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Volumn , Issue , 2006, Pages 435-466

Continuum descriptions for the dynamics in discrete lattices: Derivation and justification

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EID: 38849166067     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/3-540-35657-6_16     Document Type: Chapter
Times cited : (14)

References (67)
  • 1
    • 0035202323 scopus 로고    scopus 로고
    • Dynamics of chains with non-monotone stress-strain relations. I. Model and numerical experiments. II. Nonlinear waves and waves of phase transition
    • 149-171
    • A. M. Balk, A. V. Cherkaev, and L. I. Slepyan. Dynamics of chains with non-monotone stress-strain relations. I. Model and numerical experiments. II. Nonlinear waves and waves of phase transition. J. Mech. Phys. Solids, 49(1), 131-148, 149-171, 2001.
    • (2001) J. Mech. Phys. Solids , vol.49 , Issue.1 , pp. 131-148
    • Balk, A.M.1    Cherkaev, A.V.2    Slepyan, L.I.3
  • 2
    • 0036477755 scopus 로고    scopus 로고
    • Continuum limits of discrete systems without convexity hypotheses
    • A. Braides and M. S. Gelli. Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids, 7, 41-66, 2002.
    • (2002) Math. Mech. Solids , vol.7 , pp. 41-66
    • Braides, A.1    Gelli, M.S.2
  • 3
    • 0037005605 scopus 로고    scopus 로고
    • Limits of discrete systems with longrange interactions
    • Special issue on optimization Montpellier
    • A. Braides and M. S. Gelli. Limits of discrete systems with longrange interactions. J. Convex Anal., 9, 363-399, 2002. Special issue on optimization (Montpellier, 2000).
    • (2000) J. Convex Anal. , vol.9 , pp. 363-399
    • Braides, A.1    Gelli, M.S.2
  • 4
    • 84892819685 scopus 로고    scopus 로고
    • Two-phase structures as singular limit of a one-dimensional discrete model
    • T. Blesgen. Two-phase structures as singular limit of a one-dimensional discrete model. Cubo, 7(2), 69-79, 2005.
    • (2005) Cubo , vol.7 , Issue.2 , pp. 69-79
    • Blesgen, T.1
  • 5
    • 1642326593 scopus 로고    scopus 로고
    • Macroscopic limits of microscopic systems
    • C. Boldrighini. Macroscopic limits of microscopic systems. Rend. Mat. Appl. (7), 16, 1-107, 1996.
    • (1996) Rend. Mat. Appl. , vol.16 , Issue.7 , pp. 1-107
    • Boldrighini, C.1
  • 7
    • 5444243818 scopus 로고    scopus 로고
    • Semiclassical asymptotics for weakly nonlinear Bloch waves
    • R. Carles, P. A. Markowich, and C. Sparber. Semiclassical asymptotics for weakly nonlinear Bloch waves. J. Statist. Phys., 117(1-2), 343-375, 2004.
    • (2004) J. Statist. Phys. , vol.117 , Issue.1-2 , pp. 343-375
    • Carles, R.1    Markowich, P.A.2    Sparber, C.3
  • 8
    • 30644464329 scopus 로고    scopus 로고
    • Micro-macro transition for the atomic chain via Whitham's modulation equation
    • W. Dreyer, M. Herrmann, and A. Mielke. Micro-macro transition for the atomic chain via Whitham's modulation equation. Nonlinearity, 19, 471-500, 2006.
    • (2006) Nonlinearity , vol.19 , pp. 471-500
    • Dreyer, W.1    Herrmann, M.2    Mielke, A.3
  • 10
    • 0039600839 scopus 로고    scopus 로고
    • Cold, thermal and oscillator closure of the atomic chain
    • W. Dreyer and M. Kunik. Cold, thermal and oscillator closure of the atomic chain. J. Phys. A, 33(10), 2097-2129, 2000.
    • (2000) J. Phys. A , vol.33 , Issue.10 , pp. 2097-2129
    • Dreyer, W.1    Kunik, M.2
  • 11
    • 84892799303 scopus 로고    scopus 로고
    • Corrigendum
    • (Corrigendum: J. Phys. A 33 (2000) 2458).
