Abstract:
Solutions of a nonlinear hyperbolic system of equations describing weakly nonlinear quasitransverse waves in a weakly anisotropic elastic medium are studied. The influence of small-scale processes of dissipation and dispersion is investigated. The small-scale processes determine the structure of discontinuities (shocks) and a set of discontinuities with a stationary structure. Among the discontinuities with a stationary structure, there are special ones that, in addition to relations following from conservation laws, satisfy additional relations required for the existence of their structure. In the phase plane, the structure of such discontinuities is represented by an integral curve joining two saddles. Special discontinuities lead to nonunique self-similar solutions of the Riemann problem. Asymptotics of non-self-similar problems for equations with dissipation and dispersion are found numerically. These asymptotics correspond to self-similar solutions of the problems.
Key words:
special discontinuities, generalized KdV-Burgers equation, self-similar Riemann problem, nonuniqueness of solutions.
This publication is cited in the following 12 articles:
A. P. Chugainova, “Special discontinuities depending on dispersion processes”, Application of Mathematics in Technical and Natural Sciences (AMITANS 2020), AIP Conf. Proc., 2302, ed. M. Todorov, Amer. Inst. Phys., 2020, 100002
A. P. Chugainova, A. G. Kulikovskii, “Longitudinal and torsional shock waves in anisotropic elastic cylinders”, Z. Angew. Math. Phys., 71:1 (2020), 17
A. P. Chugainova, A. T. Il'ichev, V. A. Shargatov, “Stability of shock wave structures in nonlinear elastic media”, Math. Mech. Solids, 24:11 (2019), 3456–3471
A.P. Chugainova, V.A. Shargatov, “Analytical description of the structure of special discontinuities described by a generalized KdV–Burgers equation”, Communications in Nonlinear Science and Numerical Simulation, 66 (2019), 129
A. P. Chugainova, V. A. Shargatov, “Study of nonstationary solutions of a generalized Korteweg-de Vries-Burgers equation”, AIP Conf. Proc., 2164 (2019), 50002–8
A. G. Kulikovskii, A. P. Chugainova, “Shock waves in anisotropic cylinders”, Proc. Steklov Inst. Math., 300 (2018), 100–113
A. G. Kulikovskii, E. I. Sveshnikova, “Problem of the motion of an elastic medium formed at the solidification front”, Proc. Steklov Inst. Math., 300 (2018), 86–99
V. A. Shargatov, A. P. Chugainova, S. V. Gorkunov, S. I. Sumskoi, “Flow structure behind a shock wave in a channel with periodically arranged obstacles”, Proc. Steklov Inst. Math., 300 (2018), 206–218
Chugaynova A., 14Th International Conference on Vibration Engineering and Technology of Machinery (Vetomac Xiv), Matec Web of Conferences, 211, eds. Maia N., Dimitrovova Z., E D P Sciences, 2018
Chugainova A.P., Shargatov V.A., Gorkunov S.V., Sumskoi S.I., “Regimes of Shock Wave Propagation Through Comb-Shaped Obstacles”, AIP Conference Proceedings, 2025, ed. Todorov M., Amer Inst Physics, 2018, 080002-1
V. A. Shargatov, A. P. Chugainova, S. V. Gorkunov, S. I. Sumskoi, “Analytical and numerical solutions of the shock tube problem in a channel with a pseudo-perforated wall”, JPCS, 1099 (2018), 12013–8
A. G. Kulikovskii, A. P. Chugainova, “Long nonlinear waves in anisotropic cylinders”, Comput. Math. Math. Phys., 57:7 (2017), 1194–1200
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