11 citation statements)

References 62 publications

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“…Substituting ( 7) into (5), we obtain equation (6). Thus, the solutions of the system of equations (4.5) are also solutions of the equation ( 6) (but not vice versa).…”

Section: Joint Solution Of the Klein-gordon And Schrödinger Equations In The Rest Framementioning

confidence: 99%

“…The main successes of quantum mechanics in the quantitative description of nonrelativistic systems are connected with the Schrödinger equation. Often this equation is regarded as the nonrelativistic limit of the Klein-Gordon equation [1][2][3][4][5][6]. When considering this limit, as a rule, in the solutions of the Klein-Gordon equation, the speed of light tends to infinity.…”

Section: Introductionmentioning

confidence: 99%

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Klein‐Gordon and Schrödinger equations for a free particle in the rest frame

Salomatov

2020

phys essays

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A system of two equations is found that has solutions which coincide with the solutions of the Klein‐Gordon equation in the rest frame. This system includes the Schrödinger equation for a free neutral spinless particle. Using the Schrödinger equation as an additional condition for solving the Klein‐Gordon equation in the rest frame leads to two Helmholtz equations. Helmholtz equations can be solved by specifying a particle model and boundary conditions. One of the Helmholtz equations leads to discreteness of the rest masses of relativistic particles.

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“…Substituting ( 7) into (5), we obtain equation (6). Thus, the solutions of the system of equations (4.5) are also solutions of the equation ( 6) (but not vice versa).…”

Section: Joint Solution Of the Klein-gordon And Schrödinger Equations In The Rest Framementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Klein‐Gordon and Schrödinger equations for a free particle in the rest frame

Salomatov

2020

phys essays

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“…We also mention the recent papers [LZ16] and [Pas17], which discuss the long-time convergence of solution of NLKG in the nonrelativistic limit on R d : however, the results proved in both papers have some limitations, either on the nonlinearity (in [LZ16] the authors studied only the quadratic NLKG) or on the particular form of the solution (in [Pas17]).…”

Section: Introductionmentioning

confidence: 99%

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

Pasquali

2018

Journal of Mathematical Physics

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We study the one-dimensional nonlinear Klein-Gordon (NLKG) equation with a convolution potential, and we prove that solutions with small H s norm remain small for long times. The result is uniform with respect to c ≥ 1, which however has to belong to a set of large measure. over time scales of order O(1) through a variant of Birkhoff Normal Form theory. In [Pas17] Birkhoff Normal Form Theory was exploited in order to recover the result by Faou-Schratz, and to generalize it to all smooth compact Riemannian manifolds.Actually, [FS14] dealt with the construction of numerical schemes which are robust in the nonrelativistic limit; we refer also to [BD12], [BZ16] and to [BFS17] for the numerical analysis of the nonrelativistic limit of the NLKG equation.In this paper we generalize the techniques developed in [BG06] in order to prove a long-time existence result for the NLKG with a convolution potential with Dirichlet boundary conditions, uniformly in c ≥ 1.An immediate corollary of our result allows us to show that for any α > 0 any solution in H s with initial datum of size O(c −α ) remains of size O(c −α ) up to times of order O(c α(r+1/2) ) for any r ≥ 1; however, we have to assume that both the parameter c and the coefficients of the potential belong to a set of large measure. The main limitation of such a result is that it holds only for solutions with initial data which are small with respect to c.The new ingredient in the proof with respect to [BG06] is a diophantine type estimate for the frequencies, which holds uniformly when c → ∞.An aspect that would deserve future work is the study of the nonrelativistic limit of the NLKG without potential. This is expected to be a quite subtle problem since, for c = 0 the frequencies of NLKG are typically non resonant, while the limiting frequencies are resonant. The issue of long-time existence for small solutions of the NLKG equation without potential on compact manifolds has received a lot of interest; see for example [DS04], [Del09], [FZ10], [FHZ17] and [DI17]. However, all results in the aforementioned papers rely on a nonresonance condition which is not uniform with respect to c.The paper is organized as follows. In sect. 2 we state the results of the paper, together with some examples and comments. In sect. 3 we prove our result for the NLKG with a convolution potential on I := [0, π].Acknowledgements. This work is based on author's PhD thesis. He would like to express his thanks to his supervisor Professor Dario Bambusi.

