Dynamics of NLS solitons described by the cubic-quintic Ginzburg-Landau equation

The generalized moment method is applied to average the Ginzburg-Landau equation with quintic nonlinearity in the neighborhood of a soliton solution to the nonlinear Schrödinger equation. A qualitative analysis of the resulting dynamical system is presented. New soliton solutions bifurcating from a known exact soliton solution are obtained. The results of the qualitative analysis are compared with those obtained by direct numerical solution of the Ginzburg-Landau equation.     

Chirped Lambert W-kink solitons of the complex cubic-quintic Ginzburg-Landau equation with intrapulse Raman scattering

In this paper, an exact explicit solution for the complex cubic-quintic Ginzburg-Landau equation is obtained, by using Lambert W function or omega function. More pertinently, we term them as Lambert W-kink-type solitons, begotten under the influence of intrapulse Raman scattering. Parameter domains are delineated in which these optical solitons exit in the ensuing model. We report the effect of model coefficients on the amplitude of Lambert W-kink solitons, which enables us to control efficiently the pulse intensity and hence their subsequent evolution. Also, moving fronts or optical shock-type solitons are obtained as a byproduct of this model. We explicate the mechanism to control the intensity of these fronts, by fine tuning the spectral filtering or gain parameter. It is exhibited that the frequency chirp associated with these optical solitons depends on the intensity of the wave and saturates to a constant value as the retarded time approaches its asymptotic value.     

A collective variable approach for optical solitons in the cubic-quintic complex Ginzburg-Landau equation with third-order dispersion

Considering the theory of electromagnetic, especially from the Maxwell equations, a basic equation modeling the propagation of ultrashort optical solitons in optical fibers is derived, namely a cubic-quintic complex Ginzburg-Landau equation (CQGLE) with third-order dispersion (TOD). Considering this one-dimensional CQGLE, we derive the equations of motion of pulse parameters called collective variables (CVs), of a pulse propagating in dispersion-managed (DM) fiber optic-links. Equations obtained are investigated numerically in order to view the evolution of pulse parameters along the propagation distance. A fully numerical simulation of the CQGLE finally tests the results of the CV theory. It appears chaotic pulses, attenuate pulses and stable pulses under some parameter values.     

Interactions between two-dimensional solitons in the diffractive-diffusive Ginzburg-Landau equation with the cubic-quintic nonlinearity

We report the results of systematic numerical analysis of collisions between two and three stable dissipative solitons in the two-dimensional (2D) complex Ginzburg-Landau equation (CGLE) with the cubic-quintic (CQ) combination of gain and loss terms. The equation may be realized as a model of a laser cavity which includes the spatial diffraction, together with the anomalous group-velocity dispersion (GVD) and spectral filtering acting in the temporal direction. Collisions between solitons are possible due to the Galilean invariance along the spatial axis. Outcomes of the collisions are identified by varying the GVD coefficient, β, and the collision “velocity” (actually, it is the spatial slope of the soliton’s trajectory). At small velocities, two or three in-phase solitons merge into a single standing one. At larger velocities, both in-phase soliton pairs and pairs of solitons with opposite signs suffer a transition into a delocalized chaotic state. At still larger velocities, all collisions become quasi-elastic. A new outcome is revealed by collisions between slow solitons with opposite signs: they self-trap into persistent wobbling dipoles, which are found in two modifications — horizontal at smaller β, and vertical if β is larger (the horizontal ones resemble “zigzag” bound states of two solitons known in the 1D CGL equation of the CQ type). Collisions between solitons with a finite mismatch between their trajectories are studied too.     

Cascade replication of soliton solutions in the one-dimensional complex cubic-quintic Ginzburg-Landau equation

In the framework of the one-dimensional cubic-quintic complex Ginzburg-Landau equation, we demonstrate two methods to achieve cascade replication of dissipative solitons from a single one by using external forces whose positions and magnitudes are being precisely controlled. The first method combines high amplitude positive currents and negative currents to increase the efficiency of the process significantly and to make the process repeatable. By using two repulsive forces whose locations are time dependent, the second method enables one to obtain multiple dissipative solitons in only one cascade replication process.     

Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation

We demonstrate the existence of stable toroidal dissipative solitons with the inner phase field in the form of rotating spirals, corresponding to vorticity S=0, 1, and 2, in the complex Ginzburg-Landau equation with the cubic-quintic nonlinearity. The stable solitons easily self-trap from pulses with embedded vorticity. The stability is corroborated by accurate computation of growth rates for perturbation eigenmodes. The results provide the first example of stable vortex tori in a 3D dissipative medium, as well as the first example of higher-order tori (with S=2) in any nonlinear medium. It is found that all stable vortical solitons coexist in a large domain of the parameter space; in smaller regions, there coexist stable solitons with either S=0 and S=1, or S=1 and S=2.     

Exact solutions of discrete complex cubic-quintic Ginzburg-Landau equation with non-local quintic term

In this paper, via the extended tanh-function approach, the abundant exact solutions for discrete complex cubic-quintic Ginzburg-Landau equation, including chirpless bright soliton, chirpless dark soliton, constant magnitude solution (plane wave solution), triangular function solutions and some solutions with alternating phases, etc. are obtained. Meanwhile, the range of parameters where some exact solution exist are given. Among these solutions, solutions with alternating phases do not have continuous analogs. Moreover, in the lattice, the points of singularity of tan-type and sec-type solutions can be ‘between sites’ and thus the singularities can be avoided.     

Stability of traveling pulses of cubic-quintic complex Ginzburg-Landau equation including intrapulse Raman scattering

The complex cubic-quintic Ginzburg-Landau equation (CGLE) admits a special type of solutions called eruption solitons. Recently, the eruptions were shown to diminish or even disappear if a term of intrapulse Raman scattering (IRS) is added, in which case, self-similar traveling pulses exist. We perform a linear stability analysis of these pulses that shows that the unstable double eigenvalues of the erupting solutions split up under the effect of IRS and, following a different trajectory, they move on to the stable half-plane. The eigenfunctions characteristics explain some eruptions features. Nevertheless, for some CGLE parameters, the IRS cannot cancel the eruptions, since pulses do not propagate for the required IRS strength.     

Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimensional complex cubic-quintic Ginzburg-Landau equations

The generation and nonlinear dynamics of multidimensional optical dissipative solitonic pulses are examined. The variational method is extended to complex dissipative systems, in order to obtain steady state solutions of the (D + 1)-dimensional complex cubic-quintic Ginzburg-Landau equation (D = 1, 2, 3). A stability criterion is established fixing a domain of dissipative parameters for stable steady state solutions. Following numerical simulations, evolution of any input pulse from this domain leads to stable dissipative solitons.     

Heavily-chirped solitary pulses in the normal dispersion region: New solutions of the cubic-quintic complex Ginzburg-Landau equation

A new type of the heavily-chirped solitary pulse solutions of the nonlinear cubic-quintic complex Ginzburg-Landau equation has been found. The methodology developed provides for a systematic way to find the approximate but highly accurate analytical solutions of this equation with the generalized nonlinearities within the normal dispersion region. It is demonstrated that these solitary pulses have the extra-broadened parabolic-top or fingerlike spectra and allow compressing with more than a hundredfold growth of the pulse peak power. The obtained solutions explain the energy scalable regimes in the fiber and solid-state oscillators operating within the normal dispersion region and promise to achieve microjoules femtosecond pulses at MHz repetition rates.     

Collisions between discrete spatiotemporal dissipative Ginzburg-Landau solitons in two-dimensional photonic lattices

Systematic results of collisions between discrete spatiotemporal dissipative Ginzburg-Landau solitons in two-dimensional photonic lattices are reported. The generic outcomes are identified for (i) the collision of two identical solitons located in the corner, at the edge, and in the center of the photonic lattice, and for (ii) the collision of two non-identical corner and edge solitons located at different distances from the boundaries of the photonic lattice. Depending on the values of the kick (collision momentum) and of the nonlinear (cubic) gain, the collision scenarios include soliton merging, creation of an extra soliton, soliton bouncing, soliton spreading, and quasi-elastic (symmetric) interactions.     

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.     

