Exponentially stable breather solutions in nonautonomous dissipative nonlinear Schrödinger lattices

Abstract

We consider damped and forced discrete nonlinear Schrödinger equations on the lattice $\mathbb{Z}$. First we establish the existence of a global uniform attractor for the dissipative dynamics of the system. For strong dissipation we prove that the global uniform attractor has a finite fractal dimension and consists of a unique bounded trajectory that is confined to a finite-dimensional subspace of the infinite dimensional phase space, attracting any bounded set in phase space exponentially fast. Afterwards, we establish that for periodic, respectively, quasiperiodic forcing the unique solution is represented by a periodic, respectively, quasiperiodic breather. Notably, quasiperiodic breathers cannot exist in the system without damping and driving. Furthermore, the unique periodic (quasiperiodic) breather solution possesses a finite number of modes and is exponentially stable.

KEYWORDS:

• Lattice dynamical systems
• localised solutions
• breathers

Mathematics Subject Classifications:

• 39A23
• 39A24

Disclosure statement

No potential conflict of interest was reported by the author(s).