Double Loops and Pitchfork Symmetry Breaking Bifurcations of Optical Solitons in Nonlinear Fractional Schrödinger Equation with Competing Cubic-Quintic Nonlinearities

Symmetry breaking bifurcations of solitons are investigated in framework of a nonlinear fractional Schrödinger equation (NLFSE) with competing cubic-quintic nonlinearity. Some prototypical characteristics of the symmetry breaking, featured by transformations of symmetric and antisymmetric soliton families into asymmetric ones, are found. Stable asymmetric solitons emerge from unstable symmetric and antisymmetric ones by way of two different symmetry breaking scenarios. A twisting branch, featured with double loops bifurcation, bifurcates off from the base branch of symmetric soliton solutions and crosses it, then merges into the base branch driven by the competitive nonlinear effect. A supercritical pitchfork bifurcation is bifurcated from the branch of antisymmetric soliton solutions and gives rise to a supercritical pitchfork bifurcation. Stability of the soliton families is explored by linear stability analysis. With the increase of the Lévy index, stability region induced by the twisting loops bifurcation is expanded. However, stability region of the pitchfork bifurcation is shrunk on the parameter plane of the Lévy index and the soliton power.     

Solitons in a system of three linearly coupled fiber gratings

We introduce a model of three parallel-coupled nonlinear waveguiding cores equipped with Bragg gratings (BGs), which form an equilateral triangle. The most promising way to create multi-core BG configuration is to use inverted gratings, written on internal surfaces of relatively broad holes embedded in a photonic-crystal-fiber matrix. The objective of the work is to investigate solitons and their stability in this system. New results are also obtained for the earlier investigated dual-core system. Families of symmetric and antisymmetric solutions are found analytically, extending beyond the spectral gap in both the dual- and tri-core systems. Moreover, these families persist in the case (strong coupling between the cores) when there is no gap in the systems linear spectrum. Three different types of asymmetric solitons are found (by means of the variational approach and numerical methods) in the tri-core system. They exist only inside the spectral gap, but asymmetric solitons with nonvanishing tails are found outside the gap as well. Stability of the solitons is explored by direct simulations, and, for symmetric solitons, in a more rigorous way too, by computation of eigenvalues for small perturbations. The symmetric solitons are stable up to points at which two types of asymmetric solitons bifurcate from them. Beyond the bifurcation, one type of the asymmetric solitons is stable, and the other is not. Then, they swap their stability. Asymmetric solitons of the third type are always unstable. When the symmetric solitons are unstable, their instability is oscillatory, and, in most cases, it transforms them into stable breathers. In both the dual- and tri-core systems, the stability region of the symmetric solitons extends far beyond the gap, persisting in the case when the system has no gap at all. The whole stability region of antisymmetric solitons (a new type of solutions in the tri-core system) is located outside the gap. Thus, solitons in multi-core BGs can be observed experimentally in a much broader frequency band than in the single-core one, and in a wider parameter range than it could be expected. Asymmetric delocalized solitons, found outside the spectral gap, can be stable too.Received: 13 August 2003PACS: 42.81.Dp Propagation, scattering, and losses; solitons - 42.65.Tg Optical solitons; nonlinear guided waves - 05.45.Yv Solitons     

Multiple solutions and hysteresis in the flows driven by surface with antisymmetric velocity profile

Multiple steady solutions and hysteresis phenomenon in the square cavity flows driven by the surface with antisymmetric velocity profile are investigated by numerical simulation and bifurcation analysis.A high order spectral element method with the matrix-free pseudo-arclength technique is used for the steady-state solution and numerical continuation.The complex flow patterns beyond the symmetry-breaking at Re■320 are presented by a bifurcation diagram for Re 2500.The results of stable symmetric and asymmetric solutions are consistent with those reported in literature,and a new unstable asymmetric branch is obtained besides the stable branches.A novel hysteresis phenomenon is observed in the range of2208 Re 2262,where two pairs of stable and two pairs of unstable asymmetric steady solutions beyond the stable symmetric state coexist.The vortices near the sidewall appear when the Reynolds number increases,which correspond to the bifurcation of topology structure,but not the bifurcation of Navier-Stokes equations.The hysteresis is proposed to be the result of the combined mechanisms of the competition and coalescence of secondary vortices.     

