Riemann problem for longitudinal–torsional waves in nonlinear elastic rods

Undercompressive shocks and their role in solving Riemann problem are studied. Solutions to a special system of two hyperbolic equations representing conservation laws are investigated. On the one hand, this system of equations makes it possible to demonstrate the nonstandard solutions to the Riemann problem; on the other hand, this system of equations describes longitudinal–torsional waves in elastic rods. We use the traveling wave criterion for admissibility of shocks as the additional jump condition. If the dissipation parameters included in each of the equations of the system are different, then there are undercompressed waves.