15 research outputs found
Hamiltonian Hopf bifurcations in the discrete nonlinear Schr\"odinger trimer: oscillatory instabilities, quasiperiodic solutions and a 'new' type of self-trapping transition
Oscillatory instabilities in Hamiltonian anharmonic lattices are known to
appear through Hamiltonian Hopf bifurcations of certain time-periodic solutions
of multibreather type. Here, we analyze the basic mechanisms for this scenario
by considering the simplest possible model system of this kind where they
appear: the three-site discrete nonlinear Schr\"odinger model with periodic
boundary conditions. The stationary solution having equal amplitude and
opposite phases on two sites and zero amplitude on the third is known to be
unstable for an interval of intermediate amplitudes. We numerically analyze the
nature of the two bifurcations leading to this instability and find them to be
of two different types. Close to the lower-amplitude threshold stable
two-frequency quasiperiodic solutions exist surrounding the unstable stationary
solution, and the dynamics remains trapped around the latter so that in
particular the amplitude of the originally unexcited site remains small. By
contrast, close to the higher-amplitude threshold all two-frequency
quasiperiodic solutions are detached from the unstable stationary solution, and
the resulting dynamics is of 'population-inversion' type involving also the
originally unexcited site.Comment: 25 pages, 11 figures, to be published in J. Phys. A: Math. Gen.
Revised and shortened version with few clarifying remarks adde
Discreteness-Induced Oscillatory Instabilities of Dark Solitons
We reveal that even weak inherent discreteness of a nonlinear model can lead
to instabilities of the localized modes it supports. We present the first
example of an oscillatory instability of dark solitons, and analyse how it may
occur for dark solitons of the discrete nonlinear Schrodinger and generalized
Ablowitz-Ladik equations.Comment: 11 pages, 4 figures, to be published in Physical Review Letter
Translationally invariant nonlinear Schrodinger lattices
Persistence of stationary and traveling single-humped localized solutions in
the spatial discretizations of the nonlinear Schrodinger (NLS) equation is
addressed. The discrete NLS equation with the most general cubic polynomial
function is considered. Constraints on the nonlinear function are found from
the condition that the second-order difference equation for stationary
solutions can be reduced to the first-order difference map. The discrete NLS
equation with such an exceptional nonlinear function is shown to have a
conserved momentum but admits no standard Hamiltonian structure. It is proved
that the reduction to the first-order difference map gives a sufficient
condition for existence of translationally invariant single-humped stationary
solutions and a necessary condition for existence of single-humped traveling
solutions. Other constraints on the nonlinear function are found from the
condition that the differential advance-delay equation for traveling solutions
admits a reduction to an integrable normal form given by a third-order
differential equation. This reduction also gives a necessary condition for
existence of single-humped traveling solutions. The nonlinear function which
admits both reductions defines a two-parameter family of discrete NLS equations
which generalizes the integrable Ablowitz--Ladik lattice.Comment: 24 pages, 4 figure
Stationary Localized States Due to a Nonlinear Dimeric Impurity Embedded in a Perfect 1-D Chain
The formation of Stationary Localized states due to a nonlinear dimeric
impurity embedded in a perfect 1-d chain is studied here using the appropriate
Discrete Nonlinear Schrdinger Equation. Furthermore, the nonlinearity
has the form, where is the complex amplitude. A proper
ansatz for the Localized state is introduced in the appropriate Hamiltonian of
the system to obtain the reduced effective Hamiltonian. The Hamiltonian
contains a parameter, which is the ratio of stationary
amplitudes at impurity sites. Relevant equations for Localized states are
obtained from the fixed point of the reduced dynamical system. = 1 is
always a permissible solution. We also find solutions for which . Complete phase diagram in the plane comprising of both
cases is discussed. Several critical lines separating various regions are
found. Maximum number of Localized states is found to be six. Furthermore, the
phase diagram continuously extrapolates from one region to the other. The
importance of our results in relation to solitonic solutions in a fully
nonlinear system is discussed.Comment: Seven figures are available on reques
Strongly localized moving discrete dissipative breather-solitons in Kerr nonlinear media supported by intrinsic gain
We investigate the mobility of nonlinear localized modes in a generalized discrete Ginzburg-Landau-type model, describing a one-dimensional waveguide array in an active Kerr medium with intrinsic, saturable gain and damping. It is shown that exponentially localized, traveling discrete dissipative breather-solitons may exist as stable attractors supported only by intrinsic properties of the medium, i.e., in the absence of any external field or symmetry-breaking perturbations. Through an interplay by the gain and damping effects, the moving soliton may overcome the Peierls-Nabarro barrier, present in the corresponding conservative system, by self-induced time-periodic oscillations of its power (norm) and energy (Hamiltonian), yielding exponential decays to zero with different rates in the forward and backward directions. In certain parameter windows, bistability appears between fast modes with small oscillations and slower, large-oscillation modes. The velocities and the oscillation periods are typically related by lattice commensurability and exhibit period-doubling bifurcations to chaotically "walking" modes under parameter variations. If the model is augmented by intersite Kerr nonlinearity, thereby reducing the Peierls-Nabarro barrier of the conservative system, the existence regime for moving solitons increases considerably, and a richer scenario appears including Hopf bifurcations to incommensurately moving solutions and phase-locking intervals. Stable moving breathers also survive in the presence of weak disorder
Solitons in Triangular and Honeycomb Dynamical Lattices with the Cubic Nonlinearity
We study the existence and stability of localized states in the discrete
nonlinear Schr{\"o}dinger equation (DNLS) on two-dimensional non-square
lattices. The model includes both the nearest-neighbor and long-range
interactions. For the fundamental strongly localized soliton, the results
depend on the coordination number, i.e., on the particular type of the lattice.
