Decay of Hamiltonian Breathers Under Dissipation
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Communications in
Mathematical Physics
Decay of Hamiltonian Breathers Under Dissipation Jean-Pierre Eckmann1 , C. Eugene Wayne2 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.
E-mail: [email protected] Received: 5 August 2019 / Accepted: 30 June 2020 Published online: 3 October 2020 – © The Author(s) 2020
Abstract: We study metastable behavior in a discrete nonlinear Schrödinger equation from the viewpoint of Hamiltonian systems theory. When there are n < ∞ sites in this equation, we consider initial conditions in which almost all the energy is concentrated in one end of the system. We are interested in understanding how energy flows through the system, so we add a dissipation of size γ at the opposite end of the chain, and we show that the energy decreases extremely slowly. Furthermore, the motion is localized in the phase space near a family of breather solutions for the undamped system. We give rigorous, asymptotic estimates for the rate of evolution along the family of breathers and the width of the neighborhood within which the trajectory is confined.
Contents 1. 2. 3. 4. 5. 6. 7. 8. 9.
Introduction . . . . . . . . . . . . . . . . . . . . . The Eigenspace of the Eigenvalue 0 . . . . . . . . . Evolution Equations for γ > 0 . . . . . . . . . . . Spectral Properties of the Linearization at γ = 0 . . Complement of the Zero Eigenspace . . . . . . . . The Effect of the Dissipation on the Semigroup . . . Projecting onto the Complement of the 0 Eigenspace Bounds on the Evolution of ζ . . . . . . . . . . . . Re-orthogonalization . . . . . . . . . . . . . . . . . 9.1 The intuitive picture . . . . . . . . . . . . . . . 10. Iterating . . . . . . . . . . . . . . . . . . . . . . . 11. Conclusions and Future Directions . . . . . . . . . Dedicated to the memory of our friend and colleague, Walter Craig.
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J.-P. Eckmann, C. E. Wayne
1. Introduction In the present work we look at the problem of a finite, discrete nonlinear Schrödinger equation, with dissipation, which we considered first in [1]. We need to repeat several equations from that paper, but the aim is now to give a complete proof of the observations and assertions in that paper. One starts with −i
∂u j = −(Δu) j + |u j |2 u j , j = 1, 2, . . . , n, ∂τ
(1.1)
where we will add dissipation later. Here (Δu) j = u j−1 − 2u j + u j+1 , with free end boundary conditions for j = 1 or n, i.e., (Δu)1 = −u 1 + u 2 and (Δu)n = −u n + u n−1 . For the convenience of the reader, the following introduction repeats the setup from [1]. We will choose initial conditions for this system in which e
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