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The nonlinear Schrodinger hierarchy has a wide range of applications in modeling the propagation of light pulses in optical fibers. In this paper, we focus on the integrable nonlinear Schrodinger (NLS) equation with quintic terms, which play a prominent role when the pulse duration is very short. First, we investigate the spectral signatures of the spatial Lax pair with distinct analytical solutions and their periodized wavetrains by Fourier oscillatory method. Then, we numerically simulate the wave evolution of the quintic NLS equation from different initial conditions through the symmetrical split-step Fourier method. We find many localized high-peak structures whose profiles are very similar to the analytical solutions, and we analyze the formation of rouge waves (RWs) in different cases. These results may be helpful to understand the excitation of nonlinear waves in some nonlinear fields, such as optical fibers, oceanography and so on.

We investigate the dromion-like excitations corresponding to intramolecular chain-like proteins. In the present work, the dromion-like excitations are described by using cubic-quintic nonlinear Schrodinger equation (CQNSE) governing the dynamics of proteins and we analytically analyze the velocity (v) of dromion-like structure compared with velocity ( $$v_a$$ ) of acoustical sound waves corresponding to the longitudinal vibrations of protein molecules. Our work is motivated by the effectiveness and powerful mathematical techniques such as modified extended tanh function method and sine–cosine function method for solving CQNSE to obtain dromion-like structures.

We investigate the inhomogeneous higher-order nonlinear Schrodinger (INHLS) equation including cubic–quintic–septic (CQS) nonlinear terms and gain or loss with variable coefficients. The exact analytic solution that describes dark soliton-type pulse propagation is found for the model by employing the ansatz method. Unlike the traditional $$\text {tanh}$$ dark soliton in Kerr-type media, the functional form of this novel dark-managed soliton structure takes a $$\text {sech}^{2/3}$$ profile. Parametric conditions are presented in which these optical solitons exist. We also investigated the stability of these dark-managed solitons under some initial perturbations by employing the numerical simulation methods. Finally, the interaction dynamics of two and three dark-managed solitons has been numerically explored.

In this work, we use the inverse scattering transform method to consider the focusing Kundu–Eckhaus (KE) equation with nonzero background (NZBG) at infinity. Based on the analytical, symmetric, asymptotic properties of eigenfunctions, the inverse problem is solved via a matrix Riemann–Hilbert problem (RHP). The general multi-pole solutions are given in terms of the solution of the associated RHP. And the formula of the N simple-pole soliton solutions are obtained, too. We show that the Peregrine’s rational solution can be viewed as some appropriate limit of the simple-pole soliton solutions at branch point. Furthermore, by taking some other proper limits, the two and three simple-pole soliton solutions can yield to the double- and triple-pole solutions for focusing KE equation with NZBG. The effect of the parameter $$\beta $$ , characterizing the strength of the non-Kerr (quintic) nonlinear and the self-frequency shift effect, and some typical collisions between solutions of focusing KE equation are graphically displayed.

In this paper, a variable-coefficient cubic–quintic nonlinear Schrodinger equation involving five arbitrary real functions of space and time is analyzed from the point of view of symmetry analysis by using Lie’s invariance infinitesimal criterion. The infinitesimal generators of corresponding equivalence transformations are presented. The first-order differential invariants are constructed to identify when the equation can be mapped to a constant-coefficient cubic–quintic nonlinear Schrodinger equation. The constrained conditions on the variable coefficients we arrived here extend the cases discussed before and present more general results. Some brightlike and darklike solitary wave solutions for special potentials and cubic–quintic nonlinearities are obtained.

In this paper, we investigate the coupled cubic-quintic nonlinear Schrodinger equation, which describes the effects of quintic nonlinearity on the ultrashort optical pulse propagation in non-Kerr media. The breather wave solutions and novel bright–dark rogue wave solutions can be constructed by using the Darboux-dressing transformation and asymptotic expansion. These solutions show the spatiotemporal patterns of novel bright–dark rogue waves. It is demonstrated that the coupled or multi-component systems contain more interesting rogue wave phenomena than single component systems. Our results can be of importance in understanding and predicting the rogue waves of coupled cubic-quintic nonlinear Schrodinger equation.

