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KdV waves in atomic chains with nonlocal interactions
1. | Westfälische Wilhelms-Universität Münster, Institut für Numerische und Angewandte Mathematik, Einsteinstraße 62, 48149 Münster, Germany |
2. | Technische Universität München, Zentrum Mathematik, Boltzmannstraße 3, 85747 Garching, Germany |
References:
[1] |
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. |
[2] |
G. Friesecke and A. Mikikits-Leitner, Cnoidal waves on Fermi-Pasta-Ulam lattices, 2014, To appear in J. Dyn. Diff. Equat., available via Springer online first. |
[3] |
G. Friesecke and R. L. Pego, Solitary waves on FPU lattices. I. Qualitative properties, renormalization and continuum limit, Nonlinearity, 12 (1999), 1601-1627.
doi: 10.1088/0951-7715/12/6/311. |
[4] |
G. Friesecke and R. L. Pego, Solitary waves on FPU lattices. II. Linear implies nonlinear stability, Nonlinearity, 15 (2002), 1343-1359.
doi: 10.1088/0951-7715/15/4/317. |
[5] |
G. Friesecke and R. L. Pego, Solitary waves on Fermi-Pasta-Ulam lattices. III. Howland-type Floquet theory, Nonlinearity, 17 (2004), 207-227.
doi: 10.1088/0951-7715/17/1/013. |
[6] |
G. Friesecke and R. L. Pego, Solitary waves on Fermi-Pasta-Ulam lattices. IV. Proof of stability at low energy, Nonlinearity, 17 (2004), 229-251.
doi: 10.1088/0951-7715/17/1/014. |
[7] |
J. Gaison, S. Moskow, J. D. Wright and Q. Zhang, Approximation of polyatomic FPU lattices by KdV equations, Multiscale Model. Simul., 12 (2014), 953-995.
doi: 10.1137/130941638. |
[8] |
M. Herrmann, Unimodal wavetrains and solitons in convex Fermi-Pasta-Ulam chains, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 753-785.
doi: 10.1017/S0308210509000146. |
[9] |
M. Herrmann, K. Matthies, H. Schwetlick and J. Zimmer, Subsonic phase transition waves in bistable lattice models with small spinodal region, SIAM J. Math. Anal., 45 (2013), 2625-2645.
doi: 10.1137/120877878. |
[10] |
M. Herrmann and J. D. M. Rademacher, Heteroclinic travelling waves in convex FPU-type chains, SIAM J. Math. Anal., 42 (2010), 1483-1504.
doi: 10.1137/080743147. |
[11] |
A. Hoffman and C. E. Wayne, Counter-propagating two-soliton solutions in the Fermi-Pasta-Ulam lattice, Nonlinearity, 21 (2008), 2911-2947.
doi: 10.1088/0951-7715/21/12/011. |
[12] |
A. Hoffman and C. E. Wayne, Asymptotic two-soliton solutions in the Fermi-Pasta-Ulam model, J. Dynam. Differential Equations, 21 (2009), 343-351.
doi: 10.1007/s10884-009-9134-9. |
[13] |
A. Hoffman and C. E. Wayne, A simple proof of the stability of solitary waves in the Fermi-Pasta-Ulam model near the KdV limit, in Infinite dimensional dynamical systems, vol. 64 of Fields Inst. Commun., Springer, New York, 2013, 185-192.
doi: 10.1007/978-1-4614-4523-4_7. |
[14] |
T. Mizumachi, $N$-soliton states of the Fermi-Pasta-Ulam lattices, SIAM J. Math. Anal., 43 (2011), 2170-2210.
doi: 10.1137/100792457. |
[15] |
T. Mizumachi, Asymptotic stability of $N$-solitary waves of the FPU lattices, Arch. Ration. Mech. Anal., 207 (2013), 393-457.
doi: 10.1007/s00205-012-0564-x. |
[16] |
P. M. Morse and H. Feshbach, Methods of Theoretical Physics. 2 volumes, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. |
[17] |
G. Schneider and C. E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model, in International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), World Sci. Publ., River Edge, NJ, 2000, 390-404. |
[18] |
N. J. Zabusky and M. D. Kruskal, Interaction of 'solitons' in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15 (1965), 240-243.
