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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

Alejandro B. Aceves 1, , Luis A. Cisneros-Ake 2, and  Antonmaria A. Minzoni 3,

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.
Keywords: Modulation averaged Lagrangian, Peierls-Nabarro potential, Supersonic traveling waves..
Mathematics Subject Classification: Primary: 70K75, 74J30, 74J3.
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), 148-170. doi: doi:10.1016/S0167-2789(03)00261-6.  Google Scholar

[2]

O. M. Braun and Y. S. Kivshar, "The Frenkel-Kontorova Model, Concepts, Methods and Applications," Texts and Monographs in Physics, Springer-Verlag, Berlin-Heidelberg New York, 2004.  Google Scholar

[3]

L. A. Cisneros and A. A. Minzoni, Asymptotics for kink propagation in the discrete Sine-Gordon equation, Physica D, 237 (2008), 50-65. doi: doi:10.1016/j.physd.2007.08.005.  Google Scholar

[4]

L. A. Cisneros and A. A. Minzoni, Asymptotics for supersonic soliton propagation in the Toda lattice equation, Studies in Applied Mathematics, 120 (2008), 333-349. doi: doi:10.1111/j.1467-9590.2008.00401.x.  Google Scholar

[5]

M. Collins, A quasicontinuum approximation for solitons in an atomic chain, Chem. Phys. Lett., 77 (1981), 342-347. doi: doi:10.1016/0009-2614(81)80161-3.  Google Scholar

[6]

J. Dancz and S. A. Rice, Large amplitude vibrational motion in a one dimensional chain: Coherent state representation, J. Chem. Phys., 67 (1977), 1418-1426. doi: doi:10.1063/1.435015.  Google Scholar

[7]

H. Dym and H. P. McKean, "Fourier Series and Integrals," Probability and Mathematical Statistics, No. 14, Academic Press, New York-London, 1972.  Google Scholar

[8]

J. C. Eilbeck and R. Flesch, Calculation of families of solitary waves on discrete lattices, Phys. Lett. A, 149 (1990), 200-202. doi: doi:10.1016/0375-9601(90)90326-J.  Google Scholar

[9]

E. Fermi, J. Pasta and S. Ulam, "Los Alamos Rpt LA-1940 (1955) Collected Papers of Enrico Fermi," in Univ. of Chicago Press, Vol. II, Chicago, (1965), 978. Google Scholar

[10]

N. Flytzanis, St. Pnevmatikos and M. Peyrard, Discrete lattice solitons: Properties and stability, J. Phys. A: Math. Gen., 22 (1989), 783-801. doi: doi:10.1088/0305-4470/22/7/011.  Google Scholar

[11]

G. Friesecke and R. L. Pego, Solitary waves on FPU lattices: I. Qualitative properties, renormalization and continuum limit, Nonlinearity, 12 (1999), 1601-1627. doi: doi:10.1088/0951-7715/12/6/311.  Google Scholar

[12]

G. Friesecke and J. A. D. Wattis, Existence theorem for solitary waves on lattices, Commun. Math. Phys., 161 (1994), 391-418. doi: doi:10.1007/BF02099784.  Google Scholar

[13]

B. L. Holian, Shock waves in the Toda lattice: Analysis, Phys. Rev. A, 24 (1981), 2595-2623. doi: doi:10.1103/PhysRevA.24.2595.  Google Scholar

[14]

B. L. Holian and G. K. Straub, Molecular dynamics of shock waves in one-dimensional chains, Phys. Rev. B, 18 (1978), 1593-1608. doi: doi:10.1103/PhysRevB.18.1593.  Google Scholar

[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), 124101. doi: doi:10.1103/PhysRevLett.97.124101.  Google Scholar

[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], New York, 1984.  Google Scholar

[17]

M. Peyrard and M. Kruskal, Kink dynamics in the highly discrete sine-Gordon system, Physica D, 14 (1984), 88-102. doi: doi:10.1016/0167-2789(84)90006-X.  Google Scholar

[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), 26-33. doi: doi:10.1063/1.437242.  Google Scholar

[19]

P. Rosenau, Dynamics of dense lattice, Phys. Rev. B, 36 (1987), 5868-5876. doi: doi:10.1103/PhysRevB.36.5868.  Google Scholar

[20]

M. Toda, "Theory of Nonlinear Lattices," 2nd edition, Springer Series in Solid-State Science, 20, Springer-Verlag, Berlin, 1989.  Google Scholar