In this paper, we introduce the third-order flow equation of the Kaup–Newell (KN) system. We study this equation, and we obtain different types of solutions by using the Darboux transformation (DT) and the extended DT of the KN system, such as solitons, positons, breathers, and rogue waves. The extended DT is obtained by taking the degenerate eigenvalues $$ \lambda _{i} \rightarrow \lambda _{1} (i=3,5,7,\ldots ,2k-1)$$ and by performing the Taylor expansion near $$\lambda _{1}$$ of the determinants of DT. Some analytic expressions are explicitly given for the first-order solutions. We study the unique waveforms of both the first-order and higher-order rogue-wave solutions for special choices of parameters, and we find different types of such wave structures: fundamental pattern, triangular, modified-triangular, pentagram, ring, ring-triangular, and multi-ring wave patterns. We conclude that the third-order dispersion and quintic nonlinear term of the KN system modify both the trajectories and speeds of the solutions as compared with those corresponding to the second-order flow equation of the KN system.

In this work, the variable-coefficients complex cubic–quintic Ginzburg–Landau equation (CCQGLE) influenced by higher-order effects and nonlinear gain is considered. Based on the asymmetric method, analytic one-soliton solution for the variable-coefficients CCQGLE is constructed for the first time. In addition, with some certain conditions, the periodic wave and dromion-like structures are derived. The results obtained may be helpful in understanding the solitons amplification and solitons management in optical fiber.

Studies on the water waves contribute to the design of the related industries, such as the marine and offshore engineering, while the media with the negative refractive index can be applied as the carrier media in fiber optics. In consideration of the inhomogeneities of the media and nonuniformities of the boundaries in the real physical backgrounds, a quintic time-dependent-coefficient derivative nonlinear Schrodinger equation for certain hydrodynamic wave packets or medium with the negative refractive index is investigated in this paper. Bilinear forms and the N-soliton solutions with respect to the nonzero background, which are different from those in the existing studies, are derived under the certain constraints. Conditions for the dark/anti-dark/gray solitons are deduced due to the properties of the solitons derived via the asymptotic analysis. Effects of the dispersion coefficient $$\lambda (t)$$ , self-steepening coefficient $$\alpha (t)$$ , cubic nonlinearity $$\mu (t)$$ and quintic nonlinearity $$\nu (t)$$ on the interactions between the anti-dark and gray solitons under the certain condition are investigated. Interactions among the dark, anti-dark and gray solitons are discussed under two cases: when $${\alpha (t)}/{\lambda (t)}$$ and $${\mu (t)}/{\lambda (t)}$$ are the constants, whether the interaction is elastic or not depends on whether $$\lambda (t)$$ , $$\alpha (t)$$ and $$\mu (t)$$ are the constants or the functions of t; when $${\alpha (t)}/{\lambda (t)}$$ and $${\mu (t)}/{\lambda (t)}$$ are related to t, if the velocity of the soliton is a periodic function of t, the propagation of the corresponding soliton is periodic and the corresponding interaction is inelastic. Interactions among the three/four solitons are described to be elastic or inelastic based on the changes in the velocities and waveforms of the three/four solitons after the interactions.

Railway vehicles suffer from hunting motion, even when traveling below the critical speed obtained by linear analysis, due to the nonlinear characteristics of the wheel system. Nonlinear characteristics in Hopf bifurcations can be characterized as subcritical or supercritical, depending on whether the cubic nonlinearity is softening or hardening, respectively. In a system with softening cubic nonlinearity, third-order nonlinear analysis cannot detect nontrivial stable steady-state oscillations because they are affected by quintic nonlinearity. Therefore, in such a system, it is necessary to apply fifth-order nonlinear analysis to a system model in which quintic nonlinearity is taken into account. In this study, we investigated the cubic and quintic nonlinear phenomena in hunting motion with a roller rig that is widely used for hunting motion research. Previous experimental studies using a roller rig were restricted to the linear stability and the cubic nonlinear stability. We clarified that roller rig experiments can observe the hysteresis phenomenon and the existence of subcritical Hopf and saddle-node bifurcations, indicating that not only the cubic but also the quintic nonlinearity of the wheel system plays an important role. In addition, we obtained the normal form governing the nonlinear dynamics. We developed an experimental identification method to obtain the coefficients of the normal form. The validity of our method was confirmed by comparing the bifurcation diagrams obtained from the experimental time history and the normal form whose coefficients were experimentally identified using the proposed method.

