57 research outputs found
Thresholds for breather solutions on the Discrete Nonlinear Schr\"odinger Equation with saturable and power nonlinearity
We consider the question of existence of periodic solutions (called breather
solutions or discrete solitons) for the Discrete Nonlinear Schr\"odinger
Equation with saturable and power nonlinearity. Theoretical and numerical
results are proved concerning the existence and nonexistence of periodic
solutions by a variational approach and a fixed point argument. In the
variational approach we are restricted to DNLS lattices with Dirichlet boundary
conditions. It is proved that there exists parameters (frequency or
nonlinearity parameters) for which the corresponding minimizers satisfy
explicit upper and lower bounds on the power. The numerical studies performed
indicate that these bounds behave as thresholds for the existence of periodic
solutions. The fixed point method considers the case of infinite lattices.
Through this method, the existence of a threshold is proved in the case of
saturable nonlinearity and an explicit theoretical estimate which is
independent on the dimension is given. The numerical studies, testing the
efficiency of the bounds derived by both methods, demonstrate that these
thresholds are quite sharp estimates of a threshold value on the power needed
for the the existence of a breather solution. This it justified by the
consideration of limiting cases with respect to the size of the nonlinearity
parameters and nonlinearity exponents.Comment: 26 pages, 10 figure
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
Conservation laws, exact travelling waves and modulation instability for an extended nonlinear Schr\"odinger equation
We study various properties of solutions of an extended nonlinear
Schr\"{o}dinger (ENLS) equation, which arises in the context of geometric
evolution problems -- including vortex filament dynamics -- and governs
propagation of short pulses in optical fibers and nonlinear metamaterials. For
the periodic initial-boundary value problem, we derive conservation laws
satisfied by local in time, weak (distributional) solutions, and
establish global existence of such weak solutions. The derivation is obtained
by a regularization scheme under a balance condition on the coefficients of the
linear and nonlinear terms -- namely, the Hirota limit of the considered ENLS
model. Next, we investigate conditions for the existence of traveling wave
solutions, focusing on the case of bright and dark solitons. The balance
condition on the coefficients is found to be essential for the existence of
exact analytical soliton solutions; furthermore, we obtain conditions which
define parameter regimes for the existence of traveling solitons for various
linear dispersion strengths. Finally, we study the modulational instability of
plane waves of the ENLS equation, and identify important differences between
the ENLS case and the corresponding NLS counterpart. The analytical results are
corroborated by numerical simulations, which reveal notable differences between
the bright and the dark soliton propagation dynamics, and are in excellent
agreement with the analytical predictions of the modulation instability
analysis.Comment: 27 pages, 5 figures. To be published in Journal of Physics A:
Mathematical and Theoretica
Lower and upper estimates on the excitation threshold for breathers in DNLS lattices
We propose analytical lower and upper estimates on the excitation threshold
for breathers (in the form of spatially localized and time periodic solutions)
in DNLS lattices with power nonlinearity. The estimation depending explicitly
on the lattice parameters, is derived by a combination of a comparison argument
on appropriate lower bounds depending on the frequency of each solution with a
simple and justified heuristic argument. The numerical studies verify that the
analytical estimates can be of particular usefulness, as a simple analytical
detection of the activation energy for breathers in DNLS lattices.Comment: 10 pages, 3 figure
Stationary states of a nonlinear Schrödinger lattice with a harmonic trap
We study a discrete nonlinear Schrödinger lattice with a parabolic trapping potential. The model, describing, e.g., an array of repulsive Bose-Einstein condensate droplets confined in the wells of an optical lattice, is analytically and numerically investigated. Starting from the linear limit of the problem, we use global bifurcation theory to rigorously prove that – in the discrete regime – all linear states lead to nonlinear generalizations thereof, which assume the form of a chain of discrete dark solitons (as the density increases). The stability of the ensuing nonlinear states is studied and it is found that the ground state is stable, while the excited states feature a chain of stability/instability bands. We illustrate the mechanisms under which discreteness destabilizes the dark-soliton configurations, which become stable only in the continuum regime. Continuation from the anti-continuum limit is also considered, and a rich bifurcation structure is revealed
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