    • (2000) J. Phys. A , vol.33 , pp. 2458
  • 14
    • 0033872514 scopus 로고    scopus 로고
    • A scheme for the passage from atomic to continuum theory for thin films, nanotubes and nanorods
    • G. Friesecke and R. D. James. A scheme for the passage from atomic to continuum theory for thin films, nanotubes and nanorods. J. Mech. Phys. Solids, 48(6-7), 1519-1540, 2000.
    • (2000) J. Mech. Phys. Solids , vol.48 , Issue.6-7 , pp. 1519-1540
    • Friesecke, G.1    James, R.D.2
  • 15
    • 0036836750 scopus 로고    scopus 로고
    • Atomic-scale localization of high-energy solitary waves on lattices
    • G. Friesecke and K. Matthies. Atomic-scale localization of high-energy solitary waves on lattices. Physica D, 171, 211-220, 2002.
    • (2002) Physica D , vol.171 , pp. 211-220
    • Friesecke, G.1    Matthies, K.2
  • 17
    • 0033229031 scopus 로고    scopus 로고
    • Solitary waves on FPU lattices: I. Qualitative properties, renormalization and continuum limit
    • G. Friesecke and R. L. Pego. Solitary waves on FPU lattices: I. Qualitative properties, renormalization and continuum limit. Nonlinearity, 12, 1601-1627, 1999.
    • (1999) Nonlinearity , vol.12 , pp. 1601-1627
    • Friesecke, G.1    Pego, R.L.2
  • 18
    • 0010573332 scopus 로고    scopus 로고
    • Solitary waves on FPU lattices: II. Linear implies nonlinear stability
    • G. Friesecke and R. L. Pego. Solitary waves on FPU lattices: II. Linear implies nonlinear stability. Nonlinearity, 15, 1343-1359, 2002.
    • (2002) Nonlinearity , vol.15 , pp. 1343-1359
    • Friesecke, G.1    Pego, R.L.2
  • 19
    • 0346171294 scopus 로고    scopus 로고
    • Solitary waves on FPU lattices: III. Howland-type Floquet theory
    • G. Friesecke and R. L. Pego. Solitary waves on FPU lattices: III. Howland-type Floquet theory. Nonlinearity, 17, 207-227, 2004.
    • (2004) Nonlinearity , vol.17 , pp. 207-227
    • Friesecke, G.1    Pego, R.L.2
  • 20
    • 0346671222 scopus 로고    scopus 로고
    • Solitary waves on FPU lattices: IV. Proof of stability at low energy
    • G. Friesecke and R. L. Pego. Solitary waves on FPU lattices: IV. Proof of stability at low energy. Nonlinearity, 17, 229-251, 2004.
    • (2004) Nonlinearity , vol.17 , pp. 229-251
    • Friesecke, G.1    Pego, R.L.2
  • 21
    • 0003530086 scopus 로고
    • Studies of nonlinear problems
    • Los Alamos Scientific Laboratory of the University of California, May, Report LA-1940
    • E. Fermi, J. Pasta, and S. Ulam. Studies of nonlinear problems. Technical report, Los Alamos Scientific Laboratory of the University of California, May 1955. Report LA-1940.
    • (1955) Technical Report
    • Fermi, E.1    Pasta, J.2    Ulam, S.3
  • 22
    • 0042964912 scopus 로고    scopus 로고
    • Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice
    • G. Friesecke and F. Theil. Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice. J. Nonlin. Science, 12, 445-478, 2002.
    • (2002) J. Nonlin. Science , vol.12 , pp. 445-478
    • Friesecke, G.1    Theil, F.2
  • 23
    • 0033449491 scopus 로고    scopus 로고
    • Existence and modulation of traveling waves in particle chains
    • A.-M. Filip and S. Venakides. Existence and modulation of traveling waves in particle chains. Comm. Pure Appl. Math., 52(6), 693-735, 1999.
    • (1999) Comm. Pure Appl. Math. , vol.52 , Issue.6 , pp. 693-735
    • Filip, A.-M.1    Venakides, S.2
  • 24
    • 21344488602 scopus 로고
    • Existence theorem for solitary waves on lattices
    • G. Friesecke and J. A. D. Wattis. Existence theorem for solitary waves on lattices. Comm. Math. Phys., 161(2), 391-418, 1994.
