Abstract
Shock waves are nonlinear structures characterized by a discontinuous jump in the wave profile. There exist prototypical examples of partial differential equations (PDEs), including the well-known Burgers’ equation (without dissipation), where the relevant waveforms feature infinite derivatives arising in finite time. To study shock waves in equations like the inviscid Burgers’ equations one can study the evolution of Riemann initial conditions involving a jump in the initial data.
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References
M.J. Ablowitz, M. Hoefer, Dispersive shock waves. Scholarpedia 4, 5562 (2009)
W.A. Strauss, Partial Differential Equations: An Introduction (Wiley, Hoboken, 2008)
J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer-Verlag, New York, 1983)
P.G. Drazin, R.S. Johnson, Solitons: An Introduction (Cambridge University Press, Cambridge, UK, 1989)
G.A. El, M.A. Hoefer, M. Shearer, Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws. SIAM Review 59, 3–61 (2017)
E. Fermi, J. Pasta, S. Ulam, Studies of Nonlinear Problems. I., (Los Alamos National Laboratory, Los Alamos, NM, USA), Technical Report (1955), pp. LA–1940
M. Herrmann, J.D.M. Rademacher, Riemann solvers and undercompressive shocks of convex FPU chains. Nonlinearity 23, 277 (2010)
A. Molinari, C. Daraio, Stationary shocks in periodic highly nonlinear granular chains. Phys. Rev. E 80, 056602 (2009)
E.B. Herbold, V.F. Nesterenko, Shock wave structure in a strongly nonlinear lattice with viscous dissipation. Phys. Rev. E 75, 021304 (2007)
V.F. Nesterenko, Dynamics of Heterogeneous Materials (Springer-Verlag, New York, 2001)
H. Yasuda, C. Chong, J. Yang, P.G. Kevrekidis, Emergence of dispersive shocks and rarefaction waves in power-law contact models. Phys. Rev. E 95, 062216 (2017)
B.E. McDonald, D. Calvo, Simple waves in Hertzian chains. Phys. Rev. E 85, 066602 (2012)
Propagation of rarefaction pulses in discrete materials with strain-softening behavior. Phys. Rev. Lett. 110 144101 (2013)
F. Fraternali, G. Carpentieri, A. Amendola, R.E. Skelton, V.F. Nesterenko, Multiscale tunability of solitary wave dynamics in tensegrity metamaterials. Appl. Phys. Lett. 105, 201903 (2014)
H. Yasuda, C. Chong, E.G. Charalampidis, P.G. Kevrekidis, J. Yang, Formation of rarefaction waves in origami-based metamaterials. Phys. Rev. E 93, 043004 (2016)
C. Chong, P.G. Kevrekidis, G. Schneider, Justification of leading order quasicontinuum approximations of strongly nonlinear lattices. Disc. Cont. Dyn. Sys. A 34, 3403 (2014)
K. Atkinson, An Introduction to Numerical Analysis (Wiley, Hoboken, 1989)
R.J. LeVeque, Numerical Methods for Conservation Laws (Birkhauser, Basel, 1992)
S. Sen, J. Hong, J. Bang, E. Avalos, R. Doney, Solitary waves in the granular chain. Phys. Rep. 462, 21 (2008)
M. Toda, Theory of Nonlinear Lattices (Springer-Verlag, Heidelberg, 1989)
C.V. Turner, R.R. Rosales, The small dispersion limit for a nonlinear semidiscrete system of equations. Stud. Appl. Math. 99, 205 (1997)
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Chong, C., Kevrekidis, P.G. (2018). Dispersive Shock Waves. In: Coherent Structures in Granular Crystals. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-77884-6_2
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DOI: https://doi.org/10.1007/978-3-319-77884-6_2
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