Justification of the Nonlinear Schrödinger Approximation for a Quasilinear Klein–Gordon Equation

Düll, Wolf-Patrick

doi:10.1007/s00220-017-2966-y

Cited by 20 publications

(29 citation statements)

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“…Although there is an O(ǫ) term ǫḡ(·, ·, ·) in (3.21b), we can eliminate the O(ǫ) terms and keep only the O(ǫ 2 ) terms in the evolutionary equation of E s by the carefully constructed energy functional E s in (3.22). The strategy of definition of the energy using the normal form transformation was already used in previous papers [5,6,16,17,22,37,38]. But here we would like to remark that there are some basic differences in this paper.…”

Section: )mentioning

confidence: 99%

“…Similarly, we have Since the right-hand side of the error equation (4.29) for j 1 ∈ {±1} loses one derivative, we will need the following identities to control the time evolution of E s . See also [6].…”

Section: )mentioning

confidence: 99%

“…It was realized that it is not easy to close the energy estimates once the order of total loss of derivatives is greater than one. Recently, justification of the NLS approximation for a quasilinear Klein-Gordon equation has been obtained in [6] by using the normal form transformation to define a new energy to simplify the error estimates, in which the nonlinearities lose one derivative causing that the transformed system to lose two derivatives in total. But luckily there is no resonance in such a quasilinear Klein-Gordon model [6], i.e.…”

Section: Introductionmentioning

confidence: 99%

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Justification of the NLS Approximation for the Euler–Poisson Equation

Liu

2019

Commun. Math. Phys.

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The nonlinear Schrödinger (NLS) equation can be derived as a formal approximation equation describing the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the ion Euler-Poisson equation. In this paper, we rigorously justify such approximation by giving error estimates in Sobolev norms between exact solutions of the ion Euler-Poisson system and the formal approximation obtained via the NLS equation. The justification consists of several difficulties such as the resonances and loss of regularity, due to the quasilinearity of the problem. These difficulties are overcome by introducing normal form transformation and cutoff functions and carefully constructed energy functional of the equation. 2000 Mathematics Subject Classification. 35M20; 35Q35. 1 Jang considered the global solution with spherical symmetry initial data [23]. Furthermore, Jang, Li and Zhang obtained the smooth global solutions [24]. Finally, Li and Wu solved the Cauchy problem for the two dimensional electron Euler-Poisson system [30]. Recently, Guo, Han and Zhang [11] finally completely settled this problem of global existence and proved that no shocks form for the 1D Euler-Poisson system for electrons. For the Euler-Poisson equation for ions, Guo and Pausader [13] constructed global smooth irrotational solutions with small amplitude for ion dynamics in R 3 . For the long wave approximation, Guo and Pu [14] established rigorously the KdV limit for the ion Euler-Poisson system in 1D for both cold and hot plasma cases, where the electron density satisfies the classical Maxwell-Boltzmann law. This result was generalized to the higher dimensional cases, and the 2D Kadomtsev-Petviashvili-II (KP-II) equation and the 3D Zakharov-Kuznetsov (ZK) equation are derived under different scalings [31]. Almost at the same time, [27] also established the Zakharov-Kuznetsov equation in 3D from the Euler-Poisson system. Recently, the authors in the present paper [28,29] obtained rigorously the quantum KdV limit in 1D and the KP-I and KP-II equations in 2D for the Euler-Poisson system for cold as well as hot plasma taking quantum effect into account, where the electron equilibrium is given by a Fermi-Dirac distribution. Han-Kwan [15] also introduced a long wave scaling for the Vlasov-Poisson equation

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Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Justification of the NLS Approximation for the Euler–Poisson Equation

Liu

2019

Commun. Math. Phys.

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“…The observed convergence of the new methods can thus be explained with Theorems 3.2 and 3.4. In practice, one will choose filter functions φ as in (15) For additional numerical examples in connection with the questions studied in [4,5,9,17], we refer to a previous version of this paper [12, Section 3.5].…”

Section: Statement Of Global Error Boundsmentioning

confidence: 99%

“…The real-valued parameter κ will be used to emphasize the strength of the nonlinearities, and we will be interested both in the regime where κ is small so that the nonlinearities are small and the regime where κ is of order one. Quasilinear wave equations of this form with small |κ| have been extensively studied by Groves & Schneider [17], Chong & Schneider [5], Chirilus-Bruckner, Düll & Schneider [4] and Düll [9]: the equations from the class (1) are prototypes for models in nonlinear optics with a nonlinear Schrödinger equation as a modulation equation [17,Introduction]. Many examples from elasticity and fluid mechanics can also be reduced to quasilinear wave equations under the form (1).…”

Section: Introductionmentioning

confidence: 99%

Trigonometric integrators for quasilinear wave equations

Gauckler

Marzuola

et al. 2018

Math. Comp.

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Trigonometric time integrators are introduced as a class of explicit numerical methods for quasilinear wave equations. Second-order convergence for the semidiscretization in time with these integrators is shown for a sufficiently regular exact solution. The time integrators are also combined with a Fourier spectral method into a fully discrete scheme, for which error bounds are provided without requiring any CFL-type coupling of the discretization parameters. The proofs of the error bounds are based on energy techniques and on the semiclassical Gårding inequality.Mathematics Subject Classification (2010): 65M15, 65P10, 65L70, 65M20.

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Validity of the nonlinear Schrödinger approximation for quasilinear dispersive systems with more than one derivative

Heß

2021

Math Methods in App Sciences

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For nonlinear dispersive systems, the nonlinear Schrödinger (NLS) equation can usually be derived as a formal approximation equation describing slow spatial and temporal modulations of the envelope of a spatially and temporally oscillating underlying carrier wave. Here, we justify the NLS approximation for a whole class of quasilinear dispersive systems, which also includes toy models for the waterwave problem. This is the first time that this is done for systems, where a quasilinear quadratic term is allowed to effectively lose more than one derivative. With effective loss, we here mean the loss still present after making a diagonalization of the linear part of the system such that all linear operators in this diagonalization have the same regularity properties.

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Justification of the Nonlinear Schrödinger Approximation for a Quasilinear Klein–Gordon Equation

Cited by 20 publications

References 25 publications

Justification of the NLS Approximation for the Euler–Poisson Equation

Justification of the NLS Approximation for the Euler–Poisson Equation

Trigonometric integrators for quasilinear wave equations

Validity of the nonlinear Schrödinger approximation for quasilinear dispersive systems with more than one derivative

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