[21]

G. B. Whitham, "Linear and Nonlinear Waves," Reprint of the 1974 original, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1999.  Google Scholar

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), 148-170. doi: doi:10.1016/S0167-2789(03)00261-6.  Google Scholar

[2]

O. M. Braun and Y. S. Kivshar, "The Frenkel-Kontorova Model, Concepts, Methods and Applications," Texts and Monographs in Physics, Springer-Verlag, Berlin-Heidelberg New York, 2004.  Google Scholar

[3]

L. A. Cisneros and A. A. Minzoni, Asymptotics for kink propagation in the discrete Sine-Gordon equation, Physica D, 237 (2008), 50-65. doi: doi:10.1016/j.physd.2007.08.005.  Google Scholar

[4]

L. A. Cisneros and A. A. Minzoni, Asymptotics for supersonic soliton propagation in the Toda lattice equation, Studies in Applied Mathematics, 120 (2008), 333-349. doi: doi:10.1111/j.1467-9590.2008.00401.x.  Google Scholar

[5]

M. Collins, A quasicontinuum approximation for solitons in an atomic chain, Chem. Phys. Lett., 77 (1981), 342-347. doi: doi:10.1016/0009-2614(81)80161-3.  Google Scholar

[6]

J. Dancz and S. A. Rice, Large amplitude vibrational motion in a one dimensional chain: Coherent state representation, J. Chem. Phys., 67 (1977), 1418-1426. doi: doi:10.1063/1.435015.  Google Scholar

[7]

H. Dym and H. P. McKean, "Fourier Series and Integrals," Probability and Mathematical Statistics, No. 14, Academic Press, New York-London, 1972.  Google Scholar

[8]

J. C. Eilbeck and R. Flesch, Calculation of families of solitary waves on discrete lattices, Phys. Lett. A, 149 (1990), 200-202. doi: doi:10.1016/0375-9601(90)90326-J.  Google Scholar

[9]

E. Fermi, J. Pasta and S. Ulam, "Los Alamos Rpt LA-1940 (1955) Collected Papers of Enrico Fermi," in Univ. of Chicago Press, Vol. II, Chicago, (1965), 978. Google Scholar

[10]

N. Flytzanis, St. Pnevmatikos and M. Peyrard, Discrete lattice solitons: Properties and stability, J. Phys. A: Math. Gen., 22 (1989), 783-801. doi: doi:10.1088/0305-4470/22/7/011.  Google Scholar

[11]

G. Friesecke and R. L. Pego, Solitary waves on FPU lattices: I. Qualitative properties, renormalization and continuum limit, Nonlinearity, 12 (1999), 1601-1627. doi: doi:10.1088/0951-7715/12/6/311.  Google Scholar

[12]

G. Friesecke and J. A. D. Wattis, Existence theorem for solitary waves on lattices, Commun. Math. Phys., 161 (1994), 391-418. doi: doi:10.1007/BF02099784.  Google Scholar

[13]

B. L. Holian, Shock waves in the Toda lattice: Analysis, Phys. Rev. A, 24 (1981), 2595-2623. doi: doi:10.1103/PhysRevA.24.2595.  Google Scholar

[14]

B. L. Holian and G. K. Straub, Molecular dynamics of shock waves in one-dimensional chains, Phys. Rev. B, 18 (1978), 1593-1608. doi: doi:10.1103/PhysRevB.18.1593.  Google Scholar

[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), 124101. doi: doi:10.1103/PhysRevLett.97.124101.  Google Scholar

[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], New York, 1984.  Google Scholar

[17]

M. Peyrard and M. Kruskal, Kink dynamics in the highly discrete sine-Gordon system, Physica D, 14 (1984), 88-102. doi: doi:10.1016/0167-2789(84)90006-X.  Google Scholar

[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), 26-33. doi: doi:10.1063/1.437242.  Google Scholar

[19]

P. Rosenau, Dynamics of dense lattice, Phys. Rev. B, 36 (1987), 5868-5876. doi: doi:10.1103/PhysRevB.36.5868.  Google Scholar

[20]

M. Toda, "Theory of Nonlinear Lattices," 2nd edition, Springer Series in Solid-State Science, 20, Springer-Verlag, Berlin, 1989.  Google Scholar

[21]

G. B. Whitham, "Linear and Nonlinear Waves," Reprint of the 1974 original, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1999.  Google Scholar

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