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October  2011, 4(5): 975-994. doi: 10.3934/dcdss.2011.4.975

Asymptotics for supersonic traveling waves in the Morse lattice

1. 

Department of Mathematics, Southern Methodist University, Dallas TX 75275, United States

2. 

Department of Mathematics, ESFM-Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos Edificio 9, 07738 México D.F., Mexico

3. 

FENOMEC, Department of Mathematics and Mechanics, IIMAS-UNAM, Apdo. 20-726, 01000 México D.F., Mexico

Received  August 2009 Revised  December 2009 Published  December 2010

Supersonic traveling wave solutions to the Morse lattice are considered. By numerical means, we show that initial shock like initial values always evolve into traveling shock solutions after the emission of radiation traveling at the sound speed. Using a trial function which includes a shape modulation in the core of the shock we show how the Peierls-Nabarro self consistent potential induced by the lattice is canceled by the adjustment of the phase. We find excellent agreement between the modulation analysis and the numerical solutions.
Citation: Alejandro B. Aceves, Luis A. Cisneros-Ake, Antonmaria A. Minzoni. Asymptotics for supersonic traveling waves in the Morse lattice. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 975-994. doi: 10.3934/dcdss.2011.4.975
References:
[1]

A. A. Aigner, A. R. Champneys and V. M. Rothos, A new barrier to the existence of moving kinks in Frenkel-Kontorova lattices,, Physica D, 186 (2003), 148. doi: doi:10.1016/S0167-2789(03)00261-6.

[2]

O. M. Braun and Y. S. Kivshar, "The Frenkel-Kontorova Model, Concepts, Methods and Applications," Texts and Monographs in Physics,, Springer-Verlag, (2004).

[3]

L. A. Cisneros and A. A. Minzoni, Asymptotics for kink propagation in the discrete Sine-Gordon equation,, Physica D, 237 (2008), 50. doi: doi:10.1016/j.physd.2007.08.005.

[4]

L. A. Cisneros and A. A. Minzoni, Asymptotics for supersonic soliton propagation in the Toda lattice equation,, Studies in Applied Mathematics, 120 (2008), 333. doi: doi:10.1111/j.1467-9590.2008.00401.x.

[5]

M. Collins, A quasicontinuum approximation for solitons in an atomic chain,, Chem. Phys. Lett., 77 (1981), 342. doi: doi:10.1016/0009-2614(81)80161-3.

[6]

J. Dancz and S. A. Rice, Large amplitude vibrational motion in a one dimensional chain: Coherent state representation,, J. Chem. Phys., 67 (1977), 1418. doi: doi:10.1063/1.435015.

[7]

H. Dym and H. P. McKean, "Fourier Series and Integrals,", Probability and Mathematical Statistics, (1972).

[8]

J. C. Eilbeck and R. Flesch, Calculation of families of solitary waves on discrete lattices,, Phys. Lett. A, 149 (1990), 200. doi: doi:10.1016/0375-9601(90)90326-J.

[9]

E. Fermi, J. Pasta and S. Ulam, "Los Alamos Rpt LA-1940 (1955) Collected Papers of Enrico Fermi,", in Univ. of Chicago Press, II (1965).

[10]

N. Flytzanis, St. Pnevmatikos and M. Peyrard, Discrete lattice solitons: Properties and stability,, J. Phys. A: Math. Gen., 22 (1989), 783. doi: doi:10.1088/0305-4470/22/7/011.

[11]

G. Friesecke and R. L. Pego, Solitary waves on FPU lattices: I. Qualitative properties, renormalization and continuum limit,, Nonlinearity, 12 (1999), 1601. doi: doi:10.1088/0951-7715/12/6/311.

[12]

G. Friesecke and J. A. D. Wattis, Existence theorem for solitary waves on lattices,, Commun. Math. Phys., 161 (1994), 391. doi: doi:10.1007/BF02099784.

[13]

B. L. Holian, Shock waves in the Toda lattice: Analysis,, Phys. Rev. A, 24 (1981), 2595. doi: doi:10.1103/PhysRevA.24.2595.

[14]

B. L. Holian and G. K. Straub, Molecular dynamics of shock waves in one-dimensional chains,, Phys. Rev. B, 18 (1978), 1593. doi: doi:10.1103/PhysRevB.18.1593.

[15]

T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis and J. Cuevas, Radiationless traveling waves in saturable nonlinear Schroedinger lattices,, Phys. Rev. Lett., 97 (2006). doi: doi:10.1103/PhysRevLett.97.124101.

[16]

S. P. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov, "Theory of Solitons. The Inverse Scattering Transform,", Translated from the Russian. Contemporary Soviet Mathematics. Consultants Bureau [Plenum], (1984).

[17]

M. Peyrard and M. Kruskal, Kink dynamics in the highly discrete sine-Gordon system,, Physica D, 14 (1984), 88. doi: doi:10.1016/0167-2789(84)90006-X.