Under parity-time symmetric potentials, different-order nonlinearities such as cubic, quintic and septimal nonlinearities, altogether with their combinations and second-order and fourth-order dispersions/diffractions are simultaneously considered to form three-dimensional optical solitons. Based on some high-order nonlinear Schrodinger equations, three-dimensional analytical optical soliton solutions are found. In the defocusing cubic nonlinear case, three-dimensional optical soliton without fourth-order diffraction/dispersion is stable than that with fourth-order diffraction/dispersion. However, in the defocusing cubic and focusing quintic nonlinear case, the stability situation of soliton is just on the contrary. Among all combinations of nonlinearity, the stability of three-dimensional optical soliton in the cubic-quintic nonlinear case is better than that in the cubic nonlinear case, but worse than that in the cubic-quintic-septimal nonlinear case. In the quintic-septimal nonlinear case, three-dimensional optical soliton is unstable and will collapse ultimately.

We report on the existence and stability of Gaussian-type soliton in the nonlinear Schrodinger (NLS) equation with interplay of cubic–quintic nonlinearity, fourth-order diffraction (FOD) and novel quartic anharmonic parity-time ( $$\mathcal {PT}$$ )-symmetric Gaussian potential. We study numerically the impact of the FOD coefficient on the regions of unbroken/broken linear $$\mathcal {PT}$$ -symmetric phases. In the nonlinear domain, we derive exact soliton solutions of the one-dimensional and two-dimensional cubic–quintic NLS equation with $$\mathcal {PT}$$ -symmetric Gaussian potential and FOD coefficients. Moreover, the stability of the constructed soliton solution is investigated. The results of linear stability analysis are validated by comparison with numerical simulations. Furthermore, we also show that the relative strength of the FOD coefficient influences the direction of the power flow.

Under investigation in this paper is a quintic time-dependent coefficient derivative nonlinear Schrodinger equation for certain hydrodynamic wave packets or a medium with the negative refractive index. A gauge transformation is found to obtain the equivalent form of the equation. With respect to the wave envelope for the free water surface displacement or envelope of the electric field, Painleve integrable condition, different from that in the existing literature, is derived, with which the bilinear forms and N-soliton solutions are constructed. Asymptotic analysis illustrates that the interactions between the bright and bound solitons as well as between the bright solitons and Kuznetsov–Ma breathers are elastic with certain conditions, while some other interactions are inelastic under other conditions. Propagation paths and velocities for the solitons are both affected by the dispersion coefficient function when the relations among the coefficients are linear, or affected by the dispersion coefficient, self-steepening coefficient and cubic nonlinearity functions when the relations among the coefficients are nonlinear. Under different conditions, bell-shaped solitons can evolve into the bound solitons or Kuznetsov–Ma breathers, respectively. Interactions between the bright and parabolic (or hyperbolic) solitons are related to the dispersion coefficient, self-steepening coefficient and cubic nonlinearity functions. Compression effect on the propagation paths of the solitons, caused by the dispersion coefficient, is observed.

We investigate the generation of breathers, solitons, and rogue waves of the quintic nonlinear Schrodinger equation (QNLSE) on uniform and elliptical backgrounds. The QNLSE is the general nonlinear Schrodinger equation that includes all terms up to the fifth-order dispersion. We use Darboux transformation to construct initial conditions for the dynamical generation of localized high-intensity optical waves. The condition for the breather-to-soliton conversion is provided with the analysis of soliton intensity profiles. We discover a new class of higher-order solutions in which Jacobi elliptic functions are set as background seed solutions of the QNLSE. We also introduce a method for generating a new class of rogue waves—the periodic rogue waves—based on the matching of the periodicity of higher-order breathers with the periodicity of the background dnoidal wave.

