Nonlinear resonance for quasilinear hyperbolic equation

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Abstract
The purpose of this paper is to study the wave behavior of hyperbolic conservation laws with a moving source. Resonance occurs when the speed of the source is too close to one of the characteristic speeds of the system. For the nonlinear system characteristic speeds depend on the basic dependence variables and resonance gives rise to nonlinear interactions which lead to rich wave phenomena. Motivated by physical examples a scalar model is proposed and analyzed to describe the qualitative behavior of waves for a general system in resonance with the source. Analytical understanding is used to design a numerical scheme based on the random choice method. An important physical example is transonic gas flow through a nozzle. This analysis provides a transparent and revealing qualitative understanding of wave behavior of gas flow, including such phenomena as nonlinear stability, instability, and changing types of waves.

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