Infinitesimal group analysis for the Riemann system
Let r and s be the Riemann invariants [cf. \textit{A. Jeffrey}, Quasilinear hyperbolic systems and waves (1976; Zbl 0322.35060)] associated to the quasilinear hyperbolic system \[ (1)\quad U_ t+A(U)U_ x=0,\text{ where } U=\| \begin{matrix} u\\ v\end{matrix} \| \text{ and } A(U)=\| \begin{matrix} a_{11}\quad a_{12}\\ a_{21}\quad a_{22}\end{matrix} \|. \] Let \(\lambda\), \(\mu\) be the eigenvalues (wave speeds) of the matrix A. Then r and s satisfy the system of equations: \[ (2)\quad r_ t+\lambda (r,s)r_ x=0,\quad s_ t+\mu (r,s)s_ x=0. \] The author shows that, with no restrictions on \(\lambda\) and \(\mu\), the coefficients of the corresponding Lie group generator depend on two arbitrary functions of r and s. This fact is peculiar of the model (2); usually arbitrary functions of the independent variables x and t only can appear. The last section of the paper is devoted to an analysis of the completely exceptional case, i.e. when we have \((\partial \lambda /\partial U)\cdot d^{(\lambda)}=0=(\partial \mu /\partial U)\cdot d^{(\mu)}\), where \(d^{(\lambda)}\) and \(d^{(\mu)}\) are the right eigenvectors of A corresponding to \(\lambda\) and \(\mu\) [cf. \textit{P. G. Lax}, Commun. Pure Appl. Math. 10, 537--566 (1957; Zbl 0081.08803)].
scientific article; zbMATH DE number 4200764