Saddle-node bifurcations in burgers equation as shock wave occurrence

Abstract

The nature of the shock wave, governed by Burgers equation, is presented from viewpoint of saddle-node bifurcations. First, the inviscid Burgers equation is studied in detail, and the solution of the system with a certain initial condition is obtained. The solution in vector form is reduced into a Map in order to investigate the stability and bifurcation in the system. It has been proved that there exists a thin spatial zone where a saddle-node bifurcation occurs as time goes on, 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. Then, 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. More, the velocity distribution is obtained with a certain initial condition. The results show that the discontinuity resulting from saddle-node bifurcation is removed with the introduction of viscousity, and another kind of velocity change with strong gradient is obtained. However, the change of velocity is continuous with sharp slopes. As a conclusion, 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.