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We investigate the modulational instability (MI) of a continuous wave (cw) under the combined effects of higher-order dispersions, self steepening and self-frequency shift, cubic, quintic, and septic nonlinearities. Using Maxwell's theory, an extended nonlinear Schr\"odinger equation is derived. The linear stability analysis of the cw solution is employed to extract an expression for the MI gain, and we point out its sensitivity to both higher-order dispersions and nonlinear terms. In particular, we insist on the balance between the sixth-order dispersion and nonlinearity, septic self-steepening, and the septic self-frequency shift terms. Additionally, the linear stability analysis of cw is confronted with the stability conditions for solitons. Different combinations of the dispersion parameters are proposed that support the stability of solitons and the occurrence of MI. This is confronted with full numerical simulations where the input cw gives rise to a broad range of behaviors, mainly related to nonlinear patterns formation. Interestingly, under the activation of MI, a suitable balance between the sixth-order dispersion and the septic self-frequency shift term is found to highly influence the propagation direction of the optical wave patterns.

A one-dimensional modified Nogochi nonlinear electric transmission network with dispersive elements that consist of a large number of identical sections is theoretically studied in the present paper. The first-order nonautonomous rogue waves with quintic nonlinearity and nonlinear dispersion effects in this network are predicted and analyzed using the cubic-quintic nonlinear Schrödinger (CQ-NLS) equation with a cubic nonlinear derivative term. The results show that, in the semidiscrete limit, the voltage for the transmission network is described in some cases by the CQ-NLS equation with a derivative term that is derived employing the reductive perturbation technique. A one-parameter first-order rational solution of the derived CQ-NLS equation is presented and used to investigate analytically the dependency of the characteristics of the first-order rouge wave parameters on the electric transmission network under consideration. Our results show that when we change the quintic nonlinear and nonlinear dispersion parameter, the first-order nonautonomous rogue wave transforms into the bright-like soliton. Our results also reveal that the shape of the first-order nonautonomous rogue waves does not change when we tune the quintic nonlinear and nonlinear dispersion parameter, while the quintic nonlinear term and nonlinear dispersion effect affect the velocity of first-rogue waves and the evolution of their phase. We also show that the network parameters as well as the frequency of the carrier voltage signal can be used to manage the motion of the first-order nonautonomous rogue waves in the electrical network under consideration. Our results may help to control and manage rogue waves experimentally in electric networks.

We present breather solutions of the quintic integrable equation of the Schr\"odinger hierarchy. This equation has terms describing fifth-order dispersion and matching nonlinear terms. Using a Darboux transformation, we derive first-order and second-order breather solutions. These include first- and second-order rogue-wave solutions. To some extent, these solutions are analogous with the corresponding nonlinear Schr\"odinger equation (NLSE) solutions. However, the presence of a free parameter in the equation results in specific solutions that have no analogues in the NLSE case. We analyze new features of these solutions.

We present the fifth-order equation of the nonlinear Schrödinger hierarchy. This integrable partial differential equation contains fifth-order dispersion and nonlinear terms related to it. We present the Lax pair and use Darboux transformations to derive exact expressions for the most representative soliton solutions. This set includes two-soliton collisions and the degenerate case of the two-soliton solution, as well as beating structures composed of two or three solitons. Ultimately, the new quintic operator and the terms it adds to the standard nonlinear Schrödinger equation (NLSE) are found to primarily affect the velocity of solutions, with complicated flow-on effects. Furthermore, we present a new structure, composed of coincident equal-amplitude solitons, which cannot exist for the standard NLSE.

We consider a system of two cubic-quintic non-linear Schr\"odinger equations in two dimensions, coupled by repulsive cubic terms. We analyse situations in which a probe lump of one of the modes is surrounded by a fluid of the other one and analyse their interaction. We find a realization of D'Alembert's paradox for small velocities and non-trivial drag forces for larger ones. We present numerical analysis including the search of static and traveling form-preserving solutions along with simulations of the dynamical evolution in some representative examples.

