Quasiperiodic localized oscillating solutions in the discrete nonlinear Schrödinger equation with alternating on-site potential (original) (raw)

Bifurcations and stability of gap solitons in periodic potentials

Physical Review E, 2004

We analyze the existence, stability, and internal modes of gap solitons in nonlinear periodic systems described by the nonlinear Schrödinger equation with a sinusoidal potential, such as photonic crystals, waveguide arrays, optically-induced photonic lattices, and Bose-Einstein condensates loaded onto an optical lattice. We study bifurcations of gap solitons from the band edges of the Floquet-Bloch spectrum, and show that gap solitons can appear near all lower or upper band edges of the spectrum, for focusing or defocusing nonlinearity, respectively. We show that, in general, two types of gap solitons can bifurcate from each band edge, and one of those two is always unstable. A gap soliton corresponding to a given band edge is shown to possess a number of internal modes that bifurcate from all band edges of the same polarity. We demonstrate that stability of gap solitons is determined by location of the internal modes with respect to the spectral bands of the inverted spectrum and, when they overlap, complex eigenvalues give rise to oscillatory instabilities of gap solitons.

Bifurcations of nonlinear localized modes in disordered lattices

Physical Review A, 2009

We analyze families of localized solutions of a nonlinear Schrödinger equation in the presence of a disordered potential modeling a waveguide array. A coupled mode theory approximation reveals that the families of disordered lattice solitons follow a cascade of Hopf-like bifurcations. Using a perturbation method, we analyze the origins of this bifurcation structure. We find that each family of solutions is characterized by a constant number of nodes. These predictions are in agreement with the numerical results of the nonlinear Schrödinger equation with a disordered potential.

Solitons in one-dimensional nonlinear Schrödinger lattices with a local inhomogeneity

2008

In this paper we analyze the existence, stability, dynamical formation, and mobility properties of localized solutions in a one-dimensional system described by the discrete nonlinear Schrödinger equation with a linear point defect. We consider both attractive and repulsive defects in a focusing lattice. Among our main findings are ͑a͒ the destabilization of the on-site mode centered at the defect in the repulsive case, ͑b͒ the disappearance of localized modes in the vicinity of the defect due to saddle-node bifurcations for sufficiently strong defects of either type, ͑c͒ the decrease of the amplitude formation threshold for attractive and its increase for repulsive defects, and ͑d͒ the detailed elucidation as a function of initial speed and defect strength of the different regimes ͑trapping, trapping and reflection, pure reflection, and pure transmission͒ of interaction of a moving localized mode with the defect.

Stationary localized modes of the quintic nonlinear Schrödinger equation with a periodic potential

Physical Review A, 2007

We consider the localized modes (bright solitons) described by one-dimensional quintic nonlinear Schrödinger equation with a periodic potential. In the case of attractive nonlinearity we deduce sufficient conditions for collapse. We show that there exist spatially localized modes with arbitrarily large number of particles. We study such solutions in the semi-infinite gap (attractive case) and in the first gap (attractive and repulsive cases), and show that a nonzero minimum value of the number of particles is necessary for a localized mode to be created. In the limit of large negative frequencies (attractive case) we observe "quantization" of the number of particles of the stationary modes. Such solutions can be interpreted as coupled "Townes" solitons and appear to be stable. The modes in the first gap have numbers of particles infinitely growing with frequencies approaching one of the gap edges, which is explained by the power decay of the modes. Stability of the localized modes is discussed.

Surface solitons in quasiperiodic nonlinear photonic lattices

Physical Review A, 2012

We study discrete surface solitons in semi-infinite, one-dimensional, nonlinear (Kerr), quasiperiodic waveguide arrays of the Fibonacci and Aubry-André types, and explore different families of localized surface modes, as a function of optical power content ('nonlinearity') and quasiperiodic strength ('disorder'). We find a strong asymmetry in the power content of the mode as a function of the propagation constant, between the cases of focussing and defocussing nonlinearity, in both models. We also examine the dynamical evolution of a completely-localized initial excitation at the array surface. We find that in general, for a given optical power, a smaller quasiperiodic strength is required to effect localization at the surface than in the bulk. Also, for fixed quasiperiodic strength, a smaller optical power is needed to localize the excitation at the edge than inside the bulk.

