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December  2018, 38(12): 6091-6103. doi: 10.3934/dcds.2018136

Breathers as metastable states for the discrete NLS equation

Jean-Pierre Eckmann 1, and  C. Eugene Wayne 2,,

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

* Corresponding author: C.E. Wayne

Dedicated to Rafael de la Llave with admiration and affection on his ${{60}^{\text{th}}}$ birthday.

Received  October 2017 Revised  January 2018 Published  April 2018

Fund Project: The first author is supported ERC, "Bridges", the second is supported in part by NSF grant DMS-1311553.

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.

Keywords: Metastability, breathers, normal forms, modulation theory.
Mathematics Subject Classification: 34C45, 34C15, 37J40, 70H09.
Citation: Jean-Pierre Eckmann, C. Eugene Wayne. Breathers as metastable states for the discrete NLS equation. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 6091-6103. doi: 10.3934/dcds.2018136
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.

Figure 1.  Numerical illustration of the dynamics with (weak) dissipation. The parameters are $N = 4$, $\gamma = 0.2$, and $\epsilon = 0.01$. Shown is the ''energy'' of the degree of freedom $j$, $j = 1,\dots,4$. Note that the transients vanish after some time and then the energies settle at about $\epsilon ^{-2(j-1)}$. Here we define them as $p_j^2+q_j^2$. Note also that the dissipation is so slow that no decrease can be observed in the graph of $p_1^2+q_1^2$ over the time scale considered.
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Figure 2.  The cylinder illustrates the set of periodic solutions of Eq.(9), with $\varphi$ changing (very little) from left to right and the circle illustrating the angle $\vartheta $. The spiral illustrates the way a time-dependent solution of Eq.(3) slides along the cylinder. It actually does not converge to it but will stay at some small, finite, distance from it. So the cylinder is Lyapunov stable in the sense of [1], at least as long as $\epsilon $ stays small.
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Figure 3.  The figure shows the $\gamma$ dependence of the absolute value of the real parts of the eigenvalues, for $N = 3$ and $\epsilon = 0.01$. The three curves are linear with intercept 0 and slopes $3.2\cdot 10^{-10}$, $0.0027$, and $0.00727$. Note that the first eigenvalue has an extremely small positive real part, while the others are stable.
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Figure 4.  This graph illustrates the behavior of $p_1(t)$, for $N = 3$ and several values of $\epsilon $ and $\gamma = 0.2\epsilon $. One measures the downcrossing times $T_k$ of the $k^{\rm th}$ downcrossing of $p_1$ through 0. The theory predicts that $X = \bigl({T_k/T_{k-1}}-1\bigr)/(\sqrt{k/(k-1)}-1) = 1$. As noted in the text, the transient behavior is not yet understood.
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Figure 5.  This graph illustrates the decay properties of the $\ell_2$ norm as a function of time, for various values of $N$ and $\epsilon$. The horizontal axis is $\epsilon $ and the vertical axis is an estimate of the decay rate, obtained as follows: If $m_t$ is the $\ell_2$ norm at time $t$ and $m_{t'}$ that at time $t'$, then we compute $ k = k(\epsilon ) = \log\left( \frac{\log(m_t/m_{t'})}{\gamma\cdot (t'-t)}\right)/\log (\epsilon ). $ If $m_t$ decays like $\exp(-ct\gamma\epsilon^s)$, then the calculation will lead to $k = s$. Indeed, we see that the decay rate is $\epsilon ^{2N-1}$. The calculations shown were done for $\gamma = 0.2$.
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