On the inverse scattering transform in two spatial and one temporal dimensions

1 IlltroductionThe.goal of this paper is to develop a formalism for (a) solving inverse problem in theplane for potentials decaying at infinity (i.e. given appropriate scattering data reconstruct thepotential q(x, y)); (b) .solving the initial value problem (for appropriately decaying initial data)of certain non-linear evolution equations in two spatial and one temporal dimension (i.e. givenq(x, ys 0) find q(x, y, t)). Several results thus obtained are also formallied. This formalism hasbeen d…     

Bistability in the complex Ginzburg-Landau equation with drift

Properties of the complex Ginzburg-Landau equation with drift are studied focusing on the Benjamin-Feir stable regime. On a finite interval with Neumann boundary conditions the equation exhibits bistability between a spatially uniform time-periodic state and a variety of nonuniform states with complex time dependence. The origin of this behavior is identified and contrasted with the bistable behavior present with periodic boundary conditions and no drift.     

Stability of vortex solitons in the cubic-quintic model

We investigate one-parameter families of two-dimensional bright spinning solitons (ring vortices) in dispersive media combining cubic self-focusing and quintic self-defocusing nonlinearities. In direct simulations, the spinning solitons display a symmetry-breaking azimuthal instability, which leads to breakup of a soliton into a set of fragments, each being a stable nonspinning soliton. The fragments fly out tangentially to the circular crest of the original vortex ring. If the soliton’s energy is large enough, the instability develops so slowly that the spinning solitons may be regarded as virtually stable ones, in accord with earlier published results. Growth rates of perturbation eigenmodes with different azimuthal “quantum numbers” are calculated as a function of the soliton’s propagation constant κ from a numerical solution of the linearized equations. As a result, a narrow (in terms of κ) stability window is found for extremely broad solitons with values of the “spin” s=1 and 2. However, analytical consideration of a special perturbation mode in the form of a spontaneous shift of the soliton’s central “bubble” (core of the vortex embedded in a broad soliton) demonstrates that even extremely broad solitons are subject to an exponentially weak instability against this mode. In actual simulations, a manifestation of this instability is found in a three-dimensional soliton with s=1. In the case when the two-dimensional spinning solitons are subject to tangible azimuthal instability, the number of the nonspinning fragments into which the soliton splits is usually, but not always, equal to the azimuthal number of the instability eigenmode with the largest growth rate.     

On the invariants of the complex Ginzburg-Landau equation

Using Lie group theory, both the invariants and the similarity variables of the complex Ginzburg—Landau equation V_xx = a(V_t + bV) + cV|V|^2kwherea,b,c?Iandk?I are constructed.     

Twisted vortex filaments in the three-dimensional complex Ginzburg-Landau equation

The structure and dynamics of vortex filaments that form the cores of scroll waves in three-dimensional oscillatory media described by the complex Ginzburg-Landau equation are investigated. The study focuses on the role that twist plays in determining the bifurcation structure in various regions of the (alpha,beta) parameter space of this equation. As the degree of twist increases, initially straight filaments first undergo a Hopf bifurcation to helical filaments; further increase in the twist leads to a secondary Hopf bifurcation that results in supercoiled helices. In addition, localized states composed of superhelical segments interspersed with helical segments are found. If the twist is zero, zigzag filaments are found in certain regions of the parameter space. In very large systems disordered states comprising zigzag and helical segments with positive and negative senses exist. The behavior of vortex filaments in different regions of the parameter space is explored in some detail. In particular, an instability for nonzero twist near the alpha=beta line suggests the existence of a nonsaturating state that reduces the stability domain of straight filaments. The results are obtained through extensive simulations of the complex Ginzburg-Landau equation on large domains for long times, in conjunction with simulations on equivalent two-dimensional reductions of this equation and analytical considerations based on topological concepts.     

Partial spatial synchronization of chaotic oscillations in the Ginzburg-Landau equation

A possibility of generalized synchronization between two parts of a spatially distributed system being in space-time chaos is demonstrated with the Ginzburg-Landau equation used as an example. Regions of the distributed system parameters at which the functional relationship is established between the parts of the system are determined.     

Motion of spiral waves in the complex Ginzburg-Landau equation

Solutions of the general cubic complex Ginzburg-Landau equation comprising multiple spiral waves are considered. For parameters close to the vortex limit, and for a system of spiral waves with well-separated centres, laws of motion of the centres are found which vary depending on the order of magnitude of the separation of the centres. In particular, the direction of the interaction changes from along the line of centres to perpendicular to the line of centres as the separation increases, with the strength of the interaction algebraic at small separations and exponentially small at large separations. The corresponding asymptotic wavenumber and frequency are determined. These depend on the positions of the centres of the spirals, and so evolve slowly as the spirals move.