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“…In Section 2 below we present the detailed structure of the NLS limit system. For recent analytic approximations results we refer to [36,33,34], and in the context of Birkhoff normal form transformations in particular to the recent work [38], as well as the references therein. From a numerical point of view, however, the Klein-Gordon equation in the nonrelativistic was an open problem for a long time.…”

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confidence: 99%

“…The aim of this work is to for the first time give a comparison of the novel techniques, highlighting the gap to the KG in the asymptotic limit, reviewing their approximation validity, their geometric properties (e.g., energy conservation), the regularity requirement of each expansion to maintain the optimal asymptotic order, and the long-time behaviour of the expansion. It is worth noting that the asymptotic series [38] found by the Birkhoff normal form transformation is the same as the one derived from the modulated Fourier expansion [28]. In our analysis, we in particular focus on the three asymptotic methods: the modulated Fourier expansion [28], the multiscale expansion by frequency [5] and the Chapman-Enskog expansion [18].…”

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confidence: 99%

On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime

Schratz¹,

Zhao²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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This work concerns the time averaging techniques for the nonlinear Klein-Gordon (KG) equation in the nonrelativistic limit regime which have recently gained a lot of attention in numerical analysis. This is due to the fact that the solution becomes highly-oscillatory in time in this regime which causes the breakdown of classical integration schemes. To overcome this numerical burden various novel numerical methods with excellent efficiency were derived in recent years. The construction of each method thereby requests essentially the averaged model of the problem. However, the averaged model of each approach is found by different kinds of asymptotic approximation techniques reaching from the modulated Fourier expansion over the multiscale expansion by frequency up to the Chapman-Enskog expansion. In this work we give a first comparison of these recently introduced asymptotic series, reviewing their approximation validity to the KG in the asymptotic limit, their smoothness assumptions as well as their geometric properties, e.g., energy conservation and long-time behaviour of the remainder.2010 Mathematics Subject Classification. 35Q40, 34C29, 34E05, 81Q05. Key words and phrases. Nonlinear Klein-Gordon equation, nonrelativistic limit regime, time averaging, higher order expansion, long-time behavior. 2841 2842 KATHARINA SCHRATZ AND XIAOFEI ZHAO The so-called nonrelativistic limit ε → 0 of the KG equation (1.1) has been extensively studied in literature from a physical and mathematical point of view. Nowadays it is well understood that the KG equation converges to a nonlinear Schrödinger (NLS) equation when ε tends to zero. In Section 2 below we present the detailed structure of the NLS limit system. For recent analytic approximations results we refer to [36,33,34], and in the context of Birkhoff normal form transformations in particular to the recent work [38], as well as the references therein. From a numerical point of view, however, the Klein-Gordon equation in the nonrelativistic was an open problem for a long time. Classical methods are not able to resolve the highly-oscillatory nature of the solution which leads to severe step size restrictions (at least at order τ = O(ε −2 )) and huge computational costs ([6]). This failure of classical integration schemes triggered intensive studies on the numerical averaging of the model, serving to describe better asymptotic behaviour of the solution and to design efficient numerical approximations. Based on this idea, very recently novel numerical integrators were proposed which allow us to numerically solve the KG equation (1.1) in the non relativistic regime ε → 0, capturing the highly-oscillatory behaviour of the solution.Three novel schemes for the Klein-Gordon equation in the non relativistic limit regime were presented so far in literature [28,5,18]. Each of the proposed numerical schemes for KG is essentially based on an asymptotic expansion techinque for the averaged model, and each of the asymptotic expansion is mathematically obtained by different analytic techn...

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Dynamics of the nonlinear Klein–Gordon equation in the nonrelativistic limit

Cited by 11 publications

References 62 publications

Klein‐Gordon and Schrödinger equations for a free particle in the rest frame

Klein‐Gordon and Schrödinger equations for a free particle in the rest frame

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

On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime

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