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Some explicit resonating waves in weakly nonlinear gas dynamics. (English) Zbl 0669.76104

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.

MSC:

76Q05 Hydro- and aero-acoustics
76N15 Gas dynamics (general theory)
35Q30 Navier-Stokes equations
82B40 Kinetic theory of gases in equilibrium statistical mechanics
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References:

[1] Crandall, Bifurcation from simple eigenvalues, J. Functional Anal. 8 pp 321– (1971) · Zbl 0219.46015
[2] Majda, Resonantly interacting weakly nonlinear hyperbolic waves I, Stud. Appl. Math. 71 pp 149– (1984) · Zbl 0572.76066
[4] Vanderbauwhede, Research Notes in Mathematics 75, in: Local Bifurcation and Symmetry (1982) · Zbl 0539.58022
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