On structures of discontinuities in solutions of hyperbolic systems of equations. Waves in elastic rods. Special discontinuities.

Stationary discontinuity structures in solutions of a hyperbolic system of equations of a rather general form are considered under the assumption that viscosity is the main controlling mechanism inside the structure. Some segments of the shock adiabat are shown to correspond to evolutionary discontinuities without structure. It is also shown that there are special discontinuities on which an additional relation must hold, which arises from the condition that a discontinuity structure exists. Special discontinuities satisfy evolutionary conditions that differ from the well-known Lax conditions. Conclusions are discussed, which can also be of interest in the case of other systems of hyperbolic equations.
[1] A. G. Kulikovskii, A. P. Chugainova, "Discontinuity structures of equation solutions describing longitudinal-torsional waves in elastic rods", Doklady RAN. Fiz. Tekhn. Nauki, 2021, 497:1, 49–52
[2] A. G. Kulikovskii, A. P. Chugainova, "Structures of non-classical discontinuities in solutions of hyperbolic systems of equations", Uspekhi Mat. Nauk, 2022, 77:1(463), 55–90

References

1. A. G. Kulikovskii, A. P. Chugainova, “Struktury razryvov v resheniyakh uravnenii, opisyvayuschikh prodolno-krutilnye volny v uprugikh sterzhnyakh”, Doklady RAN. Fiz. tekhn. nauki, 497:1 (2021), 49–52; A. G. Kulikovskii, A. P. Chugainova, “Discontinuity Structures of Solutions to Equations Describing Longitudinal–Torsional Waves in Elastic Rods”, Doklady Physics, 66 (2021), 110–113  
2. A. G. Kulikovskii, A. P. Chugainova, “O strukturakh neklassicheskikh razryvov v resheniyakh giperbolicheskikh sistem uravnenii”, UMN, 77:1(463) (2022), 55–90      ; A. G. Kulikovskii, A. P. Chugainova, “Structures of non-classical discontinuities in solutions of hyperbolic systems of equations”, Russian Math. Surveys, 77:1 (2022), 47–79