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Dichotomy spectrum and almost topological conjugacy on nonautonomus unbounded difference systems
KdV-like solitary waves in two-dimensional FPU-lattices
1. | University of Münster, Institute for Analysis and Numerics, Einsteinstr. 62, 48149 Münster, Germany |
2. | Technische Universität Braunschweig, Institute for Computational Mathematics, Universitätsplatz 2, 38106 Braunschweig, Germany |
We prove the existence of solitary waves in the KdV limit of two-dimensional FPU-type lattices using asymptotic analysis of nonlinear and singularly perturbed integral equations. In particular, we generalize the existing results by Friesecke and Matthies since we allow for arbitrary propagation directions and non-unidirectional wave profiles.
References:
[1] |
F. Chen, Wandernde Wellen in FPU-Gittern, Master Thesis, Institute for Mathematics, Saarland University, Germany, 2013. Google Scholar |
[2] |
F. Chen, Traveling waves in two-dimensional FPU lattices, PhD Thesis, Institute for Applied Mathematics, University of Münster, Germany, 2017. Google Scholar |
[3] |
E. Fermi, J. Pasta and S. Ulam, Studis on nonlinear problems, Los Alamos Scientific Laboraty Report, 1940. Google Scholar |
[4] |
A.-M. Filip and S. Venakides,
Existence and modulation of traveling waves in particle chains, Comm. Pure Appl. Math., 52 (1999), 693-735.
doi: 10.1002/(SICI)1097-0312(199906)52:6<693::AID-CPA2>3.0.CO;2-9. |
[5] |
G. Friesecke and K. Matthies,
Geometric solitary waves in a 2d mass spring lattice, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 105-114.
|
[6] |
G. Friesecke and A. Mikikits-Leitner,
Cnoidal waves on Fermi-Pasta-Ulam lattices, J. Dynam. Differential Equations, 27 (2015), 627-652.
doi: 10.1007/s10884-013-9343-0. |
[7] |
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. |
[8] |
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. |
[9] |
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. |
[10] |
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. |
[11] |
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. |
[12] |
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. |
[13] |
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. |
[14] |
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. |
[15] |
M. Herrmann and A. Mikikits-Leitner,
KdV waves in atomic chains with nonlocal interactions, Discrete Contin. Dyn. Syst., 36 (2016), 2047-2067.
|
[16] |
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. |
[17] |
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. |
[18] |
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. |
[19] |
A. Hoffman and C. E. Wayne, A simple proof of the stability of solitary waves in the FermiPasta-Ulam model near the KdV limit, in Infinite dimensional dynamical systems, vol. 64 of
Fields Inst. Commun., Springer, New York, 2013,185–192. |
[20] |
A. Hoffman and J. D. Wright,
Nanopteron solutions of diatomic Fermi-Pasta-Ulam-Tsingou lattices with small mass-ratio, Phys. D, 358 (2017), 33-59.
doi: 10.1016/j.physd.2017.07.004. |
[21] |
G. Iooss,
Travelling waves in the Fermi-Pasta-Ulam lattice, Nonlinearity, 13 (2000), 849-866.
doi: 10.1088/0951-7715/13/3/319. |
[22] |
A. Pankov,
Traveling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices, Imperial College Press, London, 2005. |
[23] |
G. Schneider and C. E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit for the Fermi-Pasta-Ulam model, in International Conference on Differential
Equations, vol. 1, World Scientific, 2000,390-404. |
[24] |
H. Schwetlick and J. Zimmer,
Kinetic relations for a lattice model of phase transitions, Arch. Rational Mech. Anal., 206 (2012), 707-724.
doi: 10.1007/s00205-012-0566-8. |
[25] |
D. Smets and M. Willem,
Solitary waves with prescribed speed on infinite lattices, J. Funct. Anal., 149 (1997), 266-275.
doi: 10.1006/jfan.1996.3121. |
[26] |
L. Truskinovsky and A. Vainchtein,
Kinetics of martensitic phase transitions: Lattice model, SIAM J. Appl. Math., 66 (2005), 533-553.
