147 research outputs found
Periodic travelling wave solutions of discrete nonlinear Schr\"odinger equations
The existence of nonzero periodic travelling wave solutions for a general
discrete nonlinear Schr\"odinger equation (DNLS) on finite one-dimensional
lattices is proved. The DNLS features a general nonlinear term and variable
range of interactions going beyond the usual nearest-neighbour interaction. The
problem of the existence of travelling wave solutions is converted into a fixed
point problem for an operator on some appropriate function space which is
solved by means of Schauder's Fixed Point Theorem
Existence and congruence of global attractors for damped and forced integrable and nonintegrable discrete nonlinear Schr\"odinger equations
We study two damped and forced discrete nonlinear Schr\"odinger equations on
the one-dimensional infinite lattice. Without damping and forcing they are
represented by the integrable Ablowitz-Ladik equation (AL) featuring non-local
cubic nonlinear terms, and its standard (nonintegrable) counterpart with local
cubic nonlinear terms (DNLS). The global existence of a unique solution to the
initial value problem for both, the damped and forced AL and DNLS, is proven.
It is further shown that for sufficiently close initial data, their
corresponding solutions stay close for all times. Concerning the asymptotic
behaviour of the solutions to the damped and forced AL and DNLS, for the former
a sufficient condition for the existence of a restricted global attractor is
established while it is shown that the latter possesses a global attractor.
Finally, we prove the congruence of the restricted global AL attractor and the
DNLS attractor for dynamics ensuing from initial data contained in an
appropriate bounded subset in a Banach space
Exponentially stable breather solutions in nonautonomous dissipative nonlinear Schr\"odinger lattices
We consider damped and forced discrete nonlinear Schr\"odinger equations on
the lattice . First we establish the existence of periodic and
quasiperiodic breather solutions for periodic and quasiperiodic driving,
respectively. Notably, quasiperiodic breathers cannot exist in the system
without damping and driving. Afterwards the existence of a global uniform
attractor for the dissipative dynamics of the system is shown. For strong
dissipation we prove that the global uniform attractor has finite fractal
dimension and consists of a single trajectory that is confined to a finite
dimensional subspace of the infinite dimensional phase space, attracting any
bounded set in phase space exponentially fast. Conclusively, for strong damping
and periodic (quasiperiodic) forcing the single periodic (quasiperiodic)
breather solution possesses a finite number of modes and is exponentially
stable
Self-organized escape processes of linear chains in nonlinear potentials
An enhancement of localized nonlinear modes in coupled systems gives rise to
a novel type of escape process. We study a spatially one dimensional set-up
consisting of a linearly coupled oscillator chain of mass-points situated
in a metastable nonlinear potential. The Hamilton-dynamics exhibits breather
solutions as a result of modulational instability of the phonon states. These
breathers localize energy by freezing other parts of the chain. Eventually this
localised part of the chain grows in amplitude until it overcomes the critical
elongation characterized by the transition state. Doing so, the breathers
ignite an escape by pulling the remaining chain over the barrier. Even if the
formation of singular breathers is insufficient for an escape, coalescence of
moving breathers can result in the required concentration of energy. Compared
to a chain system with linear damping and thermal fluctuations the breathers
help the chain to overcome the barriers faster in the case of low damping. With
larger damping, the decreasing life time of the breathers effectively inhibits
the escape process.Comment: 14 pages, 13 figure
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