    • (1994) Comm. Math. Phys. , vol.161 , Issue.2 , pp. 391-418
    • Friesecke, G.1    Wattis, J.A.D.2
  • 28
    • 1642318760 scopus 로고    scopus 로고
    • The nonlinear Schrödinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities
    • J. Giannoulis and A. Mielke. The nonlinear Schrödinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities. Nonlinearity, 17, 551-565, 2004.
    • (2004) Nonlinearity , vol.17 , pp. 551-565
    • Giannoulis, J.1    Mielke, A.2
  • 29
    • 33745240977 scopus 로고    scopus 로고
    • Dispersive evolution of pulses in oscillator chains with general interaction potentials
    • J. Giannoulis and A. Mielke. Dispersive evolution of pulses in oscillator chains with general interaction potentials. Discr. Cont. Dynam. Systems Ser. B, 6, 493-523, 2006.
    • (2006) Discr. Cont. Dynam. Systems Ser. B , vol.6 , pp. 493-523
    • Giannoulis, J.1    Mielke, A.2
  • 31
    • 0011569743 scopus 로고    scopus 로고
    • Erratum
    • Erratum: Comm. Pure Appl. Math. 53 (2000) 280-281.
    • (2000) Comm. Pure Appl. Math. , vol.53 , pp. 280-281
  • 34
  • 35
    • 17744372552 scopus 로고    scopus 로고
    • Localized waves in nonlinear oscillator chains
    • G. Iooss and G. James. Localized waves in nonlinear oscillator chains. Chaos, 15, 15 pp., 2005.
    • (2005) Chaos , vol.15 , pp. 15
    • Iooss, G.1    James, G.2
  • 36
    • 0034349011 scopus 로고    scopus 로고
    • Travelling waves in a chain of coupled nonlinear oscillators
    • G. Iooss and K. Kirchgässner. Travelling waves in a chain of coupled nonlinear oscillators. Comm. Math. Phys., 211(2), 439-464, 2000.
    • (2000) Comm. Math. Phys. , vol.211 , Issue.2 , pp. 439-464
    • Iooss, G.1    Kirchgässner, K.2
  • 37
    • 0034178527 scopus 로고    scopus 로고
    • Travelling waves in the Fermi-Pasta-Ulam lattice
    • G. Iooss. Travelling waves in the Fermi-Pasta-Ulam lattice. Nonlinearity, 13(3), 849-866, 2000.
    • (2000) Nonlinearity , vol.13 , Issue.3 , pp. 849-866
    • Iooss, G.1
  • 38
    • 15944404217 scopus 로고    scopus 로고
    • Dispersive properties of a viscous numerical scheme for the Schrödinger equation
    • L. Ignat and E. Zuazua. Dispersive properties of a viscous numerical scheme for the Schrödinger equation. C. R. Math. Acad. Sci. Paris, 340, 529-534, 2005.
    • (2005) C. R. Math. Acad. Sci. Paris , vol.340 , pp. 529-534
    • Ignat, L.1    Zuazua, E.2
  • 39
    • 84867931527 scopus 로고    scopus 로고
    • Centre manifold reduction for quasilinear discrete systems
    • G. James. Centre manifold reduction for quasilinear discrete systems. J. Nonlinear Sci., 13, 27-63, 2003.
    • (2003) J. Nonlinear Sci. , vol.13 , pp. 27-63
    • James, G.1
  • 40
    • 0000576966 scopus 로고    scopus 로고
    • Small-amplitude envelope solitons in nonlinear lattices
    • V. Konotop. Small-amplitude envelope solitons in nonlinear lattices. Phys. Rev. E, 53, 2843-2858, 1996.
    • (1996) Phys. Rev. E , vol.53 , pp. 2843-2858
    • Konotop, V.1
  • 41
    • 84971162773 scopus 로고
    • The validity of modulation equations for extended systems with cubic nonlinearities
    • P. Kirrmann, G. Schneider, and A. Mielke. The validity of modulation equations for extended systems with cubic nonlinearities. Proc. Roy. Soc. Edinburgh Sect. A, 122, 85-91, 1992.
    • (1992) Proc. Roy. Soc. Edinburgh Sect. A , vol.122 , pp. 85-91
    • Kirrmann, P.1    Schneider, G.2    Mielke, A.3
  • 43
    • 0040859457 scopus 로고
    • Proof of existence of breathers for timereversible or Hamiltonian networks of weakly coupled oscillators
    • R. S. MacKay and S. Aubry. Proof of existence of breathers for timereversible or Hamiltonian networks of weakly coupled oscillators. Nonlinearity, 7(6), 1623-1643, 1994.
