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Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients
Breathers as metastable states for the discrete NLS equation
1. | Département de Physique Théorique, and Section de Mathématiques, Université de Genève, 1211 Geneva 4, Switzerland |
2. | Department of Mathematics and Statistics, Boston University, Boston, MA 02215, USA |
We study metastable motions in weakly damped Hamiltonian systems. These are believed to inhibit the transport of energy through Hamiltonian, or nearly Hamiltonian, systems with many degrees of freedom. We investigate this question in a very simple model in which the breather solutions that are thought to be responsible for the metastable states can be computed perturbatively to an arbitrary order. Then, using a modulation hypothesis, we derive estimates for the rate at which the system drifts along this manifold of periodic orbits and verify the optimality of our estimates numerically.
References:
[1] |
V. I. Arnol'd, Mathematical Methods of Classical Mechanics, vol. 60 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1989, Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition. |
[2] |
S. Aubry,
Discrete breathers: Localization and transfer of energy in discrete Hamiltonian nonlinear systems, Phys. D, 216 (2006), 1-30.
doi: 10.1016/j.physd.2005.12.020. |
[3] |
M. Beck and C. E. Wayne,
Using global invariant manifolds to understand metastability in the Burgers equation with small viscosity, SIAM J. Appl. Dyn. Syst., 8 (2009), 1043-1065.
doi: 10.1137/08073651X. |
[4] |
N. Cuneo, J.-P. Eckmann and C. Wayne,
Energy dissipation in Hamiltonian chains of rotators, Nonlinearity, 30 (2017), R81-R117.
doi: 10.1088/1361-6544/aa85d6. |
[5] |
S. Flach and C. R. Willis,
Discrete breathers, Phys. Rep., 295 (1998), 181-264.
doi: 10.1016/S0370-1573(97)00068-9. |
[6] |
E. Fontich, R. de la Llave and Y. Sire,
Construction of invariant whiskered tori by a parameterization method. Part Ⅱ: Quasi-periodic and almost periodic breathers in coupled map lattices, Journal of Differential Equations, 259 (2015), 2180-2279.
doi: 10.1016/j.jde.2015.03.034. |
[7] |
A. Haro and R. de la Llave,
New mechanisms for lack of equipartition of energy, Phys. Rev. Lett., 85 (2000), 1859-1862.
doi: 10.1103/PhysRevLett.85.1859. |
[8] |
M. Jenkinson and M. I. Weinstein,
Onsite and offsite bound states of the discrete nonlinear Schrödinger equation and the Peierls-Nabarro barrier, Nonlinearity, 29 (2016), 27-86.
doi: 10.1088/0951-7715/29/1/27. |
[9] |
S. Lepri, R. Livi and A. Politi,
Heat conduction in chains of nonlinear oscillators, Phys. Rev. Lett., 78 (1997), 1896-1899.
doi: 10.1103/PhysRevLett.78.1896. |
[10] |
R. S. MacKay and S. Aubry,
Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity, 7 (1994), 1623-1643.
doi: 10.1088/0951-7715/7/6/006. |
[11] |
R. Pego and M. Weinstein,
On asymptotic stability of solitary waves, Phys. Lett. A, 162 (1992), 263-268.
|
[12] |
K. Promislow,
A renormalization method for modulational stability of quasi-steady patterns in dispersive systems, SIAM J. Math. Anal., 33 (2002), 1455-1482.
doi: 10.1137/S0036141000377547. |
show all references
Dedicated to Rafael de la Llave with admiration and affection on his ${{60}^{\text{th}}}$ birthday.
References:
[1] |
V. I. Arnol'd, Mathematical Methods of Classical Mechanics, vol. 60 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1989, Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition. |
[2] |
S. Aubry,
Discrete breathers: Localization and transfer of energy in discrete Hamiltonian nonlinear systems, Phys. D, 216 (2006), 1-30.
doi: 10.1016/j.physd.2005.12.020. |
[3] |
M. Beck and C. E. Wayne,
Using global invariant manifolds to understand metastability in the Burgers equation with small viscosity, SIAM J. Appl. Dyn. Syst., 8 (2009), 1043-1065.
doi: 10.1137/08073651X. |
[4] |
N. Cuneo, J.-P. Eckmann and C. Wayne,
Energy dissipation in Hamiltonian chains of rotators, Nonlinearity, 30 (2017), R81-R117.
doi: 10.1088/1361-6544/aa85d6. |
[5] |
S. Flach and C. R. Willis,
Discrete breathers, Phys. Rep., 295 (1998), 181-264.
doi: 10.1016/S0370-1573(97)00068-9. |
[6] |
E. Fontich, R. de la Llave and Y. Sire,
Construction of invariant whiskered tori by a parameterization method. Part Ⅱ: Quasi-periodic and almost periodic breathers in coupled map lattices, Journal of Differential Equations, 259 (2015), 2180-2279.
doi: 10.1016/j.jde.2015.03.034. |
[7] |
A. Haro and R. de la Llave,
New mechanisms for lack of equipartition of energy, Phys. Rev. Lett., 85 (2000), 1859-1862.
doi: 10.1103/PhysRevLett.85.1859. |
[8] |
M. Jenkinson and M. I. Weinstein,
Onsite and offsite bound states of the discrete nonlinear Schrödinger equation and the Peierls-Nabarro barrier, Nonlinearity, 29 (2016), 27-86.
doi: 10.1088/0951-7715/29/1/27. |
[9] |
S. Lepri, R. Livi and A. Politi,
Heat conduction in chains of nonlinear oscillators, Phys. Rev. Lett., 78 (1997), 1896-1899.
doi: 10.1103/PhysRevLett.78.1896. |
[10] |
R. S. MacKay and S. Aubry,
Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity, 7 (1994), 1623-1643.
doi: 10.1088/0951-7715/7/6/006. |
[11] |
R. Pego and M. Weinstein,
On asymptotic stability of solitary waves, Phys. Lett. A, 162 (1992), 263-268.
|
[12] |
K. Promislow,
A renormalization method for modulational stability of quasi-steady patterns in dispersive systems, SIAM J. Math. Anal., 33 (2002), 1455-1482.
doi: 10.1137/S0036141000377547. |





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