Asymptotic behavior of solutions of general system of nonlinear hyperbolic conservation laws

The paper is concerned with the Cauchy problem for a strictly hyperbolic system of nonlinear conservation laws, where the initial values of the solution are specified on an interval around the origin and on the semiinfinite intervals to the left and right of this interval. The solution is compared with the solution of the corresponding Riemann problem. An existence theorem is proved when the initial data consist of weak shock waves and a finite number of rarefaction waves. The notion of approximate conservation laws and generalized characteristic curves due to Glimm and Lax (1970) is explained. The system is shown to be uncoupled modulo the third order of the strength of shock waves in the initial data. The nature of the solution is investigated when the corresponding Riemann problem is solved by two centered rarefaction waves and by one shock and one rarefaction wave.