Georgy L. Alfimov

Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential

In the present paper we use the Wannier function basis to construct lattice approximations of the nonlinear Schrödinger equation with a periodic potential. We show that the nonlinear Schrödinger 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 the Bose-Einstein condensate theory as well as to other physical systems, such as, for example, electromagnetic wave propagation in nonlinear photonic crystals.

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

We consider localized modes (bright solitons) of the one-dimensional quintic nonlinear Schroedinger equation with a periodic potential, describing several mean-field models of low-dimensional condensed gases. In the case of attractive nonlinearity we deduce sufficient conditions for collapse. We show that there exist spatially localized modes with arbitrarily large numbers 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.

On the existence of gap solitons

Abstract The role of high harmonics in the gap soliton evolution is studied. The problem is considered from two viewpoints. First, it is shown that being treated as purely mathematical objects (i.e., localized in space and periodic in time solutions of a nonlinear wave equation), gap solitons do not exist due to the energy transfer to high harmonics, which are spatially delocalized. Second, in spite of this fact, by direct numerical solution of the nonlinear wave equation it is found that, subject to certain conditions (namely, when the CW frequency is sufficiently close to the definite stop band edge), gap solitons can be treated as existing in the physical sense, i.e., as long-living electromagnetic pulses. Moreover, they can be described well by the usual approximation of the lowest harmonics, i.e., within the framework of the parabolic approximation. It is shown that the decay of the gap solitons is accompanied by emission of linear waves and by formation of shock waves.

Coding of nonlinear states for the Gross–Pitaevskii equation with periodic potential

Abstract We study nonlinear states for the NLS-type equation with additional periodic potential U ( x ) , also called the Gross–Pitaevskii equation, GPE, in theory of Bose–Einstein Condensate, BEC. We prove that if the nonlinearity is defocusing (repulsive, in the BEC context) then under some conditions there exists a homeomorphism between the set of all nonlinear states for GPE (i.e. real bounded solutions of some nonlinear ODE) and the set of bi-infinite sequences of numbers from 1 to N for some integer N . These sequences can be viewed as codes of the nonlinear states. We present numerical arguments that for GPE with cosine potential these conditions hold in certain areas of the plane of the external parameters. This implies that for these values of parameters all the nonlinear states can be described in terms of the coding sequences.

Oscillatory instabilities of gap solitons in a repulsive Bose–Einstein condensate

Abstract The paper is devoted to numerical study of stability of nonlinear localized modes (“gap solitons”) for the spatially one-dimensional Gross–Pitaevskii equation (1D GPE) with periodic potential and repulsive interparticle interactions. We use the Evans function approach combined with the exterior algebra formulation in order to detect and describe weak oscillatory instabilities. We show that the simplest (“fundamental”) gap solitons in the first and in the second spectral gap undergo oscillatory instabilities for certain values of the frequency parameter (i.e., the chemical potential). The number of unstable eigenvalues and the associated instability rates are described. Several stable and unstable more complex (non-fundamental) gap solitons are also discussed. The results obtained from the Evans function approach are independently confirmed using the direct numerical integration of the GPE.

Gap solitons for the repulsive Gross-Pitaevskii equation with periodic potential: Coding and method for computation

The paper is devoted to nonlinear localized modes (“gap solitons”) for the spatially one-dimensional Gross-Pitaevskii equation (1D GPE) with a periodic potential and repulsive interparticle interactions. It has been recently shown (G. L. Alfimov, A. I. Avramenko, Physica D, 254, 29 (2013)) that under certain conditions all the stationary modes for the 1D GPE can be coded by bi-infinite sequences of symbols of some finite alphabet (called “codes” of the solutions). We present and justify a numerical method which allows to reconstruct the profile of a localized mode by its code. As an example, the method is applied to compute the profiles of gap solitons for 1D GPE with a cosine potential.

Discrete set of kink velocities in Josephson structures: The nonlocal double sine–Gordon model

Abstract We study a model of Josephson layered structure which is characterized by two peculiarities: (i) superconducting layers are thin; (ii) 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 constant velocities (called “sliding” velocities) for non-radiating stationary fluxon propagation. In dynamics, the presence of this set results in quantization of fluxon velocities: in numerical experiments a traveling kink-like excitation radiates energy and slows down to one of these particular constant velocities, taking the shape of predicted 2 π -kink. We conjecture that the set of these stationary velocities is infinite and present an asymptotic formula for them.

Stable dipole solitons and soliton complexes in the nonlinear Schrödinger equation with periodically modulated nonlinearity

We develop a general classification of the infinite number of families of solitons and soliton complexes in the one-dimensional Gross-Pitaevskii/nonlinear Schrödinger 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, being 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, 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.