Pego, Robert L. Some explicit resonating waves in weakly nonlinear gas dynamics. (English) Zbl 0669.76104 Stud. Appl. Math. 79, No. 3, 263-270 (1988). Equations governing leading order wave amplitudes of resonating almost periodic wave trains in weakly nonlinear acoustics have been obtained by A. Majda and R. Rosales [ibid. 71, 149-179 (1984; Zbl 0572.76066)]. These equations consist of a pair of Burgers equations coupled through an integral term with a known kernel. Numerical experiments reported by A. Majda, R. Rosales, and M. Schonbeck [see the article reviewed above (Zbl 0669.76103)] have suggested the existence of smooth solutions of this system whose components consist of traveling waves moving in opposite directions. For the simplest cosine kernel, explicit formulae are given here for such resonating wave solutions. There is a wave of maximum amplitude with a “peak”. For more general kernels, small amplitude resonating waves are constructed via bifurcation. Cited in 10 Documents MSC: 76Q05 Hydro- and aero-acoustics 76N15 Gas dynamics (general theory) 35Q30 Navier-Stokes equations 82B40 Kinetic theory of gases in equilibrium statistical mechanics Keywords:resonating almost periodic wave trains; weakly nonlinear acoustics; Burgers equations Citations:Zbl 0572.76066; Zbl 0669.76103 PDFBibTeX XMLCite \textit{R. L. Pego}, Stud. Appl. Math. 79, No. 3, 263--270 (1988; Zbl 0669.76104) Full Text: DOI References: [1] Crandall, Bifurcation from simple eigenvalues, J. Functional Anal. 8 pp 321– (1971) · Zbl 0219.46015 · doi:10.1016/0022-1236(71)90015-2 [2] Majda, Resonantly interacting weakly nonlinear hyperbolic waves I, Stud. Appl. Math. 71 pp 149– (1984) · Zbl 0572.76066 · doi:10.1002/sapm1984712149 [4] Vanderbauwhede, Research Notes in Mathematics 75, in: Local Bifurcation and Symmetry (1982) · Zbl 0539.58022 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.