We numerically analyzed the supratransmission phenomenon in the discrete nonlinear Schrodinger equation with the cubic–quintic nonlinearity. It has been reported that the homoclinic nonlinear band-gap threshold matches very well with the model. In the case of the cooperation between the nonlinearities (self-focusing cubic and quintic terms), the train of discrete band-gap waves overcomes the potential barrier of the first sites before merging or rebounding. In the case of competing self-focusing cubic and defocusing quintic nonlinearities, it is found that the lattice induces the generation of the train of dark solitons carried by a traveling kink and the traveling kink for chosen driving amplitude.

We use the method of multiple scales to derive a sixth-order nonlinear Schrodinger equation governing the evolution of slowly modulated plane-wave solutions to the nonlinear Klein–Gordon equation with polynomial nonlinearity. The coefficients of this sixth-order equation are expressed explicitly in terms of the velocity parameter as well as linear, quadratic, cubic, quadruple, and quintic nonlinear coefficients of the original nonlinear Klein–Gordon equation.

In this paper, we consider an extended nonlinear Schrodinger equation that includes fifth-order dispersion with matching higher-order nonlinear terms. Via the modified Darboux transformation and Joukowsky transform, we present the superregular breather (SRB), multipeak soliton and hybrid solutions. The latter two modes appear as a result of the higher-order effects and are converted from a SRB one, which cannot exist for the standard NLS equation. These solutions reduce to a small localized perturbation of the background at time zero, which is different from the previous analytical solutions. The corresponding state transition conditions are given analytically. The relationship between modulation instability and state transition is unveiled. Our results will enrich the dynamics of nonlinear waves in a higher-order wave system.

Gaussian spatial soliton solutions of both the constant-coefficient and variable-coefficient (2 + 1)-dimensional nonlinear Schrodinger equations in quintic–septimal nonlinear materials with different diffractions are presented under two kinds of $${\mathcal {P}}{\mathcal {T}}$$ -symmetric potentials. The linear stability analysis and direct numerical simulation are jointly utilized to investigate the stability for analytical solutions of the constant-coefficient equation. Results from the linear stability analysis and the direct numerical simulation possess a high degree of consistency, that is, the stable case for Gaussian spatial solitons of the constant-coefficient equation appears only in the defocusing quintic and focusing septimal nonlinear material. Moreover, reconstruction of stable Gaussian spatial solitons of the variable-coefficient equation is studied based on the expression of the effective propagation distance Z(z) by choosing an appropriate form of diffraction $$\beta _1(z)$$ .

A ( $$2+1$$ )-dimensional nonlinear Schrodinger equation with cubic–quintic–septimal nonlinearities and $$\mathcal {PT}$$ -symmetric potentials is studied, and two kinds of Gaussian-type spatial soliton solutions are obtained. The compressed behaviors of spatial solitons are investigated in three different diffraction systems with Gaussian, Logarithmic and linear profiles. The real and imaginary parts of $$\mathcal {PT}$$ -symmetric potentials, the width, amplitude and phase transition of spatial solitons in different media are compared and studied.

We investigated the cubic–quintic nonlinear response in a parametrically excited simply supported beam subjected to a spring force in the axial direction. Taking into account the cubic and quintic geometric nonlinearities of curvature of the beam, the governing equation of the parametrically excited beam was derived based on Hamilton’s principle. The fifth-order approximate solution was analytically obtained using the method of multiple scales; with this calculation, the third-order nonlinear normal mode was also obtained. Its associated amplitude revealed saddle-node bifurcation and a hysteresis in the frequency response curve, which could not be predicted using the third-order approximate solution for the governing equation that included only cubic nonlinearity. Experimental results taken using a simple apparatus qualitatively verify the theoretically predicted nonlinear features in the parametric resonance caused by the cubic–quintic geometric nonlinearity of the beam.

We discuss the possibility of breather-to-soliton conversions for a higher-order nonlinear Schrodinger–Maxwell–Bloch system with quintic terms describing the propagation of ultrashort optical pulses, up to the attosecond duration, in an erbium-doped fiber. The analytic forms for the vector breather solutions of that system with complex (not pure imaginary) eigenvalues are obtained with the Darboux transformation. With the special values of the eigenvalues, we find that vector breather solutions can be converted into the vector soliton solutions on the constant (not zero) backgrounds. The condition for such conversion is explicitly derived. We show the interactions between vector breathers and vector solitons.

