
-
Previous Article
Improved results for Klein-Gordon-Maxwell systems with general nonlinearity
- DCDS Home
- This Issue
-
Next Article
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. |
[2] |
F. Chen, Traveling waves in two-dimensional FPU lattices, PhD Thesis, Institute for Applied Mathematics, University of Münster, Germany, 2017. |
[3] |
E. Fermi, J. Pasta and S. Ulam, Studis on nonlinear problems, Los Alamos Scientific Laboraty Report, 1940. |
[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. |
[2] |
F. Chen, Traveling waves in two-dimensional FPU lattices, PhD Thesis, Institute for Applied Mathematics, University of Münster, Germany, 2017. |
[3] |
E. Fermi, J. Pasta and S. Ulam, Studis on nonlinear problems, Los Alamos Scientific Laboraty Report, 1940. |
[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. |















[1] |
Dmitry Treschev. Travelling waves in FPU lattices. Discrete and Continuous Dynamical Systems, 2004, 11 (4) : 867-880. doi: 10.3934/dcds.2004.11.867 |
[2] |
Cui-Ping Cheng, Wan-Tong Li, Zhi-Cheng Wang. Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 559-575. doi: 10.3934/dcdsb.2010.13.559 |
[3] |
Hiroshi Matano, Ken-Ichi Nakamura, Bendong Lou. Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit. Networks and Heterogeneous Media, 2006, 1 (4) : 537-568. doi: 10.3934/nhm.2006.1.537 |
[4] |
Shuichi Kawashima, Shinya Nishibata, Masataka Nishikawa. Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane. Conference Publications, 2003, 2003 (Special) : 469-476. doi: 10.3934/proc.2003.2003.469 |
[5] |
Sergey Zelik. Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 351-392. doi: 10.3934/dcds.2004.11.351 |
[6] |
Fumihiko Nakamura, Michael C. Mackey. Asymptotic (statistical) periodicity in two-dimensional maps. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021227 |
[7] |
Fang-Di Dong, Wan-Tong Li, Li Zhang. Entire solutions in a two-dimensional nonlocal lattice dynamical system. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2517-2545. doi: 10.3934/cpaa.2018120 |
[8] |
Jong-Shenq Guo, Chang-Hong Wu. Front propagation for a two-dimensional periodic monostable lattice dynamical system. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 197-223. doi: 10.3934/dcds.2010.26.197 |
[9] |
Lin Yang, Yejuan Wang, Peter E. Kloeden. Exponential attractors for two-dimensional nonlocal diffusion lattice systems with delay. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1811-1831. doi: 10.3934/cpaa.2022048 |
[10] |
J. W. Choi, D. S. Lee, S. H. Oh, S. M. Sun, S. I. Whang. Multi-hump solutions of some singularly-perturbed equations of KdV type. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 5181-5209. doi: 10.3934/dcds.2014.34.5181 |
[11] |
Gabriela Planas, Eduardo Hernández. Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1245-1258. doi: 10.3934/dcds.2008.21.1245 |
[12] |
Florian Kogelbauer. On the symmetry of spatially periodic two-dimensional water waves. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7057-7061. doi: 10.3934/dcds.2016107 |
[13] |
Zhaosheng Feng, Qingguo Meng. Exact solution for a two-dimensional KDV-Burgers-type equation with nonlinear terms of any order. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 285-291. doi: 10.3934/dcdsb.2007.7.285 |
[14] |
Juncheng Wei, Jun Yang. Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 465-508. doi: 10.3934/dcds.2008.22.465 |
[15] |
Yi He, Gongbao Li. Concentrating solitary waves for a class of singularly perturbed quasilinear Schrödinger equations with a general nonlinearity. Mathematical Control and Related Fields, 2016, 6 (4) : 551-593. doi: 10.3934/mcrf.2016016 |
[16] |
Mia Jukić, Hermen Jan Hupkes. Dynamics of curved travelling fronts for the discrete Allen-Cahn equation on a two-dimensional lattice. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3163-3209. doi: 10.3934/dcds.2020402 |
[17] |
Cui-Ping Cheng, Ruo-Fan An. Global stability of traveling wave fronts in a two-dimensional lattice dynamical system with global interaction. Electronic Research Archive, 2021, 29 (5) : 3535-3550. doi: 10.3934/era.2021051 |
[18] |
Bernhard Ruf, P. N. Srikanth. Hopf fibration and singularly perturbed elliptic equations. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 823-838. doi: 10.3934/dcdss.2014.7.823 |
[19] |
Ángela Jiménez-Casas, Aníbal Rodríguez-Bernal. Dynamic boundary conditions as limit of singularly perturbed parabolic problems. Conference Publications, 2011, 2011 (Special) : 737-746. doi: 10.3934/proc.2011.2011.737 |
[20] |
Tong Zhang, Yuxi Zheng. Exact spiral solutions of the two-dimensional Euler equations. Discrete and Continuous Dynamical Systems, 1997, 3 (1) : 117-133. doi: 10.3934/dcds.1997.3.117 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]