The long-range interactions additionally destabilize the discrete soliton, or
make it more stable, if the sign of the interaction is, respectively, the same
as or opposite to the sign of the short-range interaction. We also explore more
complicated solutions, such as twisted localized modes (TLM's) and solutions
carrying multiple topological charge (vortices) that are specific to the
triangular and honeycomb lattices. In the cases when such vortices are
unstable, direct simulations demonstrate that they turn into zero-vorticity
fundamental solitons.Comment: 17 pages, 13 figures, Phys. Rev.
Modulational and Parametric Instabilities of the Discrete Nonlinear Schr\"odinger Equation
We examine the modulational and parametric instabilities arising in a
non-autonomous, discrete nonlinear Schr{\"o}dinger equation setting. The
principal motivation for our study stems from the dynamics of Bose-Einstein
condensates trapped in a deep optical lattice. We find that under periodic
variations of the heights of the interwell barriers (or equivalently of the
scattering length), additionally to the modulational instability, a window of
parametric instability becomes available to the system. We explore this
instability through multiple-scale analysis and identify it numerically. Its
principal dynamical characteristic is that, typically, it develops over much
larger times than the modulational instability, a feature that is qualitatively
justified by comparison of the corresponding instability growth rates
A Study of The Formation of Stationary Localized States Due to Nonlinear Impurities Using The Discrete Nonlinear Schr\"odinger Equation
The Discrete Nonlinear Schrdinger Equation is used to study the
formation of stationary localized states due to a single nonlinear impurity in
a Caley tree and a dimeric nonlinear impurity in the one dimensional system.
The rotational nonlinear impurity and the impurity of the form where is arbitrary and is the nonlinearity
parameter are considered. Furthermore, represents the absolute
value of the amplitude. Altogether four cases are studies. The usual Greens
function approach and the ansatz approach are coherently blended to obtain
phase diagrams showing regions of different number of states in the parameter
space. Equations of critical lines separating various regions in phase diagrams
are derived analytically. For the dimeric problem with the impurity , three values of , namely, , at and and
for are obtained. Last two values are lower than the
existing values. Energy of the states as a function of parameters is also
obtained. A model derivation for the impurities is presented. The implication
of our results in relation to disordered systems comprising of nonlinear
impurities and perfect sites is discussed.Comment: 10 figures available on reques
Time evolution of models described by one-dimensional discrete nonlinear Schr\"odinger equation
The dynamics of models described by a one-dimensional discrete nonlinear
Schr\"odinger equation is studied. The nonlinearity in these models appears due
to the coupling of the electronic motion to optical oscillators which are
treated in adiabatic approximation. First, various sizes of nonlinear cluster
embedded in an infinite linear chain are considered. The initial excitation is
applied either at the end-site or at the middle-site of the cluster. In both
the cases we obtain two kinds of transition: (i) a cluster-trapping transition
and (ii) a self-trapping transition. The dynamics of the quasiparticle with the
end-site initial excitation are found to exhibit, (i) a sharp self-trapping
transition, (ii) an amplitude-transition in the site-probabilities and (iii)
propagating soliton-like waves in large clusters. Ballistic propagation is
observed in random nonlinear systems. The effect of nonlinear impurities on the
superdiffusive behavior of random-dimer model is also studied.Comment: 16 pages, REVTEX, 9 figures available upon request, To appear in
Physical Review