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**A canonical system of integrodifferential equations arising in resonant nonlinear acoustics.**
*(English)*
Zbl 0669.76103

In general, weakly nonlinear high frequency almost periodic wave trains for systems of hyperbolic conservation laws interact and resonate to leading order. In earlier work the first two authors and J. Hunter developed simplified asymptotic equations describing this resonant interaction [ibid. 75, 187-226 (1986; Zbl 0657.35084)]. In the important special case of compressible fluid flow in one or several space dimensions, these simplified asymptotic equations are essentially two inviscid Burgers equations for the nonlinear sound waves, coupled by convolution with a known kernel given by the sum of the initial vortex strength and the derivative of the initial entropy. Here we develop some of the remarkable new properties of the solutions of this system for resonant acoustics. These new features include substantial almost periodic exchange of energy between the nonlinear sound waves, the existence of smooth periodic wave trains, and the role of such smooth wave patterns in eliminating or suppressing the strong temporal decay of sawtooth profile solutions of the decoupled inviscid Burgers equations. Our approach combines detailed numerical modeling to elucidate the new phenomena together with rigorous analysis to obtain exact solutions as well as other elementary properties of the solutions of this system.

### MSC:

76Q05 | Hydro- and aero-acoustics |

76N15 | Gas dynamics (general theory) |

35Q30 | Navier-Stokes equations |

### Keywords:

nonlinear high frequency almost periodic wave trains; systems of hyperbolic conservation laws; compressible fluid flow; two inviscid Burgers equations; nonlinear sound waves; entropy### Citations:

Zbl 0657.35084
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\textit{A. Majda} et al., Stud. Appl. Math. 79, No. 3, 205--262 (1988; Zbl 0669.76103)

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### References:

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[2] | Majda, Resonantly interacting, weakly nonlinear, hyperbolic waves. I. A single space variable, Stud. Appl. Math. 71 pp 149– (1984) · Zbl 0572.76066 |

[3] | Cehelsky, 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 |

[4] | Hunter, Resonantly interacting, weakly nonlinear hyperbolic waves. II. Several space variables, Stud. Appl. Math. 75 pp 187– (1986) · Zbl 0657.35084 |

[5] | Stud. Appl. Math. |

[6] | Lax, SIAM Regional Conf. Ser. in Appl. Math., in: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (1973) |

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