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Dispersive Shock Waves

  • Christopher ChongEmail author
  • Panayotis G. Kevrekidis
Chapter
Part of the SpringerBriefs in Physics book series (SpringerBriefs in Physics)

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.

References

  1. 1.
    M.J. Ablowitz, M. Hoefer, Dispersive shock waves. Scholarpedia 4, 5562 (2009)ADSGoogle Scholar
  2. 2.
    W.A. Strauss, Partial Differential Equations: An Introduction (Wiley, Hoboken, 2008)zbMATHGoogle Scholar
  3. 3.
    J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer-Verlag, New York, 1983)CrossRefzbMATHGoogle Scholar
  4. 4.
    P.G. Drazin, R.S. Johnson, Solitons: An Introduction (Cambridge University Press, Cambridge, UK, 1989)Google Scholar
  5. 5.
    G.A. El, M.A. Hoefer, M. Shearer, Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws. SIAM Review 59, 3–61 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    E. Fermi, J. Pasta, S. Ulam, Studies of Nonlinear Problems. I., (Los Alamos National Laboratory, Los Alamos, NM, USA), Technical Report (1955), pp. LA–1940Google Scholar
  7. 7.
    M. Herrmann, J.D.M. Rademacher, Riemann solvers and undercompressive shocks of convex FPU chains. Nonlinearity 23, 277 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    A. Molinari, C. Daraio, Stationary shocks in periodic highly nonlinear granular chains. Phys. Rev. E 80, 056602 (2009)ADSCrossRefGoogle Scholar
  9. 9.
    E.B. Herbold, V.F. Nesterenko, Shock wave structure in a strongly nonlinear lattice with viscous dissipation. Phys. Rev. E 75, 021304 (2007)ADSCrossRefGoogle Scholar
  10. 10.
    V.F. Nesterenko, Dynamics of Heterogeneous Materials (Springer-Verlag, New York, 2001)Google Scholar
  11. 11.
    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)ADSCrossRefGoogle Scholar
  12. 12.
    B.E. McDonald, D. Calvo, Simple waves in Hertzian chains. Phys. Rev. E 85, 066602 (2012)ADSCrossRefGoogle Scholar
  13. 13.
    Propagation of rarefaction pulses in discrete materials with strain-softening behavior. Phys. Rev. Lett. 110 144101 (2013)Google Scholar
  14. 14.
    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)CrossRefGoogle Scholar
  15. 15.
    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)Google Scholar
  16. 16.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    K. Atkinson, An Introduction to Numerical Analysis (Wiley, Hoboken, 1989)Google Scholar
  18. 18.
    R.J. LeVeque, Numerical Methods for Conservation Laws (Birkhauser, Basel, 1992)CrossRefzbMATHGoogle Scholar
  19. 19.
    S. Sen, J. Hong, J. Bang, E. Avalos, R. Doney, Solitary waves in the granular chain. Phys. Rep. 462, 21 (2008)Google Scholar
  20. 20.
    M. Toda, Theory of Nonlinear Lattices (Springer-Verlag, Heidelberg, 1989)Google Scholar
  21. 21.
    C.V. Turner, R.R. Rosales, The small dispersion limit for a nonlinear semidiscrete system of equations. Stud. Appl. Math. 99, 205 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Bowdoin CollegeBrunswickUSA
  2. 2.University of MassachusettsAmherstUSA

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