We investigate the behavior of the electromagnetic wave that propagates in a metamaterial for negative index regime. Second-order dispersion and cubic-quintic nonlinearities are taken into account. The behavior obtained for negative index regime is compared to that observed for absorption regime. The collective coordinates technique is used to characterize the light pulse intensity profile at some frequency ranges. Five frequency ranges have been pointed out. The perfect combination of second-order dispersion and cubic nonlinearity leads to a robust soliton at each frequency range for negative index regime. The soliton peak power progressively decreases for absorption regime. Further, this peak power also decreases with frequency. We show that absorption regime can induce rogue wave trains generation at a specific frequency range. However, this rogue wave trains generation is maintained when the quintic nonlinearity comes into play for negative index regime and amplified for absorption regime at a specific frequency range. It clearly appears that rogue wave behavior strongly depends on the frequency and the regime considered. Furthermore, the stability conditions of the electromagnetic wave have also been discussed at frequency ranges considered for both negative index and absorption regimes.

Bose-Einstein condensates with time varying two- and three-body interatomic interactions, confined in a linear potential and exchanging atoms with the thermal cloud are investigated. Using the extended tanh-function method with an auxiliary equation, i.e., the Lenard equation, many exact solutions describing the dynamics of matter-wave condensates are derived. An important issue is the time management of the cubic and the quintic nonlinearities by tuning the rate of exchange of atoms between the condensate and the thermal background. In addition, adjusting the strength of the linear potential, the rate of exchange of atoms, and many other free parameters allow one to control many features of the condensate such as its height, width, position, velocity, acceleration, and its direction, respectively. Full numerical solutions corroborate the analytical predictions.

This article analyzes the properties of rotationally symmetric self-localized solutions with different radial quantum numbers in the simplest one- and two-component reaction-diffusion systems. The consideration is made in one and two dimensions with the focus on the fundamental and first higher-order solutions showing zero and one intersections of the radial profile with zero. It is demonstrated that the solution with one intersection does not exist for the case of the quadratic-cubic nonlinearity, while the cubic-quintic extension of the models does allow existence. I show additionally that the cubic-quintic reaction diffusion system supports the existence and stability of the states with zero quantum numbers, as well as their antistates, state-antistate pairs, and clusters, which can be interpreted as the states with nonzero azimuthal quantum number.

A subset of the soliton solutions of the cubic-quintic nonlinear Schrodinger equation (NLSE) is presented in analytical form. General criteria for existence are expressed in terms of the parameters of the NLSE. The normalized momentum entering the stability criterion is evaluated explicitly.

Collapse of (1 + 1)-dimensional beams in the inhomogeneous one-dimensional quintic nonlinear Schrodinger equation is analyzed both numerically and analytically. It is shown that in the vicinity of a narrow attractive inhomogeneity, the collapse of beams in which the homogeneous medium would blow up may be delayed and even arrested. [S1063-651X(99)03610-7].

We study the nonlinear phase equation for propagating patterns. We investigate the transition from a propagating localized pattern to a space-filling spatiotemporally disordered pattern and discuss in detail to what extent there are propagating localized states that breathe in time periodically, quasiperiodically, and chaotically. Differences and similarities to the phenomena occurring for the quintic complex Ginzburg-Landau equation are elucidated. We also discuss for which experimentally accessible systems one could observe the phenomena described.

We report results of systematic simulations of interactions between dark solitons in the complex quintic Ginzburg-Landau equation. Bound states of the solitons are found. The bound states (which are not possible in the cubic equation) exist in a wide range of parameters and are highly stable, providing an example of a stable bound state of solitary pulses in a generalized Ginzburg-Landau equation.

We have performed a theoretical study of the soliton fiber laser based on the quintic complex Ginzburg-Landau equation (CGLE). This study may also apply to soliton propagation in telecommunications systems. We have developed a simple approach that allows us to obtain, in an approximate way, analytical expressions for the stable pulselike solutions of the CGLE. The method also gives an accurate estimate of the region in the parameter space where stable pulselike solutions exist. We also obtain that the minimum allowed value of the peak amplitude of the soliton solutions depends solely on the relation between the linear loss term and the quintic gain saturation term. The predictions are confirmed by numerical simulations., This work was supported by the Comunidad de Madrid under Contract No. 06T/039/96 and by the CICyT under Contract No. TIC95-0563. Part of this research was done while J.M.S.C was at the University of Cantabria. Their hospitality is appreciated.

