ИСТИНА

Weakly nonlinear multi-dimensional shock waves are characterized by small amplitude and weak curvature of the shock front. Propagation of such shock waves in a chemically reacting gas can be sustained by energy released in the reactions in which case they are called detonations. Here we derive an asymptotic model for the dynamics of detonations from the compressible reactive Navier-Stokes equations. The resultant model in two dimensions consists of a 2D forced Burgers equation, a zero-vorticity equation, and a rate equation of chemistry. In various limits, the model reduces to small-disturbance unsteady transonic flow equations of aerodynamics, weakly nonlinear models of acoustics (Zabolotskaya-Khokhlov (ZK) equation), and equations of water waves (dispersionless Kadomtsev-Petviashvili (KP) equation). The model predicts regular and irregular multi-dimensional patterns in 2D and in 1D, it exhibits transition from steady and stable traveling waves to oscillatory traveling waves through a Hopf bifurcation as a parameter characterizing sensitivity to chemical reactions is increased above a critical value. A cascade of period-doubling bifurcations leading to chaos is also observed.