Travelling lattice waves in a toy model of Lennard-Jones interaction
Authors:
Christine R. Venney and Johannes Zimmer
Journal:
Quart. Appl. Math. 72 (2014), 65-84
MSC (2010):
Primary 37L10, 82C20
DOI:
https://doi.org/10.1090/S0033-569X-2013-01320-4
Published electronically:
November 13, 2013
MathSciNet review:
3185132
Full-text PDF Free Access
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Abstract: We consider an infinite lattice model, where particles interact with nearest neighbour (NN) and next-to-nearest neighbours (NNN); the NN and NNN springs act against each other to mimic the Lennard-Jones potential. The existence of subsonic waves homoclinic to exponentially small periodic oscillations is shown as well as the existence of supersonic periodic solutions. The proofs rely on methods from normal form and centre space analysis for the homoclinic solutions and centre manifold analysis for the periodic solutions.
References
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- Percy Makita. Subharmonics and homoclinics for a class of Hamiltonian-like equations. Technical Report 37/2009, Max-Planck Institut für Mathematik in den Naturwissenschaften, Leipzig, 2009.
- Alexander Mielke, Über maximale $L^p$-Regularität für Differentialgleichungen in Banach- und Hilbert-Räumen, Math. Ann. 277 (1987), no. 1, 121–133 (German). MR 884650, DOI https://doi.org/10.1007/BF01457282
- E. K. H. Salje. Multi-scaling and mesoscopic structures. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 368(1914):1163–1174, 2010.
- A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, Dynamics reported: expositions in dynamical systems, Dynam. Report. Expositions Dynam. Systems (N.S.), vol. 1, Springer, Berlin, 1992, pp. 125–163. MR 1153030
- Christine R. Venney and Johannes Zimmer, Persistence of supersonic periodic solutions for chains with anharmonic interaction potentials between neighbours and next to nearest neighbours, Dyn. Syst. 26 (2011), no. 4, 503–518. MR 2852938, DOI https://doi.org/10.1080/14689367.2011.620565
- Jonathan A. D. Wattis, Approximations to solitary waves on lattices. III. The monatomic lattice with second-neighbour interactions, J. Phys. A 29 (1996), no. 24, 8139–8157. MR 1446913, DOI https://doi.org/10.1088/0305-4470/29/24/035
References
- Renato Calleja and Yannick Sire, Travelling waves in discrete nonlinear systems with non-nearest neighbour interactions, Nonlinearity 22 (2009), no. 11, 2583–2605. MR 2550686 (2010j:82069), DOI https://doi.org/10.1088/0951-7715/22/11/001
- N. Flytzanis, S. Pnevmatikos, and M. Remoissenet. Kink, breather and asymmetric envelope or dark solitons in nonlinear chains. I. Monatomic chain. J. Phys. C Solid State, 18(24):4603, 1985.
- G. Gaeta, C. Reiss, M. Peyrard, and T. Dauxois. Simple models of non-linear DNA dynamics. La Rivista del Nuovo Cimento (1978-1999), 17(4):1–48, 1994.
- Mariana Haragus and Gérard Iooss, Local bifurcations, center manifolds, and normal forms in infinite-dimensional dynamical systems, Universitext, Springer-Verlag London Ltd., London, 2011. MR 2759609 (2012d:37193)
- Gérard Iooss, Travelling waves in the Fermi-Pasta-Ulam lattice, Nonlinearity 13 (2000), no. 3, 849–866. MR 1759004 (2002a:37114), DOI https://doi.org/10.1088/0951-7715/13/3/319
- Gérard Iooss and Guillaume James, Localized waves in nonlinear oscillator chains, Chaos 15 (2005), no. 1, 015113, 15. MR 2133464 (2005m:37179), DOI https://doi.org/10.1063/1.1836151
- Gérard Iooss and Klaus Kirchgässner, Travelling waves in a chain of coupled nonlinear oscillators, Comm. Math. Phys. 211 (2000), no. 2, 439–464. MR 1754524 (2002b:37124), DOI https://doi.org/10.1007/s002200050821
- Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452 (96a:47025)
- Eric Lombardi, Oscillatory integrals and phenomena beyond all algebraic orders. With applications to homoclinic orbits in reversible systems, Lecture Notes in Mathematics, vol. 1741, Springer-Verlag, Berlin, 2000. MR 1770093 (2002f:34081)
- Percy D. Makita, Periodic and homoclinic travelling waves in infinite lattices, Nonlinear Anal. 74 (2011), no. 6, 2071–2086. MR 2781738 (2012c:37160), DOI https://doi.org/10.1016/j.na.2010.11.011
- Percy Makita. Subharmonics and homoclinics for a class of Hamiltonian-like equations. Technical Report 37/2009, Max-Planck Institut für Mathematik in den Naturwissenschaften, Leipzig, 2009.
- Alexander Mielke, Über maximale $L^p$-Regularität für Differentialgleichungen in Banach- und Hilbert-Räumen, Math. Ann. 277 (1987), no. 1, 121–133 (German). MR 884650 (88i:34117), DOI https://doi.org/10.1007/BF01457282
- E. K. H. Salje. Multi-scaling and mesoscopic structures. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 368(1914):1163–1174, 2010.
- A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, Dynamics reported: expositions in dynamical systems, Dynam. Report. Expositions Dynam. Systems (N.S.), vol. 1, Springer, Berlin, 1992, pp. 125–163. MR 1153030 (93f:58174)
- Christine R. Venney and Johannes Zimmer, Persistence of supersonic periodic solutions for chains with anharmonic interaction potentials between neighbours and next to nearest neighbours, Dyn. Syst. 26 (2011), no. 4, 503–518. MR 2852938
- Jonathan A. D. Wattis, Approximations to solitary waves on lattices. III. The monatomic lattice with second-neighbour interactions, J. Phys. A 29 (1996), no. 24, 8139–8157. MR 1446913 (98c:82048), DOI https://doi.org/10.1088/0305-4470/29/24/035
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Additional Information
Christine R. Venney
Affiliation:
Dept. of Mathematical Sciences, University of Bath, Bath, U. K.
Johannes Zimmer
Affiliation:
Dept. of Mathematical Sciences, University of Bath, Bath, U. K.
MR Author ID:
708969
ORCID:
0000-0002-7606-2422
Received by editor(s):
February 25, 2012
Published electronically:
November 13, 2013
Article copyright:
© Copyright 2013
Brown University