doi: 10.1137/040616942. |
[27] |
A. Vainchtein, Y. Starosvetsky, J. Wright and R. Perline, Solitary waves in diatomic chains,
Phys. Rev. E, 93 (2016), 042210.
doi: 10.1103/PhysRevE.93.042210. |
[28] |
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] |
F. Chen, Wandernde Wellen in FPU-Gittern, Master Thesis, Institute for Mathematics, Saarland University, Germany, 2013. Google Scholar |
[2] |
F. Chen, Traveling waves in two-dimensional FPU lattices, PhD Thesis, Institute for Applied Mathematics, University of Münster, Germany, 2017. Google Scholar |
[3] |
E. Fermi, J. Pasta and S. Ulam, Studis on nonlinear problems, Los Alamos Scientific Laboraty Report, 1940. Google Scholar |
[4] |
A.-M. Filip and S. Venakides,
Existence and modulation of traveling waves in particle chains, Comm. Pure Appl. Math., 52 (1999), 693-735.
doi: 10.1002/(SICI)1097-0312(199906)52:6<693::AID-CPA2>3.0.CO;2-9. |
[5] |
G. Friesecke and K. Matthies,
Geometric solitary waves in a 2d mass spring lattice, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 105-114.
|
[6] |
G. Friesecke and A. Mikikits-Leitner,
Cnoidal waves on Fermi-Pasta-Ulam lattices, J. Dynam. Differential Equations, 27 (2015), 627-652.
doi: 10.1007/s10884-013-9343-0. |
[7] |
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. |
[8] |
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. |
[9] |
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. |
[10] |
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. |
[11] |
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. |
[12] |
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. |
[13] |
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. |
[14] |
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. |
[15] |
M. Herrmann and A. Mikikits-Leitner,
KdV waves in atomic chains with nonlocal interactions, Discrete Contin. Dyn. Syst., 36 (2016), 2047-2067.
|
[16] |
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. |
[17] |
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. |
[18] |
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. |
[19] |
A. Hoffman and C. E. Wayne, A simple proof of the stability of solitary waves in the FermiPasta-Ulam model near the KdV limit, in Infinite dimensional dynamical systems, vol. 64 of
Fields Inst. Commun., Springer, New York, 2013,185–192. |
[20] |
A. Hoffman and J. D. Wright,
Nanopteron solutions of diatomic Fermi-Pasta-Ulam-Tsingou lattices with small mass-ratio, Phys. D, 358 (2017), 33-59.
doi: 10.1016/j.physd.2017.07.004. |
[21] |
G. Iooss,
Travelling waves in the Fermi-Pasta-Ulam lattice, Nonlinearity, 13 (2000), 849-866.
doi: 10.1088/0951-7715/13/3/319. |
[22] |
A. Pankov,
Traveling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices, Imperial College Press, London, 2005. |
[23] |
G. Schneider and C. E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit for the Fermi-Pasta-Ulam model, in International Conference on Differential
Equations, vol. 1, World Scientific, 2000,390-404. |
[24] |
H. Schwetlick and J. Zimmer,
Kinetic relations for a lattice model of phase transitions, Arch. Rational Mech. Anal., 206 (2012), 707-724.
doi: 10.1007/s00205-012-0566-8. |
[25] |
D. Smets and M. Willem,
Solitary waves with prescribed speed on infinite lattices, J. Funct. Anal., 149 (1997), 266-275.
doi: 10.1006/jfan.1996.3121. |
[26] |
L. Truskinovsky and A. Vainchtein,
Kinetics of martensitic phase transitions: Lattice model, SIAM J. Appl. Math., 66 (2005), 533-553.
doi: 10.1137/040616942. |
[27] |
A. Vainchtein, Y. Starosvetsky, J. Wright and R. Perline, Solitary waves in diatomic chains,
Phys. Rev. E, 93 (2016), 042210.
doi: 10.1103/PhysRevE.93.042210. |
[28] |
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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