    • (1994) Nonlinearity , vol.7 , Issue.6 , pp. 1623-1643
    • MacKay, R.S.1    Aubry, S.2
  • 44
    • 14944346675 scopus 로고    scopus 로고
    • Wigner measures in the discrete setting: High-frequency analysis of sampling and reconstruction operators
    • F. Macià. Wigner measures in the discrete setting: high-frequency analysis of sampling and reconstruction operators. SIAM J. Math. Anal., 36(2), 347-383, 2004.
    • (2004) SIAM J. Math. Anal. , vol.36 , Issue.2 , pp. 347-383
    • Macià, F.1
  • 46
    • 0012289535 scopus 로고
    • Hamiltonian and Lagrangian flows on center manifolds. With applications to elliptic variational problems
    • Springer-Verlag, Berlin
    • A. Mielke. Hamiltonian and Lagrangian flows on center manifolds. With applications to elliptic variational problems. Lecture Notes in Mathematics Vol. 1489. Springer-Verlag, Berlin, 1991.
    • (1991) Lecture Notes in Mathematics , vol.1489
    • Mielke, A.1
  • 47
    • 67649370945 scopus 로고    scopus 로고
    • The Ginzburg-Landau equation in its role as a modulation equation
    • B. Fiedler, editor, Elsevier Science B. V.
    • A. Mielke. The Ginzburg-Landau equation in its role as a modulation equation. In B. Fiedler, editor, Handbook of Dynamical Systems II, pages 759-834. Elsevier Science B. V., 2002.
    • (2002) Handbook of Dynamical Systems II , pp. 759-834
    • Mielke, A.1
  • 48
    • 33745585326 scopus 로고    scopus 로고
    • Macroscopic behavior of microscopic oscillations in harmonic lattices using Wigner-Husimi measures
    • A. Mielke. Macroscopic behavior of microscopic oscillations in harmonic lattices using Wigner-Husimi measures. Arch. Rat. Mech. Analysis, 181, 401-448, 2006.
    • (2006) Arch. Rat. Mech. Analysis , vol.181 , pp. 401-448
    • Mielke, A.1
  • 50
    • 36449006348 scopus 로고
    • A Wigner-function approach to (semi) classical limits: Electrons in a periodic potential
    • P. A. Markowich, N. J. Mauser, and F. Poupaud. A Wigner-function approach to (semi) classical limits: electrons in a periodic potential. J. Math. Phys., 35, 1066-1094, 1994.
    • (1994) J. Math. Phys. , vol.35 , pp. 1066-1094
    • Markowich, P.A.1    Mauser, N.J.2    Poupaud, F.3
  • 53
    • 0346483176 scopus 로고    scopus 로고
    • Traveling waves in lattice dynamical systems
    • A. A. Pankov and K. Pflüger. Traveling waves in lattice dynamical systems. Math. Meth. Appl. Sci., 23, 1223-1235, 2000.
    • (2000) Math. Meth. Appl. Sci. , vol.23 , pp. 1223-1235
    • Pankov, A.A.1    Pflüger, K.2
  • 54
    • 0001656215 scopus 로고
    • Error estimates for the Ginzburg-Landau approximation
    • G. Schneider. Error estimates for the Ginzburg-Landau approximation. Z. angew. Math. Phys., 45(3), 433-457, 1994.
    • (1994) Z. Angew. Math. Phys. , vol.45 , Issue.3 , pp. 433-457
    • Schneider, G.1
  • 55
    • 84985386714 scopus 로고
    • Validity and limitation of the Newell-Whitehead equation
    • G. Schneider. Validity and limitation of the Newell-Whitehead equation. Math. Nachr., 176, 249-263, 1995.
    • (1995) Math. Nachr. , vol.176 , pp. 249-263
    • Schneider, G.1
  • 56
    • 0030560044 scopus 로고    scopus 로고
    • Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation
    • G. Schneider. Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation. Comm. Math. Phys., 178(3), 679-702, 1996.