The appearance of gap solitons in a nonlinear Schrödinger lattice

Physica D: Nonlinear Phenomena, 2010

We study the appearance of discrete gap solitons in a nonlinear Schrödinger model with a periodic on-site potential that possesses a gap evacuated of plane-wave solutions in the linear limit. For finite lattices supporting an anti-phase (q = π/2) gap edge phonon as an anharmonic standing wave in the nonlinear regime, gap solitons are numerically found to emerge via pitchfork bifurcations from the gap edge. Analytically, modulational instabilities between pair of bifurcation points on this "nonlinear gap boundary" are found in terms of critical gap widths, turning to zero in the infinite size limit, which are associated with the birth of the localized soliton as well as discrete multisolitons in the gap. Such tunable instabilities can be of relevance in exciting soliton states in modulated arrays of nonlinear optical waveguides or Bose-Einstein condensates in periodic potentials. For lattices whose gap edge phonon only asymptotically approaches the anti-phase solution, the nonlinear gap boundary splits in a bifurcation scenario leading to the birth of the discrete gap soliton as a continuable orbit to the gap edge in the linear limit. The instability-induced dynamics of the localized soliton in the gap regime is found to thermalize according to the Gibbsian equilibrium distribution, while the spontaneous formation of persisting intrinsically localized modes (discrete breathers) from the extended out-gap soliton reveals a phase transition of the solution.

Stability of discrete solitons in nonlinear Schrödinger lattices

We consider the discrete solitons bifurcating from the anti-continuum limit of the discrete nonlinear Schrödinger (NLS) lattice. The discrete soliton in the anti-continuum limit represents an arbitrary finite superposition of in-phase or anti-phase excited nodes, separated by an arbitrary sequence of empty nodes. By using stability analysis, we prove that the discrete solitons are all unstable near the anti-continuum limit, except for the solitons, which consist of alternating anti-phase excited nodes. We classify analytically and confirm numerically the number of unstable eigenvalues associated with each family of the discrete solitons.

Oscillatory Instabilities of Standing Waves in One-Dimensional Nonlinear Lattices

Physical Review Letters, 2000

In one-dimensional anharmonic lattices, we construct nonlinear standing waves (SWs) reducing to harmonic SWs at small amplitude. For SWs with spatial periodicity incommensurate with the lattice period, a transition by breaking of analyticity versus wave amplitude is observed. As a consequence of the discreteness, oscillatory linear instabilities, persisting for arbitrarily small amplitude in infinite lattices, appear for all wave numbers Q fi 0, p. Incommensurate analytic SWs with jQj . p͞2 may however appear as "quasistable," as their instability growth rate is of higher order. 42.65.Sf, 45.05. + x, 63.20.Ry A well known and rather spectacular phenomenon occurring in many nonlinear media (e.g., fluids or optical waveguides) is the modulational (Benjamin-Feir) instability (MI), by which a traveling plane wave breaks up into a train of solitary waves (see, e.g., ). It is also well known that wave propagation in many continuous nonlinear media is well described by nonlinear Schrödingertype equations, where the solitary wave trains are described by spatially periodic and stable standing wave (SW) solutions, the so-called cnoidal envelope waves .

Localized modes in two-dimensional Schro¨dinger lattices with a pair of nonlinear sites

Optics Communications, 2014

We address the existence and stability of localized modes in the two-dimensional (2D) linear Schrödinger lattice with two symmetric nonlinear sites embedded into it, and a generalization for moderately localized nonlinearity featuring two distinct symmetric maxima. The latter setting admits a much greater variety of localized modes. Symmetric, antisymmetric, and asymmetric discrete solitons are found, and a subcritical bifurcation, accounting for the spontaneous symmetry breaking (SSB) of the symmetric modes and transition to asymmetric ones, is identified. Existence and stability of more complex 2D solutions in the form of discrete symmetric and asymmetric vortices are also discussed.