Vakhitov Kolokolov Research Articles

Raman perturbation and surface core soliton in hollow photonic crystal fiber

Effect of Raman scattering and higher order dispersion are investigated in case of surface core or gap solitons, known to exist in hollow core photonic crystal fibers. The necessary conditions for the existence of the soliton are analyzed in the parameter plane. It is observed that these conditions are compatible with the famous Vakhitov–Kolokolov (VK) criterion. In the later part the system of partial differential equations are solved by ETDRK method from which the shapes of the pulses are investigated as it travels along the z direction. Both the terms, intrapulse Raman scattering and higher order dispersion have important effect on the propagation characteristics

Soliton dynamics in the complex modified KdV equation with a periodic potential

We study the existence and stability of stationary and moving solitary waves in a periodically modulated system governed by an extended cmKdV (complex modified Korteweg–de Vries) equation. The proposed equation describes, in particular, the co-propagation of two electromagnetic waves with different amplitudes and orthogonal linear polarizations in a liquid crystal waveguide, the stronger (nonlinear) wave actually carrying the soliton, while the other (a nearly linear one) creates an effective periodic potential. A variational analysis predicts solitons pinned at minima and maxima of the periodic potential, and the Vakhitov–Kolokolov criterion predicts that some of them may be stable. Numerical simulations confirm the existence of stable stationary solitary waves trapped at the minima of the potential, and show that persistently moving solitons exist too. The dynamics of pairs of interacting solitons is also studied. In the absence of the potential, the interaction is drastically different from the behavior known in the NLS (nonlinear Schrödinger) equation, as the force of the interaction between the cmKdV solitons is proportional to the sine, rather than cosine, of the phase difference between the solitons. In the presence of the potential, two solitons placed in one potential well form a persistently oscillating bound state.

The black solitons of one-dimensional NLS equations

In this paper, we prove a criterion to determine if a black soliton solution (which is an odd solution that does not vanish at infinity) to a one-dimensional nonlinear Schrödinger equation is linearly stable or not. This criterion handles the sign of the limit at 0 of the Vakhitov–Kolokolov function. For some nonlinearities, we numerically compute the black soliton and the Vakhitov–Kolokolov function in order to investigate linear stability of black solitons. We then show that linearly unstable black solitons are also orbitally unstable. In the Gross–Pitaevskii case, we rigorously prove the linear stability of the black soliton. Finally, we numerically study the dynamical stability of these solutions solving both linearized and fully nonlinear equations with a finite differences algorithm.

On stability of vortices in three-dimensional self-attractive Bose–Einstein condensates

Abstract Results of accurate analysis of stability are reported for localized vortices in the Bose–Einstein condensate (BEC) with the negative scattering length, trapped in an anisotropic potential with the aspect ratio Ω . The cases of Ω ≫ 1 and Ω ≪ 1 correspond to the “pancake” (nearly-2D) and “cigar-shaped” (nearly-1D) configurations, respectively (in the latter limit, the vortices become “tubular” solitons). The analysis is based on the 3D Gross–Pitaevskii equation. The family of solutions with vorticity S = 1 is accurately predicted by the variational approximation. The relative size of the stability area for the vortices with S = 1 (which was studied, in a part, before) increases with the decrease of Ω in terms of the number of atoms, but decreases in terms of the chemical potential. All states with S ⩾ 2 are unstable, while the stability of the ordinary solitons ( S = 0 ) obeys the Vakhitov–Kolokolov criterion. The stability predictions are verified by direct simulations of the full 3D equation.

Defect solitons in photonic lattices

Nonlinear defect modes (defect solitons) and their stability in one-dimensional photonic lattices with focusing saturable nonlinearity are investigated. It is shown that defect solitons bifurcate out from every infinitesimal linear defect mode. Low-power defect solitons are linearly stable in lower bandgaps but unstable in higher bandgaps. At higher powers, defect solitons become unstable in attractive defects, but can remain stable in repulsive defects. Furthermore, for high-power solitons in attractive defects, we found a type of Vakhitov-Kolokolov (VK) instability which is different from the usual VK instability based on the sign of the slope in the power curve. Lastly, we demonstrate that in each bandgap, in addition to defect solitons which bifurcate from linear defect modes, there is also an infinite family of other defect solitons which can be stable in certain parameter regimes.

Four-wave solitons in waveguides with a cross-grating

A model of a two-dimensional optical waveguide with the Kerr nonlinearity and two transversal (cross-)Bragg gratings (BGs) is considered. Four waves trapped in the waveguide are coupled linearly by reflections on the cross-BG, and nonlinearly by self-phase-modulation, cross-phase-modulation, and four-wave-mixing. One-dimensional gap solitons (GSs) in the model are found by means of the variational approximation and numerical methods, the analytical and numerical results being in good agreement. The solitons fall into three distinct categories, which are identified as symmetric (S), anti-symmetric (anti-S), and asymmetric (aS) ones at the center of the bandgap, and those which are obtained by continuation of these three types in the general case. The stability of the GSs is studied in the spatial domain. All the solitons of the S and anti-S types are unstable (their instability modes are different), while the aS solitons have a well-defined stability region. The latter is identified by means of the Vakhitov–Kolokolov (VK) criterion, which is verified by direct simulations. It is demonstrated too that stable breathers, which are close to strongly asymmetric solitons, readily self-trap from a two-component input that corresponds to a physically relevant boundary condition in the spatial domain (the development of the instability of solitons of the S type gives rise to breathers of a different kind, in which the field periodically switches between two aS configurations that are mirror images to each other). Tilted spatial solitons of the aS type are found too; they are stable for relatively small values of the tilt.

