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Analysis of supercontinuum generation under general dispersion characteristics and beyond the slowly varying envelope approximation
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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
H. Dym and H. P. McKean, "Fourier Series and Integrals," Probability and Mathematical Statistics, No. 14, Academic Press, New York-London, 1972. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
P. Rosenau, Dynamics of dense lattice, Phys. Rev. B, 36 (1987), 5868-5876.
doi: doi:10.1103/PhysRevB.36.5868. |
[20] |
M. Toda, "Theory of Nonlinear Lattices," 2nd edition, Springer Series in Solid-State Science, 20, Springer-Verlag, Berlin, 1989. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
H. Dym and H. P. McKean, "Fourier Series and Integrals," Probability and Mathematical Statistics, No. 14, Academic Press, New York-London, 1972. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
P. Rosenau, Dynamics of dense lattice, Phys. Rev. B, 36 (1987), 5868-5876.
doi: doi:10.1103/PhysRevB.36.5868. |
[20] |
M. Toda, "Theory of Nonlinear Lattices," 2nd edition, Springer Series in Solid-State Science, 20, Springer-Verlag, Berlin, 1989. |
[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. |
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