Generalized characteristic analysis and Guckenheimer structure
The authors study the two-dimensional scalar conservation law \(u_t+f(u)_x+g(u)_y=0\) and by transformation the Riemann problem corresponding to \(-\xi u_{\xi}-\eta u_{\eta}+f(u)_{\xi}+g(u)_{\eta}=0\) with \(\lim_{r \rightarrow \infty}u(\xi,\eta)=u_0(\theta)\), where \((r,\theta)\) are the polar coordinates of \((\xi,\eta)\). The authors develop a generalized characteristic analysis method and study the intersections of elementary waves and construct the Riemann solutions for the conservation law. Finally, necessary and sufficient conditions for the appearance of Guckenheimer structure is given. This structure was shown by Guckenheimer to arise in the example \(f(u)=u^2/2\), \(g(u)=u^3/3\) and \(u_0\) is \(1,-1\) or \(0\), and is similar to Mach reflection in gas dynamics.
scientific article; zbMATH DE number 1283670