Nonlinear dual-core photonic crystal fiber couplers

We study nonlinear modes of dual-core photonic crystal fiber couplers made of a material with the focusing Kerr nonlinearity. We find numerically the profiles of symmetric, antisymmetric, and asymmetric nonlinear modes and analyze all-optical switching generated by the instability of the symmetric mode. We also describe elliptic spatial solitons controlled by the waveguide boundaries.     

Nonlinear guided waves and symmetry breaking in left-handed waveguides

We analyze nonlinear guided waves in a planar waveguide made of a left-handed material surrounded by a Kerr-like nonlinear dielectric, and predict that such a waveguide can support fast and slow symmetric and antisymmetric nonlinear modes. We study the symmetry breaking bifurcation and asymmetric modes in such a symmetric structure. The analysis of nonlinear dispersion properties of the guided waves shows that the modes can be either forward or backward.     

Symmetric and antisymmetric vector solitons for the fractional quadric-cubic coupled nonlinear Schrödinger equation

The fractional quadric-cubic coupled nonlinear Schrödinger equation is concerned, and vector symmetric and antisymmetric soliton solutions are obtained by the square operator method. The relationship between the Lévy index and the amplitudes of vector symmetric and antisymmetric solitons is investigated. Two components of vector symmetric and antisymmetric solitons show a positive and negative trend with the Lévy index, respectively. The stability intervals of these solitons and the propagation constants corresponding to the maximum and minimum instability growth rates are studied. Results indicate that vector symmetric solitons are more stable and have better interference resistance than vector antisymmetric solitons.     

Propagation and stability of the stationary stripe nonlinear guided waves in a symmetric structure with a nonlinear film

Abstact The propagation and stability of the stationary stripe nonlinear guided waves in an antiwaveguide symmetric layered structure containing a nonlinear film are simulated numerically. The symmetric, antisymmetric and asymmetric modes are considered, in both Kerr-law and saturable instantaneous self-focusing nonlinearity cases. The numerical calculations support the statement that stationary stripe nonlinear guided waves are always unstable on propagation for Kerr-law nonlinearities and the use of saturable nonlinear films leads to stability on propagation for effective indices corresponding to the ascending branches of the nonlinear dispersion curves.     

Families of spatial solitons in a two-channel waveguide with the cubic-quintic nonlinearity

We present eight types of spatial optical solitons which are possible in a model of a planar waveguide that includes a dual-channel trapping structure and competing (cubic-quintic) nonlinearity. The families of trapped beams include “broad” and “narrow” symmetric and antisymmetric solitons, composite states, built as combinations of broad and narrow beams with identical or opposite signs (“unipolar” and “bipolar” states, respectively), and “single-sided” broad and narrow beams trapped, essentially, in a single channel. The stability of the families is investigated via the computation of eigenvalues of small perturbations, and is verified in direct simulations. Three species-narrow symmetric, broad antisymmetric, and unipolar composite states-are unstable to perturbations with real eigenvalues, while the other five families are stable. The unstable states do not decay, but, instead, spontaneously transform themselves into persistent breathers, which, in some cases, demonstrate dynamical symmetry breaking and chaotic internal oscillations. A noteworthy feature is a stability exchange between the broad and narrow antisymmetric states: in the limit when the two channels merge into one, the former species becomes stable, while the latter one loses its stability. Different branches of the stationary states are linked by four bifurcations, which take different forms in the model with the strong and weak coupling between the channels.     

