Long-Time Analysis of Nonlinearly Perturbed Wave Equations Via Modulated Fourier Expansions

Cohen, David E.; Hairer, Ernst; Lubich, Christian

doi:10.1007/s00205-007-0095-z

Cited by 64 publications

(102 citation statements)

References 17 publications

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“…With this scaling there is no factor √ 2N in the system (3) and no factor 2N in the potential (6), but the factor 2N appears in the energies (7) and (8).…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…Our principal tool for studying the long-time behaviour of mode energies is a modulated Fourier expansion, which was originally introduced for the study of numerical energy conservation in Hamiltonian ordinary differential equations in the presence of high oscillations [17,8]. This technique was also successfully applied to the long-time analysis of weakly nonlinear Hamiltonian partial differential equations [9, 15,16].…”

Section: Modulated Fourier Expansionmentioning

confidence: 99%

“…with τ = N −3 t, so that y k j depends only on the equivalence class [k] = k+M of k. The construction of the modulation functions z k j has been modified in such a way that in the modulation equations (70) for the above-defined y k j , the sum is now over equivalence classes of multi-indices [ [8] for the construction of resonant modulated Fourier expansions). Consequently, in the extended potential U of (71) the sum is now over k 1 + k 2 + k 3 ∼ 0.…”

Section: Including the First Near-resonancementioning

confidence: 99%

“…This permits us to explain the preservation of the low-frequency packet over time scales that are much longer than the time scale for the formation of the packet. We mention that modulated Fourier expansions have been used similarly before in the analysis of numerical methods for oscillatory Hamiltonian differential equations [17,8] and in the long-time analysis of non-linearly perturbed wave equations [9] and Schrödinger equations [15,16].…”

Section: Introductionmentioning

confidence: 99%

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On the Energy Distribution in Fermi–Pasta–Ulam Lattices

Hairer

Lubich

2012

Arch Rational Mech Anal

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For FPU chains with large particle numbers, the formation of a packet of modes with geometrically decaying harmonic energies from an initially excited single low-frequency mode and the metastability of this packet over longer time scales are rigorously studied in this paper. The analysis uses modulated Fourier expansions in time of solutions to the FPU system and exploits the existence of almost-invariant energies in the modulation system. The results and techniques apply to the FPU α-and β-models as well as to higher-order nonlinearities. They are valid in the regime of scaling between particle number and total energy in which the FPU system can be viewed as a perturbation to a linear system, considered over time scales that go far beyond standard perturbation theory. Weak non-resonance estimates for the almost-resonant frequencies determine the time scales that can be covered by this analysis.

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“…With this scaling there is no factor √ 2N in the system (3) and no factor 2N in the potential (6), but the factor 2N appears in the energies (7) and (8).…”

Section: Formulation Of the Problemmentioning

confidence: 99%

Section: Modulated Fourier Expansionmentioning

confidence: 99%

Section: Including the First Near-resonancementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Energy Distribution in Fermi–Pasta–Ulam Lattices

Hairer

Lubich

2012

Arch Rational Mech Anal

Self Cite

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“…Eventually, Section 5 is a collection of technical lemmas needed in the course of the analysis. Related works and tools, that are related with the techniques used in this article and the questions we raise, may be found in [1,8,15] (where KAM tools are developed in the infinite dimensional setting), in [3,4,12,14] (where numerical methods in the Hamiltonian setting are developed and analyzed), or in [11] (where considerations on splitting schemes are developed).…”

Section: Consider the Schrödinger Equation With Potential I∂ T U(t Xmentioning

confidence: 99%

Propagation of Gevrey regularity over long times for the fully discrete Lie Trotter splitting scheme applied to the linear Schrödinger equation

Castella¹,

Dujardin²

2009

ESAIM: M2AN

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Abstract. In this paper, we study the linear Schrödinger equation over the d-dimensional torus,with small values of the perturbing potential. We consider numerical approximations of the associated solutions obtained by a symplectic splitting method (to discretize the time variable) in combination with the Fast Fourier Transform algorithm (to discretize the space variable). In this fully discrete setting, we prove that the regularity of the initial datum is preserved over long times, i.e. times that are exponentially long with the time discretization parameter. We here refer to Gevrey regularity, and our estimates turn out to be uniform in the space discretization parameter. This paper extends [G. Dujardin and E. Faou, Numer. Math. 97 (2004) 493-535], where a similar result has been obtained in the semi-discrete situation, i.e. when the mere time variable is discretized and space is kept a continuous variable.Mathematics Subject Classification. 65P10, 37M15, 37K55.

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Stroboscopic averaging of highly oscillatory nonlinear wave equations

Leboucher

2014

Math Methods in App Sciences

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International audienceIn this paper, we are concerned with stroboscopic averaging for highly oscillatory evolution equations posed in a Banach space. Using Taylor expansion, we construct a non-oscillatory high-order system whose solution remains exponentially close to the exact one over a long time. We then apply this result to the nonlinear wave equation in one dimension. We present the stroboscopic averaging method, which is a numerical method introduced by Chartier, Murua and Sanz-Serna, and apply it to our problem. Finally, we conclude by presenting the qualitative and quantitative efficiency of this numerical method for some nonlinear wave problem

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Long-Time Analysis of Nonlinearly Perturbed Wave Equations Via Modulated Fourier Expansions

Cited by 64 publications

References 17 publications

On the Energy Distribution in Fermi–Pasta–Ulam Lattices

On the Energy Distribution in Fermi–Pasta–Ulam Lattices

Propagation of Gevrey regularity over long times for the fully discrete Lie Trotter splitting scheme applied to the linear Schrödinger equation

Stroboscopic averaging of highly oscillatory nonlinear wave equations

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