Three-wave solitons and continuous waves in media with competing quadratic and cubic nonlinearities.

We formulate a general model of three-wave optical interactions (in the spatial domain), which combines quadratic (chi(2)) and cubic (chi(3)) nonlinearities, the latter including four-wave mixing. The model can be realized in chi(2) materials where an effective chi(3) nonlinearity is engineered by means of the quasi-phase-matching technique. Both self-focusing and self-defocusing chi(3) nonlinearities are considered. The birefringence of the two fundamental-frequency (FF) waves is taken into regard. Several types of solitons in this system are found, by means of the variational approximation and numerical methods. These are exact single-component solitons and generic three-wave (3W) ones, which are classified by relative signs of their components. Stability of the solitons is investigated by means of the Vakhitov-Kolokolov (VK) criterion, and then tested by direct simulations. One type of the single-component FF solitons (the "fast" one, in terms of the known two-component birefringent chi(3) model) is, chiefly, unstable, as in that model, but nevertheless a stability interval is found for it, which provides for the first example of stable fast solitons. The other FF soliton (the "slow" one, in terms of the same chi(3) model, where it is always stable) has its stability and instability regions. A single-component soliton in the second harmonic (SH) is found too; it also has its stability region, contrary to the common belief that such a soliton must always be unstable due to the parametric interaction. The 3W solitons are stable indeed if this is predicted by the VK condition, in the case when all the three components are positive. Following variation of the chi(2) mismatch parameter, the 3W soliton bifurcates from the SH one, and at another point it bifurcates back into the slow-FF single-component soliton; conjectured normal forms of the respective bifurcations are given. 3W solitons with different signs of their components may be unstable contrary to the VK criterion, which is explained by consideration of the chi(2) term in the system's Hamiltonian. In direct simulations, unstable solitons evolve into stable breathers. A different instability takes place in the case of the self-defocusing chi(3) nonlinearity, when all the solitons blow up into a turbulent state. Parallel to the solitons, continuous-wave solutions are studied too. In terms of the existence and stability, they resemble solitons of similar types.

Four-wave solitons in Bragg cross-gratings

A temporal-domain model for a two-dimensional (2D) waveguide with two transversal Bragg gratings (BGs) is introduced. In this medium, four waves are coupled linearly by the Bragg reflections, and nonlinearly by XPM and four-wave mixing. The same model in the spatial domain applies to a 3D photonic crystal. The subject of the work is (effectively) 1D solitons in this system. Equations which govern their dynamics also describe the evolution of two circular polarizations in a fibre BG. Depending on the relative strength of the two gratings, the system may or may not have a gap in its spectrum, and it always has 'semi-gaps', where embedded solitons (ES) may be found. In the case when the propagation constant k (frustration parameter) is zero, the system admits two reductions, symmetric (S) and anti-symmetric (anti-S), to two-wave systems, which are equivalent to the standard fibre-BG model. These systems give rise to exact solutions, which, in terms of the full system, may be gap solitons (GS) or ES. Asymmetric (aS) solitons for k = 0, and solutions generated by the continuation of the S, anti-S and aS solitons to , are found by means of the variational approximation and in a numerical form. In the semi-gaps, these solutions may be both ES and quasi-solitons (QS) with nonvanishing tails. Stability of the solitons is first investigated by means of the Vakhitov–Kolokolov (VK) criterion, which predicts no instability for the extended solution families of the S and anti-S type. For the aS solitons, the VK criterion predicts a nontrivial stability border inside the bandgap, the location of which is controlled by k: at k = 0 the entire bandgap is VK-stable, while for large k the area of the VK stability shrinks to nothing. This intra-gap stability border is a new feature of GS models. Direct simulations demonstrate that, at k = 0, the S and anti-S reductions are always stable against perturbations violating the reduction, hence the stability of the exact S and anti-S solitons can be predicted on the basis of known results for the standard fibre-BG model. Continuing these solutions in k and testing their stability by simulations, we conclude that the solution branches stemming from the stable and unstable exact solitons at k = 0 always remain, respectively, stable or unstable. As a result, the extended soliton family of the S type is chiefly (but not exactly) stable inside the bandgap and unstable in the semi-gaps, while the opposite is true for the extended family of the anti-S type solutions. The aS solitons which are predicted to be VK-unstable quickly decay indeed, while their VK-stable counterparts survive for a long time, but are finally destroyed by oscillatory perturbations, which the VK criterion does not take into consideration. Moving solitons are constructed too, and a limit velocity is found for them.

