Dobrokhotov, S. Yu.; Nazaikinskii, V. E. Propagation of a linear wave created by a spatially localized perturbation in a regular lattice and punctured Lagrangian manifolds. (English) Zbl 1376.37114 Russ. J. Math. Phys. 24, No. 1, 127-133 (2017). Summary: The following results are obtained for the Cauchy problem with localized initial data for the crystal lattice vibration equations with continuous and discrete time: (i) the asymptotics of the solution is determined by Lagrangian manifolds with singularities (“punctured” Lagrangian manifolds); (ii) Maslov’s canonical operator is defined on such manifolds as a modification of a new representation recently obtained for the canonical operator by the present authors together with A. I. Shafarevich [Dokl. Math. 93, No. 1, 99–102 (2016; Zbl 1345.53083); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 466, No. 6, 641–644 (2016)]; (iii) the projection of the Lagrangian manifold onto the configuration plane specifies a bounded oscillation region, whose boundary (which is naturally referred to as the leading edge front) is determined by the Hamiltonians corresponding to the limit wave equations; (iv) the leading edge front is a special caustic, which possibly contains stronger focal points. These observations, together with earlier results, lead to efficient formulas for the wave field in a neighborhood of the leading edge front. Cited in 8 Documents MSC: 37K60 Lattice dynamics; integrable lattice equations 53D12 Lagrangian submanifolds; Maslov index 35S10 Initial value problems for PDEs with pseudodifferential operators 39A12 Discrete version of topics in analysis Keywords:Cauchy problem; crystal lattice vibration; Lagrangian manifold; wave equation Citations:Zbl 1345.53083 PDFBibTeX XMLCite \textit{S. Yu. Dobrokhotov} and \textit{V. E. Nazaikinskii}, Russ. J. Math. Phys. 24, No. 1, 127--133 (2017; Zbl 1376.37114) Full Text: DOI References: [1] V. P. Maslov, “Nonstandard Characteristics in Asymptotic Problems,” Uspekhi Mat. Nauk 38 6 (234), 3-36 (1983); Russian Math. Surveys 38 (6), 1-42 (1983). [2] S. Yu. Dobrokhotov, A. I. Shafarevich, and B. Tirozzi, “Localized Wave and Vortical Solutions to Linear Hyperbolic Systems and Their Application to Linear Shallow-Water Equations,” Russ. J. Math. Phys. 15 (2), 192-221 (2008). · Zbl 1180.35336 [3] S. Yu. Dobrokhotov, R. V. Nekrasov, and B. Tirozzi, “Asymptotic Solutions of the Linear Shallow-Water Equations with Localized Initial Data,” J. Engng. Math. 69 (2-3), 225-242 (2011). · Zbl 1325.76036 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.