Travelling lattice waves in a toy model of Lennard-Jones interaction

Venney, Christine; Zimmer, Johannes

doi:10.1090/S0033-569X-2013-01320-4

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

Abstract | References | Similar Articles | Additional Information

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

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, 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; EDP Sciences, Les Ulis, 2011. MR 2759609

Gérard Iooss, Travelling waves in the Fermi-Pasta-Ulam lattice, Nonlinearity 13 (2000), no. 3, 849–866. MR 1759004, 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, 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, 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

Eric Lombardi, Oscillatory integrals and phenomena beyond all algebraic orders, Lecture Notes in Mathematics, vol. 1741, Springer-Verlag, Berlin, 2000. With applications to homoclinic orbits in reversible systems. MR 1770093

Percy D. Makita, Periodic and homoclinic travelling waves in infinite lattices, Nonlinear Anal. 74 (2011), no. 6, 2071–2086. MR 2781738, 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, 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