A. V. Yulin

Dissipative periodic waves, solitons, and breathers of the nonlinear Schrödinger equation with complex potentials.

Exact solutions for the generalized nonlinear Schrödinger equation with inhomogeneous complex linear and nonlinear potentials are found. We have found localized and periodic solutions for a wide class of localized and periodic modulations in the space of complex potentials and nonlinearity coefficients. Examples of stable and unstable solutions are given. We also demonstrated numerically the existence of stable dissipative breathers in the presence of an additional parabolic trap.

Newton's cradles in optics: From N-soliton fission to soliton chains

A mechanism for creating a Newton’s cradle (NC) in nonlinear light wave trains under the action of the third-order dispersion (TOD) is demonstrated. The formation of the NC structure plays an important role in the process of fission of higher-order (N ) solitons in optical fibers. After the splitting of the initial N soliton into a nonuniform chain of fundamental quasisolitons, the tallest one travels along the entire chain, through consecutive collisions with other solitons, and then escapes, while the remaining chain of pulses stays as a bound state, due to the radiation-mediated interaction between them. Increasing the initial soliton’s order, N , leads to the transmission through, and release of additional solitons with enhanced power, along with the emission of radiation, which may demonstrate a broadband supercontinuum spectrum. The NC dynamical regime remains robust in the presence of extra perturbations, such as the Raman and self-steepening effects, and dispersion terms above the third order. It is demonstrated that essentially the same NC mechanism is induced by the TOD in finite segments of periodic wave trains (in particular, soliton chains). A difference from the mechanical NC is that the TOD-driven pulse passing through the soliton array collects energy and momentum from other solitons. Thus, uniform and nonuniform arrays of nonlinear wave pulses offer an essential extension of the mechanical NC, in which the quasiparticles, unlike mechanical beads, interact inelastically, exchanging energy and generating radiation. Nevertheless, the characteristic phenomenology of NC chains may be clearly identified in these nonlinear-wave settings too.

Trapping of light in solitonic cavities and its role in the supercontinuum generation

We demonstrate that the fission of higher-order N-solitons with a subsequent ejection of fundamental quasi-solitons creates cavities formed by a pair of solitary waves with dispersive light trapped between them. As a result of multiple reflections of the trapped light from the bounding solitons which act as mirrors, they bend their trajectories and collide. In the spectral domain, the two solitons receive blue and red wavelength shifts, and the spectrum of the trapped light alters as well. This phenomenon strongly affects spectral characteristics of the generated supercontinuum. Consideration of the systems parameters which affect the creation of the cavity reveals possibilities of predicting and controlling soliton-soliton collisions induced by multiple reflections of the trapped light.

Localized modes in chi((2)) media with PT-symmetric localized potential

We study the existence and stability of solitons in the quadratic nonlinear media with spatially localized parity-time- (PT)-symmetric modulation of the linear refractive index. Families of stable one- and two-hump solitons are found. The properties of nonlinear modes bifurcating from a linear limit of small fundamental harmonic fields are investigated. It is shown that the fundamental branch, bifurcating from the linear mode of the fundamental harmonic is limited in power. The power maximum decreases with the strength of the imaginary part of the refractive index. The modes bifurcating from the linear mode of the second harmonic can exist even above the PT-symmetry-breaking threshold. We found that the fundamental branch bifurcating from the linear limit can undergo a secondary bifurcation colliding with a branch of two-hump soliton solutions. The stability intervals for different values of the propagation constant and gain or loss gradient are obtained. The examples of dynamics and excitations of solitons obtained by numerical simulations are also given.

Two-dimensional superfluid flows in inhomogeneous Bose-Einstein condensates.

We report an algorithm of constructing linear and nonlinear potentials in the two-dimensional Gross-Pitaevskii equation subject to given boundary conditions, which allow for exact analytic solutions. The obtained solutions represent superfluid flows in inhomogeneous Bose-Einstein condensates. The method is based on the combination of the similarity reduction of the two-dimensional Gross-Pitaevskii equation to the one-dimensional nonlinear Schrödinger equation, the latter allowing for exact solutions, with the conformal mapping of the given domain, where the flow is considered, to a half space. The stability of the obtained flows is addressed. A number of stable and physically relevant examples are described.

Conservative and PT-symmetric compactons in waveguide networks.

Stable discrete compactons in interconnected three-line waveguide arrays are found in linear and nonlinear limits in conservative and in parity-time (PT)-symmetric models. The compactons result from the interference of the fields in the two lines of waveguides ensuring that the third (middle) line caries no energy. PT-symmetric compactons require not only the presence of gain and losses in the two lines of the waveguides but also complex coupling, i.e., gain and losses in the coupling between the lines carrying the energy and the third line with zero field. The obtained compactons can be stable and their branches can cross the branches of the dissipative solitons. Unusual bifurcations of branches of solitons from linear compactons are described.

Tuning resonant interaction of orthogonally polarized solitons and dispersive waves with the soliton power.

We demonstrate that the relatively small power induced changes in the soliton wavenumber comparable with splitting of the effective indexes of the orthogonally polarized waveguide modes result in significant changes of the efficiency of the interaction between solitons and dispersive waves and can be used to control energy transfer between the soliton and newly generated waves and to delay or accelerate solitons.

Bloch oscillations sustained by nonlinearity

We demonstrate that nonlinearity may play a constructive role in supporting Bloch oscillations in a model which is discrete, in one dimension and continuous in the orthogonal one. The model can be experimentally realized in several fields of physics such as optics and Bose-Einstein condensates. We demonstrate that designing an optimal relation between the nonlinearity and the linear gradient strength provides extremely long-lived Bloch oscillations with little degradation. Such robust oscillations can be observed for a broad range of parameters and even for moderate nonlinearities and large enough values of linear potential. We also present an approximate analytical description of the wave packet’s evolution featuring a hybrid Bloch oscillating wave-soliton behavior that excellently corresponds to the direct numerical simulations.