31 research outputs found
Periodic and compacton travelling wave solutions of discrete nonlinear Klein-Gordon lattices
We prove the existence of periodic travelling wave solutions for general
discrete nonlinear Klein-Gordon systems, considering both cases of hard and
soft on-site potentials. In the case of hard on-site potentials we implement a
fixed point theory approach, combining Schauder's fixed point theorem and the
contraction mapping principle. This approach enables us to identify a ring in
the energy space for non-trivial solutions to exist, energy (norm) thresholds
for their existence and upper bounds on their velocity. In the case of soft
on-site potentials, the proof of existence of periodic travelling wave
solutions is facilitated by a variational approach based on the Mountain Pass
Theorem. The proof of the existence of travelling wave solutions satisfying
Dirichlet boundary conditions establishes rigorously the presence of compactons
in discrete nonlinear Klein-Gordon chains. Thresholds on the averaged kinetic
energy for these solutions to exist are also derived.Comment: 21 pages, 1 figur
Self trapping transition for a nonlinear impurity within a linear chain
In the present work we revisit the issue of the self-trapping dynamical
transition at a nonlinear impurity embedded in an otherwise linear lattice. For
our Schr\"odinger chain example, we present rigorous arguments that establish
necessary conditions and corresponding parametric bounds for the transition
between linear decay and nonlinear persistence of a defect mode. The proofs
combine a contraction mapping approach applied in the fully dynamical problem
in the case of a 3D-lattice, together with variational arguments for the
derivation of parametric bounds for the creation of stationary states
associated with the expected fate of the self-trapping dynamical transition.
The results are relevant for both power law nonlinearities and saturable ones.
The analytical results are corroborated by numerical computations.Comment: 16 pages, 7 figures. To be published in Journal of Mathematical
Physic
Global existence and compact attractors for the discrete nonlinear Schrödinger equation
AbstractWe study the asymptotic behavior of solutions of discrete nonlinear Schrödinger-type (DNLS) equations. For a conservative system, we consider the global in time solvability and the question of existence of standing wave solutions. Similarities and differences with the continuous counterpart (NLS-partial differential equation) are pointed out. For a dissipative system we prove existence of a global attractor and its stability under finite-dimensional approximations. Similar questions are treated in a weighted phase space. Finally, we propose possible extensions for various types of DNLS equations