Fundamental modes in waveguide pipe twisted by saturated double-well potential

We study fundamental modes trapped in a rotating ring with a saturated nonlinear double-well potential. This model, which is based on the nonlinear Schrödinger equation, can be constructed in a twisted waveguide pipe in terms of light propagation, or in a Bose–Einstein condensate (BEC) loaded into a toroidal trap under a combination of a rotating π-out-of-phase linear potential and nonlinear pseudopotential induced by means of a rotating optical field and the Feshbach resonance. Three types of fundamental modes are identified in this model, one symmetric and the other two asymmetric. The shape and stability of the modes and the transitions between different modes are investigated in the first rotational Brillouin zone. A similar model used a Kerr medium to build its nonlinear potential, but we replace it with a saturated nonlinear medium. The model exhibits not only symmetry breaking, but also symmetry recovery. A specific type of unstable asymmetric mode is also found, and the evolution of the unstable asymmetric mode features Josephson oscillation between two linear wells. By considering the model as a configuration of a BEC system, the ground state mode is identified among these three types, which characterize a specific distribution of the BEC atoms around the trap.     

Solitons pinned to hot spots

We generalize a recently proposed model based on the cubic complex Ginzburg-Landau (CGL) equation, which gives rise to stable dissipative solitons supported by localized gain applied at a “hot spot” (HS), in the presence of the linear loss in the bulk. We introduce a model with the Kerr nonlinearity concentrated at the HS, together with the local gain and, possibly, with the local nonlinear loss. The model, which may be implemented in laser cavities based on planar waveguides, gives rise to exact solutions for pinned dissipative solitons. In the case when the HS does not include the localized nonlinear loss, numerical tests demonstrate that these solitons are stable/unstable if the localized nonlinearity is self-defocusing/focusing. Another new setting considered in this work is a pair of two symmetric HSs. We find exact asymmetric solutions for it, although they are unstable. Numerical simulations demonstrate that stable modes supported by the HS pair tend to be symmetric. An unexpected conclusion is that the interaction between breathers pinned to two broad HSs, which are the only stable modes in isolation in that case, transforms them into a static symmetric mode.     

Trapped states and transitions between them in double-well pseudopotentials

We analyze a model of a double-well pseudopotential (DWPP), based in the 1D Gross-Pitaevskii equation with a spatially modulated self-attractive nonlinearity. In the limit case when the DWPP structure reduces to the local nonlinearity coefficient represented by a set of two delta-functions, analytical solutions are obtained for symmetric, antisymmetric and asymmetric states. In this case, the transition from symmetric to asymmetric states, i.e., a spontaneous-symmetry-breaking (SSB) bifurcation, is subcritical. Numerical analysis demonstrates that the symmetric states are stable up to the SSB point, while emerging asymmetric states (together with all antisymmetric solutions) are unstable in the delta-function model. In a general model, which features a finite width of the nonlinear-potential wells, the asymmetric states quickly become stable, simultaneously with the switch of the bifurcation into the supercritical type. Antisymmetric solutions may also enjoy stabilization in the finite-width DWPP structure, demonstrating a bistability involving the asymmetric states. The symmetric states require a finite norm for their existence. A full diagram for the existence and stability of the trapped states is produced for the general model.     

In-plane antisymmetric response of cables through bifurcation under symmetric sinusoidally time-varying load

Analysis and results for in-plane non-linear antisymmetric responses of a cable, supported at the same level, through bifurcation under in-plane symmetric sinusoidally time-varying load are presented. The non-linear equation of the in-plane motion of the cable is solved by a Galerkin method and the harmonic balance method. From the computed results the frequency range, where the antisymmetric response occurs, varies with the sag-to-span ratio of the cable and is broad in the particular sag-to-span ratios. The second unstable region is important compared with the principal unstable region. Strong coupling between symmetric and antisymmetric modes is observed in the unstable regions for the particular sag-to-span ratios.     