Time-localized solitary wave solutions of the one-dimensional complex Ginzburg-Landau equation (CGLE) are analyzed for the case of normal group-velocity dispersion. Exact solutions solutions are found for both the cubic and the quintic CGLE. The stability of these solutions is investigated numerically. The regions in the parameter space in which stable pulselike solutions of the quintic CGLE exist are numerically determined. These regions contain subspaces where analytical solutions may be found. An investigation of the role of group-velocity dispersion changes in magnitude and sign on the spectral and temporal characteristics of the stable pulse solutions is also carried out., The work of J.M.S.-C. was supported by the CICyT under Contract No. TIC95-0563-C05-03 and by the Comunidad de Madrid under Contract No. 06T/039/96. N.A. would like to thank the Australian Academy of Sciences for financial support.

The stability and collision dynamics of moving solitons in Bragg gratings with cubic-quintic nonlinearity and dispersive reflectivity are investigated. Two disjoint families of solitons are found on the plane of the coefficient of quintic nonlinearity versus the normalized frequency $(\ensuremath{\eta},{\ensuremath{\Omega}}_{\mathrm{norm}})$. Through numerical stability analysis, we have identified stability regions on the $(\ensuremath{\eta},{\ensuremath{\Omega}}_{\mathrm{norm}})$ plane for various values of dispersive reflectivity parameter ($m$) and velocity ($v$). The size of stability regions is found to be dependent on $m$ and $v$. Collisions of counterpropgating Type 1 and Type 2 solitons have been systematically investigated. It is found that for low to moderate values of dispersive reflectivity, the collisions of Type 1 solitons can result in various outcomes such as separation of solitons with reduced, increased, unchanged, or asymmetric velocities and generation of a quiescent soliton by merger or formation of three solitons. For strong dispersive reflectivity (e.g., $m=0.5$), the collisions of low-velocity in-phase Type 1 solitons may lead to repulsion of solitons, asymmetric separation, merger into a single soliton, or formation of three solitons (one quiescent and two moving solitons). At higher velocities collisions predominantly lead to the formation of three solitons. For $m=0.5$, in-phase Type 2 solitons may repel or form a temporary bound state of quiescent Type 1 solitons that subsequently splits into two asymmetrically separating Type 1 solitons. $\ensuremath{\pi}$-out-of-phase Type 2 solitons may also merge to form a quiescent Type 1 soliton.

A subset of the exact analytical traveling-wave solutions of the equation i${\mathrm{\ensuremath{\Psi}}}_{\mathit{t}}$+${\mathrm{\ensuremath{\Psi}}}_{\mathit{zz}}$=${\mathit{a}}_{1}$\ensuremath{\Psi}|\ensuremath{\Psi}${\mathrm{|}}^{2}$+${\mathit{a}}_{2}$\ensuremath{\Psi}|\ensuremath{\Psi}${\mathrm{|}}^{4}$ is presented in compact form. The solutions are expressed in terms of Weierstrass's elliptic function \ensuremath{\wp} and include periodic and solitary-wave-like solutions. The algebraic conditions for both cases are given. Examples are presented for simple cases. \textcopyright{} 1996 The American Physical Society.

We consider travelling wave solutions of the reaction diffusion equation with quintic nonlinearities $u_t = u_{xx} + \mu u (1 -u ) ( 1 +\alpha u + \beta u^2 +\gamma u^3)$. If the parameters $\alpha , \beta$ and $\gamma$ obey a special relation, then the criterion for the existence of a strong heteroclinic connection can be expressed in terms of two of these parameters. If an additional restriction is imposed, explicit front solutions can be obtained. The approach used can be extended to polynomials whose highest degree is odd., Comment: Revtex, 5 pages