In this paper, we present a method to compute focal values and periodic constants at infinity of a class of switching systems and apply it to study a cubic system. We prove that such a cubic system can have 7 limit cycles in the sufficiently small neighborhood of infinity. Moreover, we consider a quintic switching system to obtain 14 limit cycles at infinity, while continuous quintic systems can have only 11 limit cycles in the sufficiently small neighborhood of infinity. This indicates that switching systems or discontinuous systems can exhibit more complex dynamics compared to smooth systems.

The nonlinear dynamics of a hybrid Rayleigh–Van der Pol–Duffing oscillator includes pure and impure quadratic damping are investigated. The multiple timescales method is used to study exhaustively various resonances states. It is noticed that the system presents nine resonances states. The frequency response curves of quintic, third and second superharmonic, and subharmonic resonances states are obtained. Bistability, hysteresis, and jump phenomenon are also obtained. It is pointed out that these resonance phenomena are strongly related to the nonlinear cubic and quadratic damping and to the external force. The numerical simulations are used to make bifurcation sequences displayed by the model for each type of oscillatory. It is noticed that the pure quadratic, impure cubic damping, and external excitation affect regular and chaotic states.

A (3+1)-dimensional nonlinear Schrodinger equation with variable-coefficient dispersion/diffraction and cubic-quintic-septimal nonlinearities is studied, two families of analytical light bullet solutions with two types of $${{\mathcal {PT}}}$$ -symmetric potentials are obtained. The coefficient of the septimal nonlinear term strongly influences the form of light bullet. The direct numerical simulation indicates that light bullet solutions in different cubic-quintic-septimal nonlinear media exhibit different property of stability, and under different $${\mathcal {PT}}$$ -symmetric potentials they also show different stability against white noise. These stabilities of evolution originate from subtle interplay among dispersion, diffraction, nonlinearity and $${\mathcal {PT}}$$ -symmetric potential. Moreover, compression and expansion of light bullets in the hyperbolic dispersion/diffraction system and periodic modulation system are investigated numerically. The evolution of light bullet in periodic modulation system is more stable than that in the hyperbolic dispersion/diffraction system.

A ( $$1+1$$ )-dimensional inhomogeneous cubic–quintic–septimal nonlinear Schrodinger equation with $$\mathcal {PT}$$ -symmetric potentials is studied, and two families of soliton solutions are obtained. From soliton solutions, the amplitude of soliton is independent of the $$\mathcal {PT}$$ -symmetric potential parameter k; however, the phase depends on the parameter k. The phase of soliton alters from negative to positive values at the location of center. Moreover, the evolutional behaviors of these solitons are discussed.

Analytical spatiotemporal localized soliton solutions in cubic and power-law competing nonlinear media with different diffractions and $$\mathcal {PT}$$ -symmetric potentials are derived. The compression and expansion of localized soliton structures in the diffraction decreasing system and periodic modulation system are studied analytically and numerically. The numerical simulation shows that exact solutions exist stable region in the defocusing cubic and focusing quintic nonlinear medium. However, exact solutions in the focusing cubic and defocusing quintic nonlinear medium are unstable, namely they cannot maintain their original shapes, they are distorted and collapse, and they ultimately decay into noise. These results might stimulate the related experiment to observe localized structures of nonlinear Schrodinger equation in $$\mathcal {PT}$$ -symmetric systems.

Using similarity transformations, the analytical solutions to the quintic nonlinear Schrodinger equation with potentials and nonlinearities depending both on time and space are constructed. The stability of resonance solitons is examined by direct numerical simulation. It is found that stable fundamental soliton states with $$m=0$$ and low-order soliton states with $$m=1$$ can be supported both by self-focusing and self-defocusing materials. Higher-order solitons are found unstable, however, displaying quasi-stable propagation over prolonged distances.

