Resonantly interacting, weakly nonlinear hyperbolic waves. II: Several space variables. (English) Zbl 0657.35084

[For part I see ibid. 71, 149-179 (1984; Zbl 0572.76066).]
A uniformly valid asymptotic theory of resonantly interacting high- frequency waves for nonlinear hyperbolic systems in several space dimensions is developed. When applied to the equations of two-dimensional compressible fluid flow, this theory both predicts the geometric location of the new sound waves produced from the resonant interaction of sound waves and vorticity waves as well as yielding a simplified system which governs the evolution of the amplitudes. In this special case, this system is two Bugers equations coupled by a linear integral operator with known kernel given by the vortex strength of the shear wave. Several inherently multidimensional assumptions for the phases are needed in this theory, and theoretical examples are given which delineate these assumptions. Furthermore, explicit necessary and sufficient conditions for the validity of the earlier noninteracting wave theory of Hunter and Keller are derived; these explicit conditions indicate that generally waves resonate and interact in several dimensions.


35L65 Hyperbolic conservation laws
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics


Zbl 0572.76066
Full Text: DOI


[1] Choquet-Bruhat, Ondes asymptotiques et approchées pur systèmes d’èquations aux derivées partielles nonlinèaires, J. Math. Pures Appl. 48 pp 117– (1969)
[2] Majda, Appl. Math. Science Ser. 53 (1984)
[3] Hunter, Weakly nonlinear, high-frequency waves, Comm. Pure Appl. Math. 36 pp 547– (1983) · Zbl 0547.35070 · doi:10.1002/cpa.3160360502
[4] Majda, Resonantly interacting weakly nonlinear hyperbolic waves. I. A single space variable, Stud. Appl. Math. 71 pp 149– (1984) · Zbl 0572.76066 · doi:10.1002/sapm1984712149
[5] Hunter, Weakly nonlinear wave propagation (1981)
[6] Proceedings of IMA Symposium on Oscillations, May, 1985
[7] Ablowitz, Solitons and the Inverse Scattering Transform (1981) · doi:10.1137/1.9781611970883
[8] Craik, Nonlinear Waves (1983)
[9] Majda, A theory for spontaneous Mach stem formation in reacting shock fronts. I. The basic perturbation analysis, SIAM J. Appl. Math. 43 pp 1310– (1983) · Zbl 0544.76135 · doi:10.1137/0143088
[10] 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 pp 117– (1984) · Zbl 0584.76075 · doi:10.1002/sapm1984712117
[11] Cehelskly, Resonantly interacting weakly nonlinear hyperbolic waves in the presence of shocks: A single space variable in a homogeneous time independent medium, Stud. Appl. Math. 74 pp 117– (1986) · Zbl 0652.76046 · doi:10.1002/sapm1986742117
[12] SIAM J. Appl. Math.
[13] DiPerna, The validity of geometrical optics for weak solutions of conservation laws, Comm. Math. Phys. 98 pp 313– (1985) · Zbl 0582.35081 · doi:10.1007/BF01205786
[14] Proceedings of the International Congress, Warsaw, 1983
[15] Melrose, Interaction of nonlinear progressing waves for semilinear wave equations, Ann. Math. 121 pp 187– (1985) · Zbl 0575.35063 · doi:10.2307/1971196
[16] Katznelson, An Introduction to Harmonic Analysis (1976)
[17] Hunter, A ray method for slowly modulated nonlinear waves, SIAM J. Appl. Math. 45 pp 735– (1985) · Zbl 0582.76012 · doi:10.1137/0145043
[18] Papanicolau, Coll. Math. Societahs János Bólyai #27, 1979 pp 835– (1981)
[19] Fulton, Algebraic Curves (1969)
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