[18]

T. J. Rolfe, S. A. Rice and J. Dancz, A numerical study of large amplitude motion on a chain of coupled nonlinear oscillators,, J. Chem. Phys., 70 (1979), 26. doi: doi:10.1063/1.437242.

[19]

P. Rosenau, Dynamics of dense lattice,, Phys. Rev. B, 36 (1987), 5868. doi: doi:10.1103/PhysRevB.36.5868.

[20]

M. Toda, "Theory of Nonlinear Lattices,", 2nd edition, 20 (1989).

[21]

G. B. Whitham, "Linear and Nonlinear Waves,", Reprint of the 1974 original, (1974).

show all references

References:
[1]

A. A. Aigner, A. R. Champneys and V. M. Rothos, A new barrier to the existence of moving kinks in Frenkel-Kontorova lattices,, Physica D, 186 (2003), 148. doi: doi:10.1016/S0167-2789(03)00261-6.

[2]

O. M. Braun and Y. S. Kivshar, "The Frenkel-Kontorova Model, Concepts, Methods and Applications," Texts and Monographs in Physics,, Springer-Verlag, (2004).

[3]

L. A. Cisneros and A. A. Minzoni, Asymptotics for kink propagation in the discrete Sine-Gordon equation,, Physica D, 237 (2008), 50. doi: doi:10.1016/j.physd.2007.08.005.

[4]

L. A. Cisneros and A. A. Minzoni, Asymptotics for supersonic soliton propagation in the Toda lattice equation,, Studies in Applied Mathematics, 120 (2008), 333. doi: doi:10.1111/j.1467-9590.2008.00401.x.

[5]

M. Collins, A quasicontinuum approximation for solitons in an atomic chain,, Chem. Phys. Lett., 77 (1981), 342. doi: doi:10.1016/0009-2614(81)80161-3.

[6]

J. Dancz and S. A. Rice, Large amplitude vibrational motion in a one dimensional chain: Coherent state representation,, J. Chem. Phys., 67 (1977), 1418. doi: doi:10.1063/1.435015.

[7]

H. Dym and H. P. McKean, "Fourier Series and Integrals,", Probability and Mathematical Statistics, (1972).

[8]

J. C. Eilbeck and R. Flesch, Calculation of families of solitary waves on discrete lattices,, Phys. Lett. A, 149 (1990), 200. doi: doi:10.1016/0375-9601(90)90326-J.

[9]

E. Fermi, J. Pasta and S. Ulam, "Los Alamos Rpt LA-1940 (1955) Collected Papers of Enrico Fermi,", in Univ. of Chicago Press, II (1965).

[10]

N. Flytzanis, St. Pnevmatikos and M. Peyrard, Discrete lattice solitons: Properties and stability,, J. Phys. A: Math. Gen., 22 (1989), 783. doi: doi:10.1088/0305-4470/22/7/011.

[11]

G. Friesecke and R. L. Pego, Solitary waves on FPU lattices: I. Qualitative properties, renormalization and continuum limit,, Nonlinearity, 12 (1999), 1601. doi: doi:10.1088/0951-7715/12/6/311.

[12]

G. Friesecke and J. A. D. Wattis, Existence theorem for solitary waves on lattices,, Commun. Math. Phys., 161 (1994), 391. doi: doi:10.1007/BF02099784.

[13]

B. L. Holian, Shock waves in the Toda lattice: Analysis,, Phys. Rev. A, 24 (1981), 2595. doi: doi:10.1103/PhysRevA.24.2595.

[14]

B. L. Holian and G. K. Straub, Molecular dynamics of shock waves in one-dimensional chains,, Phys. Rev. B, 18 (1978), 1593. doi: doi:10.1103/PhysRevB.18.1593.

[15]

T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis and J. Cuevas, Radiationless traveling waves in saturable nonlinear Schroedinger lattices,, Phys. Rev. Lett., 97 (2006). doi: doi:10.1103/PhysRevLett.97.124101.

[16]

S. P. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov, "Theory of Solitons. The Inverse Scattering Transform,", Translated from the Russian. Contemporary Soviet Mathematics. Consultants Bureau [Plenum], (1984).

[17]

M. Peyrard and M. Kruskal, Kink dynamics in the highly discrete sine-Gordon system,, Physica D, 14 (1984), 88. doi: doi:10.1016/0167-2789(84)90006-X.

[18]

T. J. Rolfe, S. A. Rice and J. Dancz, A numerical study of large amplitude motion on a chain of coupled nonlinear oscillators,, J. Chem. Phys., 70 (1979), 26. doi: doi:10.1063/1.437242.

[19]

P. Rosenau, Dynamics of dense lattice,, Phys. Rev. B, 36 (1987), 5868. doi: doi:10.1103/PhysRevB.36.5868.

[20]

M. Toda, "Theory of Nonlinear Lattices,", 2nd edition, 20 (1989).

[21]

G. B. Whitham, "Linear and Nonlinear Waves,", Reprint of the 1974 original, (1974).

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