We explore the existence of moving nonradiating kinks in nonlocal generalizations of ${\ensuremath{\varphi}}^{4}$ and ${\ensuremath{\varphi}}^{4}$-${\ensuremath{\varphi}}^{6}$ models. These models are described by nonlocal nonlinear Klein-Gordon equation, ${u}_{tt}\ensuremath{-}\mathcal{L}u+F(u)=0$, where $\mathcal{L}$ is a Fourier multiplier operator of a specific form and $F(u)$ includes either just a cubic term (${\ensuremath{\varphi}}^{4}$ case) or cubic and quintic (${\ensuremath{\varphi}}^{4}$-${\ensuremath{\varphi}}^{6}$ case) terms. The general mechanism responsible for the discretization of kink velocities in the nonlocal model is discussed. We report numerical results obtained for these models. It is shown that, contrary to the traditional ${\ensuremath{\varphi}}^{4}$ model, the nonlocal ${\ensuremath{\varphi}}^{4}$ model does not admit moving nonradiating kinks but admits solitary waves that do not exist in the local model. At the same time the nonlocal ${\ensuremath{\varphi}}^{4}$-${\ensuremath{\varphi}}^{6}$ model describes moving nonradiating kinks. The set of velocities allowed for these kinks is discrete with the highest possible velocity ${c}_{1}$. This set of velocities is unambiguously determined by the parameters of the model. Numerical simulations show that a kink launched at the velocity $c$ higher than ${c}_{1}$ starts to decelerate, and its velocity settles down to the highest value of the discrete spectrum ${c}_{1}$.

We consider the quintic one dimensional nonlinear Schr\"odinger equation with forcing and both linear and nonlinear dissipation. Quintic nonlinearity results in multiple collapse events randomly distributed in space and time forming forced turbulence. Without dissipation each of these collapses produces finite time singularity but dissipative terms prevents actual formation of singularity. In statistical steady state of the developed turbulence the spatial correlation function has a universal form with the correlation length determined by the modulational instability scale. The amplitude fluctuations at that scale are nearly-Gaussian while the large amplitude tail of probability density function (PDF) is strongly non-Gaussian with power-like behavior. The small amplitude nearly-Gaussian fluctuations seed formation of large collapse events. The universal spatio-temporal form of these events together with the PDF for their maximum amplitudes define the power-like tail of PDF for large amplitude fluctuations, i.e., the intermittency of strong turbulence., Comment: 14 pages, 17 figures

A large family of analytical solitary wave solutions to the generalized nonautonomous cubic-quintic nonlinear Schr\"odinger equation with time- and space-dependent distributed coefficients and external potentials are obtained by using a similarity transformation technique. We use the cubic nonlinearity as an independent parameter function, where a simple procedure is established to obtain different classes of potentials and solutions. The solutions exist under certain conditions and impose constraints on the coefficients depicting dispersion, cubic and quintic nonlinearities, and gain (or loss). We investigate the space-quadratic potential, optical lattice potential, flying bird potential, and potential barrier (well). Some interesting periodic solitary wave solutions corresponding to these potentials are then studied. Also, properties of a few solutions and physical applications of interest to the field are discussed. Finally, the stability of the solitary wave solutions under slight disturbance of the constraint conditions and initial perturbation of white noise is discussed numerically; the results reveal that the solitary waves can propagate in a stable way under slight disturbance of the constraint conditions and the initial perturbation of white noise.

We deal with the three-dimensional Gross-Pitaevskii equation which is used to describe a cloud of dilute bosonic atoms that interact under competing two- and three-body scattering potentials. We study the case where the cloud of atoms is strongly confined in two spatial dimensions, allowing us to build an unidimensional nonlinear equation,controlled by the nonlinearities and the confining potentials that trap the system along the longitudinal coordinate. We focus attention on specific limits dictated by the cubic and quintic coefficients, and we implement numerical simulations to help us to quantify the validity of the procedure.