    • (1996) Comm. Math. Phys. , vol.178 , Issue.3 , pp. 679-702
    • Schneider, G.1
  • 57
    • 0007015954 scopus 로고    scopus 로고
    • Justification of modulation equations for hyperbolic systems via normal forms
    • G. Schneider. Justification of modulation equations for hyperbolic systems via normal forms. NoDEA Nonlinear Differential Equations Appl., 5(1), 69-82, 1998.
    • (1998) NoDEA Nonlinear Differential Equations Appl. , vol.5 , Issue.1 , pp. 69-82
    • Schneider, G.1
  • 58
    • 17744387967 scopus 로고    scopus 로고
    • Mathematical derivation of the continuum limit of the magnetic force between two parts of a rigid crystalline material
    • A. Schlömerkemper. Mathematical derivation of the continuum limit of the magnetic force between two parts of a rigid crystalline material. Arch. Ration. Mech. Anal., 176, 227-269, 2005.
    • (2005) Arch. Ration. Mech. Anal. , vol.176 , pp. 227-269
    • Schlömerkemper, A.1
  • 59
    • 24144491340 scopus 로고    scopus 로고
    • Justification and failure of the nonlinear Schrödinger equation in case of non-trivial quadratic resonances
    • G. Schneider. Justification and failure of the nonlinear Schrödinger equation in case of non-trivial quadratic resonances. J. Diff. Eqns., 216, 354-386, 2005.
    • (2005) J. Diff. Eqns. , vol.216 , pp. 354-386
    • Schneider, G.1
  • 60
    • 33745640744 scopus 로고    scopus 로고
    • Effective mass theorems for nonlinear schrödinger equations
    • C. Sparber. Effective mass theorems for nonlinear schrödinger equations. SIAM J. Appl. Math., 66, 820-842, 2006.
    • (2006) SIAM J. Appl. Math. , vol.66 , pp. 820-842
    • Sparber, C.1
  • 62
    • 33745629140 scopus 로고    scopus 로고
    • The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics
    • To appear arXiv: math-ph/0505025
    • H. Spohn. The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics. J. Stat. Physics, 2005. To appear (arXiv: math-ph/0505025).
    • (2005) J. Stat. Physics
    • Spohn, H.1
  • 63
    • 0001259642 scopus 로고    scopus 로고
    • Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model
    • B. Fiedler, K. Gröger, and J. Sprekels, editors, World Scientific
    • G. Schneider and C. E. Wayne. Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model. In B. Fiedler, K. Gröger, and J. Sprekels, editors, International Conference on Differential Equations, volume 1, pages 390-404. World Scientific, 2000.
    • (2000) International Conference on Differential Equations , vol.1 , pp. 390-404
    • Schneider, G.1    Wayne, C.E.2
  • 64
    • 0242458488 scopus 로고    scopus 로고
    • Estimates for the three-wave interaction of surface water waves
    • G. Schneider and C. E. Wayne. Estimates for the three-wave interaction of surface water waves. European J. Appl. Math., 14, 547-570, 2003.
    • (2003) European J. Appl. Math. , vol.14 , pp. 547-570
    • Schneider, G.1    Wayne, C.E.2
  • 65
    • 33745611597 scopus 로고    scopus 로고
    • Propagation of Wigner functions for the Schrödinger equation with a perturbed periodic potential
    • Birkhäuser Boston
    • S. Teufel and G. Panati. Propagation of Wigner functions for the Schrödinger equation with a perturbed periodic potential. In Multiscale methods in quantum mechanics, Trends Math., pages 207-220. Birkhäuser Boston, 2004.
    • (2004) Multiscale Methods in Quantum Mechanics, Trends Math. , pp. 207-220
    • Teufel, S.1    Panati, G.2
  • 66
    • 0004181419 scopus 로고
    • Wiley-InterscienceJohn Wiley & Sons, New York, Pure and Applied Mathematics
    • G. B. Whitham. Linear and nonlinear waves. Wiley-Interscience[John Wiley & Sons], New York, 1974. Pure and Applied Mathematics.
    • (1974) Linear and Nonlinear Waves
    • Whitham, G.B.1
  • 67
    • 21244472382 scopus 로고    scopus 로고
    • Propagation, observation, control and numerical approximation of waves approximated by finite difference method
    • E. Zuazua. Propagation, observation, control and numerical approximation of waves approximated by finite difference method. SIAM Review, 47, 197-243, 2005.
    • (2005) SIAM Review , vol.47 , pp. 197-243
    • Zuazua, E.1


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