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On the microscopic spacetime convexity principle of fully nonlinear parabolic equations I: Spacetime convex solutions
Justification of leading order quasicontinuum approximations of strongly nonlinear lattices
1. | Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-9305, United States |
2. | Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-9315 |
3. | Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany |
References:
[1] |
K. Ahnert and A. Pikovsky, Compactons and chaos in strongly nonlinear lattices, Phys. Rev. E, 79 (2009), 026209.
doi: 10.1103/PhysRevE.79.026209. |
[2] |
M. Chirilus-Bruckner, C. Chong, O. Prill and G. Schneider, Rigorous description of macroscopic wave packets in infinite periodic chains of coupled oscillators by modulation equations, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 879-901.
doi: 10.3934/dcdss.2012.5.879. |
[3] |
J. M. English and R. L. Pego, On the solitary wave pulse in a chain of beads, Proceedings of the AMS, 133 (2005), 1763-1768 (electronic).
doi: 10.1090/S0002-9939-05-07851-2. |
[4] |
G. Friesecke and J. A. D. Wattis, Existence theorem for solitary waves on lattices, Comm. Math. Phys., 161 (1994), 391-418.
doi: 10.1007/BF02099784. |
[5] |
J. Goodman and P. Lax, On dispersive difference schemes, Comm. Pure. Appl. Math., 41 (1988), 591-613.
doi: 10.1002/cpa.3160410506. |
[6] |
E. B. Herbold and V. F. Nesterenko, Shock wave structure in a strongly nonlinear lattice with viscous dissipation, Phys. Rev. E, 75 (2007), 021304.
doi: 10.1103/PhysRevE.75.021304. |
[7] |
M. Herrmann, Oscillatory waves in discrete scalar conservation laws, Math. Models Methods Appl. Sci., 22 (2012), 1150002.
doi: 10.1142/S021820251200585X. |
[8] |
M. Herrmann and J. D. M Rademacher, Riemann solvers and undercompressive shocks of convex FPU chains, Nonlinearity, 23 (2010), 277-304.
doi: 10.1088/0951-7715/23/2/004. |
[9] |
H. Hertz, On the contact of elastic solids, J. Reine Angew. Math., 92 (1881), 156-171. |
[10] |
T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205.
doi: 10.1007/BF00280740. |
[11] |
P. G. Kevrekidis, Non-linear waves in lattices: Past, present, future, IMA Journal of Applied Mathematics, 76 (2011), 389-423.
doi: 10.1093/imamat/hxr015. |
[12] |
P. G. Kevrekidis, I. G. Kevrekidis, A. R. Bishop and E. S. Titi, Continuum approach to discreteness, Phys. Rev. E, 65 (2002), 046613.
doi: 10.1103/PhysRevE.65.046613. |
[13] |
P. D. Lax, On dispersive difference schemes, Physica D, 18 (1986), 250-254.
doi: 10.1016/0167-2789(86)90185-5. |
[14] |
R. J. Leveque, Numerical Methods for Conservation Laws (Lectures in Mathematics), Birkhauser, 1992.
doi: 10.1007/978-3-0348-8629-1. |
[15] |
A. Molinari and C. Daraio, Stationary shocks in periodic highly nonlinear granular chains, Phys. Rev. E, 80 (2009), 056602.
doi: 10.1103/PhysRevE.80.056602. |
[16] |
V.F. Nesterenko, Dynamics of Heterogeneous Materials, Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4757-3524-6. |
[17] |
P. Rosenau, Dynamics of nonlinear mass-spring chains near the continuum limit, Physics Letters A, 118 (1986), 222-227.
doi: 10.1016/0375-9601(86)90170-2. |
[18] |
P. Rosenau, Dynamics of dense lattices, Phys. Rev. B, 36 (1987), 5868-5876.
doi: 10.1103/PhysRevB.36.5868. |
[19] |
G. Schneider, Bounds for the nonlinear Schrödinger approximation of the Fermi-Pasta-Ulam system, Appl. Anal., 89 (2010), 1523-1539.
doi: 10.1080/00036810903277150. |
[20] |
G. Schneider and C. E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model, International Conference on Differential Equations. Proceedings of the Conference, Equadiff '99, Berlin, Germany, (ed. B. Fiedler et al.), (1999). 1.Singapore: World Scientific. 390-404, (2000). |
[21] |
S. Sen, J. Hong, J. Bang, E. Avalos and R. Doney, Solitary waves in the granular chain, Physics Reports, 462 (2008), 21-66.
doi: 10.1016/j.physrep.2007.10.007. |
[22] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, Heidelberg, Berlin, 1983. |
[23] |
A. Stefanov and P. Kevrekidis, On the existence of solitary traveling waves for generalized hertzian chains, J. Nonlin. Sci., 22 (2012), 327-349.
