Lamacz, Agnes; Schweizer, Ben Representation of solutions to wave equations with profile functions. (English) Zbl 1452.35089 Anal. Appl., Singap. 18, No. 6, 1001-1024 (2020). Summary: In many applications, solutions to wave equations can be represented in Fourier space with the help of a dispersion function. Examples include wave equations on periodic lattices with spacing \(\varepsilon>0\), wave equations on \(\mathbb{R}^d\) with constant coefficients, and wave equations on \(\mathbb{R}^d\) with coefficients of periodicity \(\varepsilon>0\). We characterize such solutions for large times \(t=\tau/ \varepsilon^{- 2} \). We establish a reconstruction formula that yields approximations for solutions in three steps: (1) From given initial data \(u_0\), appropriate initial data for a profile equation are extracted. (2) The dispersion function determines a profile evolution equation, which, in turn, yields the shape of the profile at time \(\tau= \varepsilon^2t\). (3) A shell reconstruction operator transforms the profile to a function on \(\mathbb{R}^d\). The resulting function is a good approximation of the solution \(u(.,\tau/ \varepsilon^2)\). MSC: 35L05 Wave equation 35B40 Asymptotic behavior of solutions to PDEs 35C05 Solutions to PDEs in closed form 35Q60 PDEs in connection with optics and electromagnetic theory Keywords:large time asymptotics; effective equation; dispersion; representation formula; Fourier analysis PDFBibTeX XMLCite \textit{A. Lamacz} and \textit{B. Schweizer}, Anal. Appl., Singap. 18, No. 6, 1001--1024 (2020; Zbl 1452.35089) Full Text: DOI arXiv References: [1] Abdulle, A. and Pouchon, T., Effective models for the multidimensional wave equation in heterogeneous media over long time and numerical homogenization, Math. Models Methods Appl. Sci.26(14) (2016) 2651-2684. · Zbl 1362.35033 [2] Abdulle, A. and Pouchon, T., Effective models for long time wave propagation in locally periodic media, SIAM J. Numer. Anal.56(5) (2018) 2701-2730. · Zbl 1404.35020 [3] Allaire, G., Dispersive limits in the homogenization of the wave equation, Ann. Fac. Sci. Toulouse Math. (6)12(4) (2003) 415-431. · Zbl 1070.35006 [4] Allaire, G. and Yamada, T., Optimization of dispersive coefficients in the homogenization of the wave equation in periodic structures, Numer. Math.140(2) (2018) 265-326. · Zbl 1404.35021 [5] Benoit, A. and Gloria, A., Long-time homogenization and asymptotic ballistic transport of classical waves, Ann. Sci. Éc. Norm. Supér. (4)52(3) (2019) 703-759. · Zbl 1437.35030 [6] Dohnal, T., Lamacz, A. and Schweizer, B., Bloch-wave homogenization on large time scales and dispersive effective wave equations, Multiscale Model. Simul.12(2) (2014) 488-513. · Zbl 1320.35043 [7] Dohnal, T., Lamacz, A. and Schweizer, B., Dispersive homogenized models and coefficient formulas for waves in general periodic media, Asymptot. Anal.93(1-2) (2015) 21-49. · Zbl 1332.35024 [8] Lamacz, A., Dispersive effective models for waves in heterogeneous media, Math. Models Methods Appl. Sci.21(9) (2011) 1871-1899. · Zbl 1252.35067 [9] López, J. L. and Soler, J., A space-time Wigner function approach to long time Schrödinger-Poisson dynamics, SIAM J. Math. Anal.49(6) (2017) 4915-4941. · Zbl 1379.35266 [10] Nédélec, J.-C., Acoustic and Electromagnetic Equations, , Vol. 144 (Springer-Verlag, New York, 2001). Integral representations for harmonic problems. [11] Santosa, F. and Symes, W. W., A dispersive effective medium for wave propagation in periodic composites, SIAM J. Appl. Math.51(4) (1991) 984-1005. · Zbl 0741.73017 [12] Schweizer, B. and Theil, F., Lattice dynamics on large time scales and dispersive effective equations, SIAM J. Appl. Math.78(6) (2018) 3060-3086. · Zbl 1400.37097 [13] Wong, R., Asymptotic Approximations of Integrals, , Vol. 34 (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001). Corrected reprint of the 1989 original. · Zbl 1078.41001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.