We investigate the two-dimensional cubic–quintic nonlinear Schrodinger (CQNLS) equation with space-modulated potential and nonlinearities, which describes the nonlinear wave interactions in nonlinear optics and Bose–Einstein condensates. We use the modified conformal mapping, the Cauchy–Riemann equation, and its extension to present many types of new wave structures of this CQNLS model for different external potentials and nonlinearities, which exhibit bright-like and dark-like solitons, vortex solitons, and other types of wave solutions. Moreover, we also analyze the velocity fields and transverse power-flow or Poynting vectors related to the obtained wave solutions and find the direct relationships between the velocity fields and transverse power-flow or Poynting vectors.

A neural network scheme based on quintic B-spline functions of optimality systems arising from linear parabolic optimal control problems is discussed. It is also shown that the proposed neural network model is stable in the sense of Lyapunov and it is globally convergent to the optimal solution of the original problem. Theoretical and experimental results show the advantages of B-spline functions and demonstrate that the suggested scheme is able to efficiently solve constrained optimal control problems. The resulting scheme also shows that robustness with respect to changes of the value of $$\nu $$ , the weight of the cost of the control, is sufficiently small.

Using the self-similarity transformation, we find analytical spatial bright and dark self-similar solitons, i.e., the similaritons, of the generalized (2+1)-dimensional quintic nonlinear Schrodinger equation with varying diffraction, nonlinearity, and gain. Characteristic examples with physically relevant behavior of these similaritons are studied, and the stability of these solutions is verified with numerical integration.

We study two kinds of the generalized (3 $$+$$ 1)-dimensional cubic-quintic Schrodinger equation in $$\mathcal {PT}$$ -symmetric potentials and obtain two families (sech-type and Gaussian-type) and four kinds of analytical light bullet (LB) solutions. The stability of these solutions is tested by the linear stability analysis and the direct numerical simulation. Results imply that sech-type LB solutions are unstable for all parameters only in the extended Rosen–Morse potentials. Sech-type and Gaussian-type LB solutions are both stable below some thresholds for the imaginary part of other $$\mathcal {PT}$$ -symmetric potentials in the defocusing cubic and focusing quintic medium, while they are always unstable for all parameters in other media. Moreover, we discuss the broadened and compressed behaviors of LBs in inhomogeneous hyperbolic system and periodic amplification system.

Based on the Kundu–Eckhaus equation with variable coefficients, the exact one-soliton and two-soliton solutions have been explicitly given by an appropriate similarity transformation method. As an example, an exponential distributed fiber control system, nonlinearity management system and dispersion management system have been considered, and the propagation characteristics of the exact soliton solutions in the three nonuniform management systems have been investigated in detail. Especially, the dynamic properties of the amplitude, pulse width and the central position of the soliton with transmission distance have been studied. The results have some guiding significance for soliton amplification, compression and control management, and can provide some theoretical analysis for carrying out optical soliton communication experiment.

Inspired by Llibre and Vallls (in J. Math. Anal. Appl. 357:427–437, 2009), the conditions of center and isochronous center at the origin for a class of non-analytic quintic systems are studied in this paper. By a transformation, we first transform the systems into analytic systems, then sufficient and necessary conditions for the origin of the systems being a center are obtained. The fact that 11 limit circles could be bifurcated is proved. A complete classification of the sufficient and necessary conditions is given for the origin of the systems being an isochronous center.

In this paper, we study a class of cubic Z 2-equivariant polynomial Hamiltonian systems under the perturbation of Z 2-equivariant polynomial of degree 5. First, we consider the unperturbed system and obtain necessary and sufficient conditions for the critical point (0,1) to be a nilpotent saddle, center, or cusp. We show that it can have 14 different phase portraits. Using the methods of Hopf and homoclinic bifurcation theory, we study the bifurcation problem of the perturbed system and prove that there exist 12 limit cycles.

In this paper, center conditions and bifurcation of limit cycles at the nilpotent critical point in a class of quintic polynomial differential system are investigated. With the help of the computer algebra system MATHEMATICA, the first 11 quasi-Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact that there exist 11 small amplitude limit cycles created from the three-order nilpotent critical point is also proved. Henceforth, we give a lower bound of cyclicity of three-order nilpotent critical point for quintic Lyapunov systems. The results of Jiang et al. (Int. J. Bifurcation Chaos 19:2107–2113, 2009) are improved.