doi: 10.1007/s00332-011-9119-9. |
[24] |
A. Stefanov and P. Kevrekidis, Traveling waves for monomer chains with precompression, Nonlinearity, 26 (2013), 539-564.
doi: 10.1088/0951-7715/26/2/539. |
show all references
References:
[1] |
K. Ahnert and A. Pikovsky, Compactons and chaos in strongly nonlinear lattices, Phys. Rev. E, 79 (2009), 026209.
doi: 10.1103/PhysRevE.79.026209. |
[2] |
M. Chirilus-Bruckner, C. Chong, O. Prill and G. Schneider, Rigorous description of macroscopic wave packets in infinite periodic chains of coupled oscillators by modulation equations, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 879-901.
doi: 10.3934/dcdss.2012.5.879. |
[3] |
J. M. English and R. L. Pego, On the solitary wave pulse in a chain of beads, Proceedings of the AMS, 133 (2005), 1763-1768 (electronic).
doi: 10.1090/S0002-9939-05-07851-2. |
[4] |
G. Friesecke and J. A. D. Wattis, Existence theorem for solitary waves on lattices, Comm. Math. Phys., 161 (1994), 391-418.
doi: 10.1007/BF02099784. |
[5] |
J. Goodman and P. Lax, On dispersive difference schemes, Comm. Pure. Appl. Math., 41 (1988), 591-613.
doi: 10.1002/cpa.3160410506. |
[6] |
E. B. Herbold and V. F. Nesterenko, Shock wave structure in a strongly nonlinear lattice with viscous dissipation, Phys. Rev. E, 75 (2007), 021304.
doi: 10.1103/PhysRevE.75.021304. |
[7] |
M. Herrmann, Oscillatory waves in discrete scalar conservation laws, Math. Models Methods Appl. Sci., 22 (2012), 1150002.
doi: 10.1142/S021820251200585X. |
[8] |
M. Herrmann and J. D. M Rademacher, Riemann solvers and undercompressive shocks of convex FPU chains, Nonlinearity, 23 (2010), 277-304.
doi: 10.1088/0951-7715/23/2/004. |
[9] |
H. Hertz, On the contact of elastic solids, J. Reine Angew. Math., 92 (1881), 156-171. |
[10] |
T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205.
doi: 10.1007/BF00280740. |
[11] |
P. G. Kevrekidis, Non-linear waves in lattices: Past, present, future, IMA Journal of Applied Mathematics, 76 (2011), 389-423.
doi: 10.1093/imamat/hxr015. |
[12] |
P. G. Kevrekidis, I. G. Kevrekidis, A. R. Bishop and E. S. Titi, Continuum approach to discreteness, Phys. Rev. E, 65 (2002), 046613.
doi: 10.1103/PhysRevE.65.046613. |
[13] |
P. D. Lax, On dispersive difference schemes, Physica D, 18 (1986), 250-254.
doi: 10.1016/0167-2789(86)90185-5. |
[14] |
R. J. Leveque, Numerical Methods for Conservation Laws (Lectures in Mathematics), Birkhauser, 1992.
doi: 10.1007/978-3-0348-8629-1. |
[15] |
A. Molinari and C. Daraio, Stationary shocks in periodic highly nonlinear granular chains, Phys. Rev. E, 80 (2009), 056602.
doi: 10.1103/PhysRevE.80.056602. |
[16] |
V.F. Nesterenko, Dynamics of Heterogeneous Materials, Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4757-3524-6. |
[17] |
P. Rosenau, Dynamics of nonlinear mass-spring chains near the continuum limit, Physics Letters A, 118 (1986), 222-227.
doi: 10.1016/0375-9601(86)90170-2. |
[18] |
P. Rosenau, Dynamics of dense lattices, Phys. Rev. B, 36 (1987), 5868-5876.
doi: 10.1103/PhysRevB.36.5868. |
[19] |
G. Schneider, Bounds for the nonlinear Schrödinger approximation of the Fermi-Pasta-Ulam system, Appl. Anal., 89 (2010), 1523-1539.
doi: 10.1080/00036810903277150. |
[20] |
G. Schneider and C. E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model, International Conference on Differential Equations. Proceedings of the Conference, Equadiff '99, Berlin, Germany, (ed. B. Fiedler et al.), (1999). 1.Singapore: World Scientific. 390-404, (2000). |
[21] |
S. Sen, J. Hong, J. Bang, E. Avalos and R. Doney, Solitary waves in the granular chain, Physics Reports, 462 (2008), 21-66.
doi: 10.1016/j.physrep.2007.10.007. |
[22] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, Heidelberg, Berlin, 1983. |
[23] |
A. Stefanov and P. Kevrekidis, On the existence of solitary traveling waves for generalized hertzian chains, J. Nonlin. Sci., 22 (2012), 327-349.
doi: 10.1007/s00332-011-9119-9. |
[24] |
A. Stefanov and P. Kevrekidis, Traveling waves for monomer chains with precompression, Nonlinearity, 26 (2013), 539-564.
doi: 10.1088/0951-7715/26/2/539. |
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