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We consider standing lattice solitons for discrete nonlinear Schrodinger equation with saturation (NLSS), where so-called transparent points were recently discovered. These transparent points are the values of the governing parameter (e.g., the lattice spacing) for which the Peierls-Nabarro barrier vanishes. In order to explain the existence of transparent points, we study a solitary wave solution in the continuous NLSS and analyse the singularities of its analytic continuation in the complex plane. The existence of a quadruplet of logarithmic singularities nearest to the real axis is proven and applied to two settings: (i) the fourth-order differential equation arising as the next-order continuum approximation of the discrete NLSS and (ii) the advance-delay version of the discrete NLSS., Comment: 38 pages, 12 figures, 2 tables

We present a new approach for search of coexisting classes of localised modes admitted by the repulsive (defocusing) scalar or vector nonlinear Schr\"odinger-type equations. The approach is based on the observation that generic solutions of the corresponding stationary system have singularities at finite points on the real axis. We start with establishing conditions on the initial data of the associated Cauchy problem that guarantee the formation of a singularity. Making use of these sufficient conditions, we identify the bounded, nonsingular, solutions --- and then classify them according to their asymptotic behaviour. To determine the bounded solutions, a properly chosen space of initial data is scanned numerically. Due to asymptotic or symmetry considerations, we can limit ourselves to a one- or two-dimensional space. For each set of initial conditions we compute the distances $X^{\pm}$ to the nearest forward and backward singularities; large $X^+$ or $X^-$ indicate the proximity to a bounded solution. We illustrate our method with the Gross-Pitaevskii equation with a $\PT$-symmetric complex potential, a system of coupled Gross-Pitaevskii equations with real potentials, and the Lugiato-Lefever equation with normal dispersion., Comment: 28 pages, 14 figures; accepted for Physica D

We study localized modes (LMs) of the one-dimensional Gross-Pitaevskii/nonlinear Schr\"{o}dinger equation with a harmonic-oscillator (parabolic) confining potential, and a periodically modulated coefficient in front of the cubic term (nonlinear lattice pseudopotential). The equation applies to a cigar-shaped Bose-Einstein condensate loaded in the combination of a magnetic trap and an optical lattice which induces the periodic pseudopotential via the Feshbach resonance. Families of stable LMs in the model feature specific properties which result from the interplay between spatial scales introduced by the parabolic trap and the period of the nonlinear pseudopotential. Asymptotic results on the shapes and stability of LMs are obtained for small-amplitude solutions and in the limit of a rapidly oscillating nonlinear pseudopotential. We show that the presence of the lattice pseudopotential may result in: (i) creation of new LM families which have no counterparts in the case of the uniform nonlinearity; (ii) stabilization of some previously unstable LM species; (iii) evolution of unstable LMs into a pulsating mode trapped in one well of the lattice pseudopotential., Comment: to be published in Commun. Nonlin. Sci. Numer. Simul

We develop a general classification of the infinite number of families of solitons and soliton complexes in the one-dimensional Gross-Pitaevskii/nonlinear Schrodinger equation with a nonlinear lattice pseudopotential, i.e., periodically modulated coefficient in front of the cubic term, which takes both positive and negative local values. This model finds direct implementations in atomic Bose-Einstein condensates and nonlinear optics. The most essential finding is the existence of two branches of dipole solitons (DSs), which feature an antisymmetric shape, essentially squeezed into a single cell of the nonlinear lattice. This soliton species was not previously considered in nonlinear lattices. We demonstrate that one branch of the DS family (namely, the one which obeys the Vakhitov-Kolokolov criterion) is stable, while unstable DSs spontaneously transform into stable fundamental solitons (FSs). The results are obtained in numerical and approximate analytical forms, the latter based on the variational approximation. Some stable bound states of FSs are found too., Comment: 12 pages, Chaos, in press