Bistable guided solitons in the cubic–quintic medium

We consider spatial solitons in a channel waveguide with uniform cubic–quintic nonlinearity. By means of the variational approximation and numerical methods, two branches of the soliton solutions are found, which are in contrast to their free-space counterparts; the bistability occurs exactly at those values of the propagation constant where the free-space solitons do not exist. The Vakhitov–Kolokolov criterion formally predicts that one branch should be unstable; however, direct numerical tests disprove this expectation, showing that all the solitons are completely stable. Besides the bistability, another feature of the solitons which is promising for applications is that their edges can be made much sharper than in the free-space solitons. Systematic simulations of the evolution of an initial Gaussian beam are also performed, showing that it quickly self-traps into a soliton of either type, or into a breather, but never decays into radiation.

Self-trapping of light in a two-dimensional photonic lattice

We study wave propagation in a two-dimensional photonic lattice with focusing Kerr nonlinearity, and report on the existence of various nonlinear localized structures in the form of fundamental, dipole, and vortex solitons. First, the linear bandgap structure induced by the two-dimensional photonic crystal is determined, and solitons are found to exist in the photonic bandgap. Next, structures of these solitons and their stability properties are analyzed in detail. When the propagation constant is not close to the edge of the bandgap, the fundamental soliton is largely confined to one lattice site; the dipole soliton consists of two π-out-of-phase, Gaussian-like humps, whereas the vortex comprises four fundamental modes superimposed in a square configuration with a phase structure that is topologically equivalent to the conventional homogeneous-bulk vortex. At high lattice potential, all these soliton states are stable against small perturbations. However, among the three states, the fundamental solitons are the most robust, whereas vortices are the least. If the propagation constant is close to the edge of the bandgap, then all three soliton states spread over many lattice sites and become linearly unstable as a result of the Vakhitov–Kolokolov instability.

Stability of vortex solitons in a photorefractive optical lattice

Stability of on- and off-site vortex solitons with unit charge in a photorefractive optical lattice is analysed. It is shown that both types of vortex solitons are linearly unstable in the high- and low-intensity regimes. In the high-intensity regime, the vortex appears as a familiar ring vortex, and it suffers oscillatory instabilities. In the low-intensity regime, the vortex suffers both oscillatory and Vakhitov–Kolokolov instabilities. However, in the intermediate-intensity regime, both types of vortices could be stable if the lattice intensity is moderate, and the applied DC field is above a certain threshold value. Under the same physical conditions, on-site vortices are more stable than off-site ones. Increasing the DC field stabilizes both types of vortex solitons.

Solitons in Bose–Einstein condensates trapped in a double-well potential

We investigate, analytically and numerically, families of bright solitons in a system of two linearly coupled nonlinear Schrödinger/Gross–Pitaevskii equations, describing two Bose–Einstein condensates trapped in an asymmetric double-well potential, in particular, when the scattering lengths in the condensates have arbitrary magnitudes and opposite signs. The solitons are found to exist everywhere where they are permitted by the dispersion law. Using the Vakhitov–Kolokolov criterion and numerical methods, we show that, except for small regions in the parameter space, the solitons are stable to small perturbations. Some of them feature self-trapping of almost all the atoms in the condensate with no atomic interaction or weak repulsion is coupled to the self-attractive condensate. An unusual bifurcation is found, when the soliton bifurcates from the zero solution with vanishing amplitude and width simultaneously diverging but at a finite number of atoms in the soliton. By means of numerical simulations, it is found that, depending on values of the parameters and the initial perturbation, unstable solitons either give rise to breathers or completely break down into incoherent waves (“radiation”). A version of the model with the self-attraction in both components, which applies to the description of dual-core fibers in nonlinear optics, is considered too, and new results are obtained for this much studied system.

Energy of the soliton internal modes and broken symmetries in nonlinear optics

Applications of the concepts of the energy and, in particular, of the negative energy of the internal modes are described in the context of nonlinear optics. It is shown that the transformation of a soliton propagating in a bulk into solitons propagating in a coherently pumped cavity is followed by emergence of the negative-energy modes. These modes are directly associated with a locking of the soliton phase to the phase of the coherent pump, i.e., with a breaking of the phase symmetry. Energy of the Vakhitov–Kolokolov modes is also calculated. This energy is positive, providing that conditions of the Vakhitov–Kolokolov stability criterion are met. Mutual annihilation of the positive- and the negative-energy modes leads to oscillatory (Hopf) instability of the cavity solitons.

Stable two-dimensional soliton supported by a local nonlinearity

We show that an attractive nonlinear potential in the form of a modified Azbel two-dimensional (2D) \ensuremath{\delta} function (that, unlike the traditional 2D \ensuremath{\delta} function, gives rise to a well-defined bound state in quantum mechanics) supports a 2D localized state (``soliton''), that may be stable according to the Vakhitov-Kolokolov (VK) criterion, provided that the soliton's amplitude is not too small. A direct, although not general, dynamical stability consideration yields results in compliance with the VK criterion.