Formations of Transitional Zones in Shock Wave with Saddle-Node Bifurcations

Abstract

The formations of transitional zones in shock wave, governed by Burgers’ equation, are studied from viewpoint of saddle-node bifurcations. First, the inviscid Burgers’ equation is studied in detail, the solution of the system with a certain smooth initial condition is obtained, and the solution in vector form is reduced into a Map in order to investigate the stability and bifurcation in the system. It is proved that there exists a thin spatial zone where a saddle-node bifurcation occurs in finite time, and the velocity of the fluid behaves as jumping, namely, the characteristic of shock wave. Further, the period-doubling bifurcation is captured, that means there exist multiple states as time increases, and the complicated spatio-temporal pattern is formatted. In addition to above, the viscous Burgers’ equation is further studied to extend to dissipative systems. By traveling wave transformation, the governing equation is reduced into an ordinary differential equation. More, the instability or bifurcation condition is obtained, and it is proved that there are three singular points in the system as the bifurcation condition is satisfied. The results show that the discontinuity resulting from saddle-node bifurcations is removed with the introduction of viscosity, and another kind of velocity change with strong gradient is obtained. However, the change of velocity is continuous with sharp slopes. As a conclusion, it can be drawn that all results can provide a fundamental understanding of the nonlinear phenomena relevant to shock wave and other complicated nonlinear phenomena, from viewpoint of nonlinear dynamics.