Higher-order surface solitons in one-dimensional Bessel optical lattices

We demonstrate the existence of higher-order solitons occurring at an interface separating two one-dimensional (1D) Bessel optical lattices with different orders or modulation depths in a defocusing medium. We show that, in contrast to homogeneous waveguides where higher-order solitons are always unstable, the Bessel lattices with an interface support branches of higher-order structures bifurcating from the corresponding linear modes. The profiles of solitons depend remarkably on the lattice parameters and the stability can be enhanced by increasing the lattice depth and selecting higher-order lattices. We also reveal that the interface model with defocusing saturable Kerr nonlinearity can support stable multi-peaked solitons. The uncovered phenomena may open a new way for soliton control and manipulation.     

Generation of spatial solitons using non-linear guided modes

We investigate how two-dimensional spatial optical solitons can be generated in a non-linear Kerr medium using the non-linear guided modes of a weakly-guiding slab waveguide with a linear core and a non-linear cladding as the source of excitation. Symmetric, antisymmetric and asymmetric non-linear modes are considered, from which we determine the parameters of single solitons, oscillating two-soliton bound states, and two repelling solitons, respectively. Both the beam propagation method and inverse scattering transform are used.     

Interface kink solitons in defocusing saturable nonlinear media

We report on the dynamics of semi-localized nonlinear optical modes supported by an interface separating a uniform defocusing saturable medium and an imprinted semi-infinite photonic lattice. Out-of-phase and in-phase kink solitons composed by dark-soliton-like pedestals and oscillatory tails are found. Two branches of out-of-phase kink solitons exist in shallow lattices. Saturable nonlinearity enhances the pedestal height and renormalized energy flow of kink solitons evidently. While in-phase kink solitons are always unstable, out-of-phase kink solitons will be completely stable provided that lattice depth exceeds a critical value. Furthermore, stable kink solitons in the higher band gaps are also possible. Our results may give a helpful hint for understanding the dynamics of kink solitons with high pedestals in other fields.     

Bifurcations and stability of internal solitary waves

We study both supercritical and subcritical bifurcations of internal solitary waves propagating along the interface between two deep ideal fluids. We derive a generalized nonlinear Schrödinger equation to describe solitons near the critical density ratio corresponding to transition from subcritical to supercritical bifurcation. This equation takes into account gradient terms for the four-wave interactions (the so-called Lifshitz term and a nonlocal term analogous to that first found by Dysthe for pure gravity waves), as well as the six-wave nonlinear interaction term. Within this model, we find two branches of solitons and analyze their Lyapunov stability.     

Bifurcation,stability and symmetry of nonlinear waves

The bifurcation of wave-like spatio-temporal structures due to a hard-mode instability at non-zero wave number is investigated for a simple class of driven systems in one space dimension. We find generically a bifurcation of two branches of waves, travelling waves and standing waves, characterized by nontrivial subgroups of the symmetry group of the system. If both branches are supercritical, the wave with the larger amplitude is found to be stable. In all other cases, both waves are unstable for small amplitudes. At the common boundary of the stability regions of the two wave types in parameter space we find a bifurcation of a branch of modulated waves involving two independent frequencies, connecting the branches of travelling waves and standing waves.Work supported by the Swiss National Science Foundation     

Solitons in a medium with linear dissipation and localized gain

We present a variety of dissipative solitons and breathing modes in a medium with localized gain and homogeneous linear dissipation. The system possesses a number of unusual properties, like exponentially localized modes in both focusing and defocusing media, existence of modes in focusing media at negative propagation constant values, simultaneous existence of stable symmetric and antisymmetric localized modes when the gain landscape possesses two local maxima, as well as the existence of stable breathing solutions.     

Vector solitons in the dynamics of anharmonic monatomic lattices

It is shown that three types of solitary acoustic waves can develop in anharmonic crystal lattices corresponding to the three branches of acoustic phonons. A system of three nonlinear Schrödinger equations is derived to describe this situation. For greatly different group velocities, the interaction between solitons reduces collisions between them. When the group velocities of the different acoustic modes in a lattice are close to one another, bound states of the corresponding types of solitary waves occur. Bound states of this sort are vector solitons, whose polarization varies along the pulse. If the transverse acoustic modes are degenerate in velocity, the situation is extremely similar to the propagation of pulses in optical fibers.