The problem of wave-number selection by a ramp, i.e., a region smoothly matching sub- and supercritical domains, is considered within the framework of the cubic and quintic Ginzburg-Landau (GL) equations with a sign-changing overcriticality parameter. A local frequency is also allowed to be a smooth function of the spatial coordinate. For the cubic model, a unique value of the selected wave number is found by means of an asymptotic procedure valid when the imaginary parts of coefficients in the GL equation are small, while the group velocity is arbitrary. Under certain conditions, the wave number may lie outside the stability band, which is expected to give rise to a dynamical chaos. In the quintic model, which describes a system with the inverted bifurcation, the selection scenario is much simpler: A front separating a traveling wave and the trivial state is expected to be pinned at the point of the ramp where its velocity, regarded as a function of the local overcriticality, vanishes. Eventually, the wave-number selection is performed by the pinned front. Experimentally, the selection may be realized in the traveling-wave convection, or in a self-oscillatory chemical system.

We analyze the occurrence of vibrational resonance in a damped quintic oscillator with three cases of single well of the potential V(x)=1/2omega(0)(2)x(2)+1/4betax(4)+1/6gammax(6) driven by both low-frequency force f cos omegat and high-frequency force g cos Omegat with Omegaomega. We restrict our analysis to the parametric choices (i) omega(0)(2), beta, gamma0 (single well), (ii) omega(0)(2), gamma0, beta0, beta(2)4omega(0)(2)gamma (single well), and (iii) omega(0)(2)0, beta arbitrary, gamma0 (double-hump single well). From the approximate theoretical expression of response amplitude Q at the low-frequency omega we determine the values of omega and g (denoted as omega(VR) and g(VR)) at which vibrational resonance occurs. We show that for fixed values of the parameters of the system when omega is varied either resonance does not occur or it occurs only once. When the amplitude g is varied for the case of the potential with the parametric choice (i) at most one resonance occur while for the other two choices (ii) and (iii) multiple resonance occur. Further, g(VR) is found to be independent of the damping strength d while omega(VR) depends on d. The theoretical predictions are found to be in good agreement with the numerical result. We illustrate that the vibrational resonance can be characterized in terms of width of the orbit also.

We investigate analytically and numerically the modulational instability of a Bose-Einstein condensate with both two- and three-body interatomic interactions and trapped in an external parabolic potential. Analytical investigations performed lead us to establish an explicit time-dependent criterion for the modulational instability of the condensate. The effects of the potential as well as of the quintic nonlinear interaction are studied. Direct numerical simulations of the Gross-Pitaevskii equation with two- and three-body interactions describing the dynamics of the condensate agree with the analytical predictions.

Propagation of an optical high-power cylindrically symmetric beam in a material characterized by cubic-quintic nonlinearity is studied both analytically and numerically. In this case we have to consider the self-defocusing effect caused by the presence of free electrons produced due to plasma formation. The variational method is used to study the system analytically. The finite-difference beam propagation method is used for the numerical analysis. Stable (2+1) D spatial solitons are observed. The analytical results are found to be in very good agreement with the numerical results.

Systematic results are reported for dynamics of circular patterns (“necklaces”), composed of fundamental solitons and carrying orbital angular momentum, in the two-dimensional model, which describes the propagation of light beams in bulk media combining self-focusing cubic and self-defocusing quintic nonlinearities. Semianalytical predictions for the existence of quasistable necklace structures are obtained on the basis of an effective interaction potential. Then, direct simulations are run. In the case when the initial pattern is far from an equilibrium size predicted by the potential, it cannot maintain its shape. However, a necklace with the initial shape close to the predicted equilibrium survives very long evolution, featuring persistent oscillations. The quasistable evolution is not essentially disturbed by a large noise component added to the initial configuration. Basic conclusions concerning the necklace dynamics in this model are qualitatively the same as in a recently studied one which combines quadratic and self-defocusing cubic nonlinearities. Thus, we infer that a combination of competing self-focusing and self-defocusing nonlinearities enhances the robustness not only of vortex solitons but also of vorticity-carrying necklace patterns.

The nonlinear dynamics of laser beams carrying phase singularity in media with cubic-quintic nonlinearity changing from self-focusing to self-defocusing is examined. A novel kind of stable nonlocalized optical vortices appears in such media as well as localized vortex solitons. Linear stability analysis and numerical simulations show the stability of localized vortices only in the defocusing region.