×

Resonantly interacting weakly nonlinear hyperbolic waves. I: A single space variable. (English) Zbl 0572.76066

A systematic asymptotic theory is presented for resonantly interacting weakly nonlinear hyperbolic waves in one space variable. The theory includes as a special case the theory of nonresonant interacting waves for general hyperbolic systems developed recently by J. K. Hunter and J. B. Keller [see e.g.: Commun. Pure Appl. Math. 36, 547-569 (1983; Zbl 0547.35070)], when specialized to one space variable. Here resonances are included. Such resonances are typical when the hyperbolic system has at least three equations and when, for example, small- amplitude periodic initial data are given.

MSC:

76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q30 Navier-Stokes equations

Citations:

Zbl 0547.35070
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ablowitz, SIAM Studies in Applied Mathematics, in: Solitons and the Inverse Scattering Transform (1981) · Zbl 0472.35002 · doi:10.1137/1.9781611970883
[2] Courant, Supersonic Flow and Shock Waves (1948)
[3] Hunter, Weakly nonlinear high frequency waves, Comm. Pure Appl. Math. 36 (5) pp 547– (1983) · Zbl 0547.35070 · doi:10.1002/cpa.3160360502
[4] Nayfeh, Singular Perturbations and Asymptotics pp 223– (1980) · doi:10.1016/B978-0-12-493260-9.50013-8
[5] J. Differential Equations
[7] Majda, A theory for spontaneous Mach stem formation in reacting shock fronts, I: The basic perturbation analysis, SIAM J. Appl. Math. 43 (6) pp 1310– (1983) · Zbl 0544.76135 · doi:10.1137/0143088
[8] Majda, A theory for spontaneous Mach stem formation in reacting shock fronts, II: Steady wave bifurcations and the evidence for breakdown, Stud. Appl. Math. 71 (2) pp 117– (1984) · Zbl 0584.76075 · doi:10.1002/sapm1984712117
[9] Whitham, Linear and Nonlinear Waves (1974)
[10] Lax, SIAM Regional Conf. Ser. in Appl. Math, in: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (1973) · doi:10.1137/1.9781611970562
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.