We study a model of Josephson layered structure which is characterized by two peculiarities: (i) superconducting layers are thin; (ii) due to suppression of superconducting states in superconducting layers the current-phase relation is non-sinusoidal and is described by two sine harmonics. The governing equation is a nonlocal generalization of double sine-Gordon (NDSG) equation. We argue that the dynamics of fluxons in the NDSG model is unusual. Specifically, we show that there exists a set of particular velocities for non-radiating fluxon propagation. In dynamics the presence of these ``priveleged'' velocitied results in phenomenon of quantization of fluxon velocities: in our numerical experiments a travelling kink-like excitation radiates energy and slows down to one of these particular velocities, taking a shape of predicted 2-pi-kink. This situation differs from both, double sine-Gordon local model and the nonlocal sine-Gordon model, considered before. We conjecture that the set of these velocities is infinite and present an asymptotic formula for them., Comment: 18 pages, 6 figures; submitted to Physica D

A method for the study of steady-state nonlinear modes for Gross-Pitaevskii equation (GPE) is described. It is based on exact statement about coding of the steady-state solutions of GPE which vanish as $x\to+\infty$ by reals. This allows to fulfill {\it demonstrative computation} of nonlinear modes of GPE i.e. the computation which allows to guarantee that {\it all} nonlinear modes within a given range of parameters have been found. The method has been applied to GPE with quadratic and double-well potential, for both, repulsive and attractive nonlinearities. The bifurcation diagrams of nonlinear modes in these cases are represented. The stability of these modes has been discussed., Comment: 21 pages, 6 figures

We study localized modes in binary mixtures of Bose-Einstein condensates embedded in one-dimensional optical lattices. We report a diversity of asymmetric modes and investigate their dynamics. We concentrate on the cases where one of the components is dominant, i.e. has much larger number of atoms than the other one, and where both components have the numbers of atoms of the same order but different symmetries. In the first case we propose a method of systematic obtaining the modes, considering the "small" component as bifurcating from the continuum spectrum. A generalization of this approach combined with the use of the symmetry of the coupled Gross-Pitaevskii equations allows obtaining breather modes, which are also presented., Comment: 11 pages, 16 figures

We consider the localized modes (bright solitons) described by one-dimensional quintic nonlinear Schrodinger 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., Comment: 8 pages, 6 figures

We consider localized modes (discrete breathers) of the discrete nonlinear Schr\"{o}dinger equation $i\frac{d\psi_n}{dt}=\psi_{n+1}+\psi_{n-1}-2\psi_n+\sigma|\psi_n|^2\psi_n$, $\sigma=\pm1$, $n\in \mathbb{Z}$. We study the diversity of the steady-state solutions of the form $\psi_n(t)=e^{i\omega t}v_n$ and the intervals of the frequency, $\omega$, of their existence. The base for the analysis is provided by the anticontinuous limit ($\omega$ negative and large enough) where all the solutions can be coded by the sequences of three symbols "-", "0" and "+". Using dynamical systems approach we show that this coding is valid for $\omega<\omega^*\approx -3.4533$ and the point $\omega^*$ is a point of accumulation of saddle-node bifurcations. Also we study other bifurcations of intrinsic localized modes which take place for $\omega>\omega^*$ and give the complete table of them for the solutions with codes consisting of less than four symbols., Comment: 33 pages, 14 figures. To appear in Physica D

In the present Letter we use the Wannier function basis to construct lattice approximations of the nonlinear Schr\"{o}dinger equation with a periodic potential. We show that the nonlinear Schr\"{o}dinger equation with a periodic potential is equivalent to a vector lattice with long-range interactions. For the case-example of the cosine potential we study the validity of the so-called tight-binding approximation i.e., the approximation when nearest neighbor interactions are dominant. The results are relevant to Bose-Einstein condensate theory as well as to other physical systems like, for example, electromagnetic wave propagation in nonlinear photonic crystals., Comment: 5 pages, 1 figure, submitted to Phys. Rev. E