doi: 10.1103/PhysRevLett.15.240. |
show all references
References:
[1] |
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. |
[2] |
G. Friesecke and A. Mikikits-Leitner, Cnoidal waves on Fermi-Pasta-Ulam lattices, 2014, To appear in J. Dyn. Diff. Equat., available via Springer online first. |
[3] |
G. Friesecke and R. L. Pego, Solitary waves on FPU lattices. I. Qualitative properties, renormalization and continuum limit, Nonlinearity, 12 (1999), 1601-1627.
doi: 10.1088/0951-7715/12/6/311. |
[4] |
G. Friesecke and R. L. Pego, Solitary waves on FPU lattices. II. Linear implies nonlinear stability, Nonlinearity, 15 (2002), 1343-1359.
doi: 10.1088/0951-7715/15/4/317. |
[5] |
G. Friesecke and R. L. Pego, Solitary waves on Fermi-Pasta-Ulam lattices. III. Howland-type Floquet theory, Nonlinearity, 17 (2004), 207-227.
doi: 10.1088/0951-7715/17/1/013. |
[6] |
G. Friesecke and R. L. Pego, Solitary waves on Fermi-Pasta-Ulam lattices. IV. Proof of stability at low energy, Nonlinearity, 17 (2004), 229-251.
doi: 10.1088/0951-7715/17/1/014. |
[7] |
J. Gaison, S. Moskow, J. D. Wright and Q. Zhang, Approximation of polyatomic FPU lattices by KdV equations, Multiscale Model. Simul., 12 (2014), 953-995.
doi: 10.1137/130941638. |
[8] |
M. Herrmann, Unimodal wavetrains and solitons in convex Fermi-Pasta-Ulam chains, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 753-785.
doi: 10.1017/S0308210509000146. |
[9] |
M. Herrmann, K. Matthies, H. Schwetlick and J. Zimmer, Subsonic phase transition waves in bistable lattice models with small spinodal region, SIAM J. Math. Anal., 45 (2013), 2625-2645.
doi: 10.1137/120877878. |
[10] |
M. Herrmann and J. D. M. Rademacher, Heteroclinic travelling waves in convex FPU-type chains, SIAM J. Math. Anal., 42 (2010), 1483-1504.
doi: 10.1137/080743147. |
[11] |
A. Hoffman and C. E. Wayne, Counter-propagating two-soliton solutions in the Fermi-Pasta-Ulam lattice, Nonlinearity, 21 (2008), 2911-2947.
doi: 10.1088/0951-7715/21/12/011. |
[12] |
A. Hoffman and C. E. Wayne, Asymptotic two-soliton solutions in the Fermi-Pasta-Ulam model, J. Dynam. Differential Equations, 21 (2009), 343-351.
doi: 10.1007/s10884-009-9134-9. |
[13] |
A. Hoffman and C. E. Wayne, A simple proof of the stability of solitary waves in the Fermi-Pasta-Ulam model near the KdV limit, in Infinite dimensional dynamical systems, vol. 64 of Fields Inst. Commun., Springer, New York, 2013, 185-192.
doi: 10.1007/978-1-4614-4523-4_7. |
[14] |
T. Mizumachi, $N$-soliton states of the Fermi-Pasta-Ulam lattices, SIAM J. Math. Anal., 43 (2011), 2170-2210.
doi: 10.1137/100792457. |
[15] |
T. Mizumachi, Asymptotic stability of $N$-solitary waves of the FPU lattices, Arch. Ration. Mech. Anal., 207 (2013), 393-457.
doi: 10.1007/s00205-012-0564-x. |
[16] |
P. M. Morse and H. Feshbach, Methods of Theoretical Physics. 2 volumes, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. |
[17] |
G. Schneider and C. E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model, in International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), World Sci. Publ., River Edge, NJ, 2000, 390-404. |
[18] |
N. J. Zabusky and M. D. Kruskal, Interaction of 'solitons' in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15 (1965), 240-243.
doi: 10.1103/PhysRevLett.15.240. |
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