• Previous Article
    Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients
  • DCDS Home
  • This Issue
  • Next Article
    Diverging period and vanishing dissipation: Families of periodic sinks in the quasi-conservative case
December 2018, 38(12): 6091-6103. doi: 10.3934/dcds.2018136

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

* 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.

Citation: Jean-Pierre Eckmann, C. Eugene Wayne. Breathers as metastable states for the discrete NLS equation. Discrete & Continuous Dynamical Systems - A, 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. CuneoJ.-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. FontichR. 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. LepriR. 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

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. CuneoJ.-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. FontichR. 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. LepriR. 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.
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.
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.
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.
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 $\gamma = 0.2$.
[1]

Luigi Chierchia, Gabriella Pinzari. Planetary Birkhoff normal forms. Journal of Modern Dynamics, 2011, 5 (4) : 623-664. doi: 10.3934/jmd.2011.5.623

[2]

Shui-Nee Chow, Kening Lu, Yun-Qiu Shen. Normal forms for quasiperiodic evolutionary equations. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 65-94. doi: 10.3934/dcds.1996.2.65

[3]

Xingwu Chen, Weinian Zhang. Normal forms of planar switching systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6715-6736. doi: 10.3934/dcds.2016092

[4]

A. Katok and R. J. Spatzier. Nonstationary normal forms and rigidity of group actions. Electronic Research Announcements, 1996, 2: 124-133.

[5]

Marco Abate, Francesca Tovena. Formal normal forms for holomorphic maps tangent to the identity. Conference Publications, 2005, 2005 (Special) : 1-10. doi: 10.3934/proc.2005.2005.1

[6]

Boris Kalinin, Victoria Sadovskaya. Normal forms for non-uniform contractions. Journal of Modern Dynamics, 2017, 11: 341-368. doi: 10.3934/jmd.2017014

[7]

P. De Maesschalck. Gevrey normal forms for nilpotent contact points of order two. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 677-688. doi: 10.3934/dcds.2014.34.677

[8]

Weigu Li, Jaume Llibre, Hao Wu. Polynomial and linearized normal forms for almost periodic differential systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 345-360. doi: 10.3934/dcds.2016.36.345

[9]

Ricardo Miranda Martins. Formal equivalence between normal forms of reversible and hamiltonian dynamical systems. Communications on Pure & Applied Analysis, 2014, 13 (2) : 703-713. doi: 10.3934/cpaa.2014.13.703

[10]

Vincent Naudot, Jiazhong Yang. Finite smooth normal forms and integrability of local families of vector fields. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 667-682. doi: 10.3934/dcdss.2010.3.667

[11]

Tomas Johnson, Warwick Tucker. Automated computation of robust normal forms of planar analytic vector fields. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 769-782. doi: 10.3934/dcdsb.2009.12.769

[12]

Majid Gazor, Mojtaba Moazeni. Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 205-224. doi: 10.3934/dcds.2015.35.205

[13]

Alessandro Fortunati, Stephen Wiggins. Normal forms à la Moser for aperiodically time-dependent Hamiltonians in the vicinity of a hyperbolic equilibrium. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1109-1118. doi: 10.3934/dcdss.2016044

[14]

Teresa Faria. Normal forms for semilinear functional differential equations in Banach spaces and applications. Part II. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 155-176. doi: 10.3934/dcds.2001.7.155

[15]

Andreas Henrici. Symmetries of the periodic Toda lattice, with an application to normal forms and perturbations of the lattice with Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2949-2977. doi: 10.3934/dcds.2015.35.2949

[16]

Qingxu Dou, Jesús Cuevas, J. C. Eilbeck, Francis Michael Russell. Breathers and kinks in a simulated crystal experiment. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1107-1118. doi: 10.3934/dcdss.2011.4.1107

[17]

Jiecheng Chen, Dashan Fan, Lijing Sun. Asymptotic estimates for unimodular Fourier multipliers on modulation spaces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 467-485. doi: 10.3934/dcds.2012.32.467

[18]

Michael Kastner, Jacques-Alexandre Sepulchre. Effective Hamiltonian for traveling discrete breathers in the FPU chain. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 719-734. doi: 10.3934/dcdsb.2005.5.719

[19]

Panayotis Panayotaros. Continuation and bifurcations of breathers in a finite discrete NLS equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1227-1245. doi: 10.3934/dcdss.2011.4.1227

[20]

Dario Bambusi, D. Vella. Quasi periodic breathers in Hamiltonian lattices with symmetries. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 389-399. doi: 10.3934/dcdsb.2002.2.389

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (115)
  • HTML views (378)
  • Cited by (0)

Other articles
by authors

[Back to Top]