Quintic Nonlinearity Research Articles

All meromorphic traveling waves of cubic and quintic complex Ginzburg-Landau equations

For both cubic and quintic nonlinearities of the one-dimensional complex Ginzburg-Landau evolution equation, we prove by a theorem of Eremenko the finiteness of the number of traveling waves whose squared modulus has only poles in the complex plane, and we provide all their closed form expressions. Among these eleven solutions, five are provided by the method used. This allows us to complete the list of solutions previously obtained by other authors.

Exact solutions of Euler–Bernoulli beams

In numerous real-world applications, transverse vibrations of beams are nonlinear in nature. It is a task to solve nonlinear beam systems due to their substantial dependence on the 4 variables of the system and the boundary conditions. To comprehend the nonlinear vibration characteristics, it is essential to do a precise parametric analysis. This research demonstrates an approximation solution for odd and even nonlinear transverse vibrating beams using the Laplace-based variation iteration method, and the formulation of the beams depends on the Galerkin approximation. For the solution of the nonlinear differential equation, this method is efficient as compared to the existing methods in the literature because the solutions exactly match with the numerical solutions. The Laplace-based variation iteration method has been used for the first time to obtain the solution to this important problem. To demonstrate the applicability and precision of the Laplace-based iteration method, several initial conditions are applied to the governing equation for nonlinearly vibrating transverse beams. The natural frequencies and periodic response curves are computed using Laplace-based VIM and compared with the Runge–Kutta RK4 method. In contrast to the RK4, the results demonstrate that the proposed method yields excellent consensus. The Lagrange multiplier is widely regarded as one of the most essential concepts in variational theory. The result obtained are displayed in the table form. Highlights The highlights of the solution of the Euler–Bernoulli beam equation with quintic nonlinearity using Lagrange multiplier are: 1. Introducing the constraint of the boundary conditions into the equation using Lagrange multipliers. 2. Formulating the equations for the Lagrange multipliers and the deflection of the beam. 3. Solving the resulting system of algebraic equations using numerical methods. 4. Obtaining the deflection of the beam as a function of its length and the applied load. 5. Analyzing the behavior of the beam under different loads and boundary conditions.

Dynamics of Gaussian and Super-Gaussian optical beams in weakly nonlinear nonlocal media

We investigate the dynamics of Gaussian and Super-Gaussian optical beams in weakly nonlocal nonlinear media with cubic quintic nonlinearities. Through the variational method, a set of internal parameters of beam propagation are constructed and their properties, illustrated from numerical simulations. We discover that Gaussian and Super-Gaussian optical beams generally perform stable propagation with different dynamics depending on whether beams order, quintic nonlinearity and/or nonlocality are taken into account. The evolution of the light beams is periodic due to the competition between nonlinearities and diffraction. The phase front curvature increases with the beams order when the width decreases. The numerical study of the interaction between a Gaussian and Super-Gaussian optical beams, shows that with or without the quintic nonlinearity and/or nonlocality, various behaviours such as repulsion, attraction and soft shock waves are observed. Moreover, new phenomena are exhibited in the case of the Super-Gaussian, according to their shape.

Reversible global centres with quintic homogeneous nonlinearities

A centre of a differential system in the plane is a singular point having a neighbourhood U such that is filled of periodic orbits. A global centre is a centre such that is filled of periodic orbits. To determine if a given differential system has a centre is in general a difficult problem, but it is even harder to know if it has a global centre. In the present paper we deal with the class of polynomial differential systems of the form (1) where P and Q are homogeneous polynomials of degree n. It is known that these systems can have global centres only if n is odd and the global centres in the cases n = 1 and n = 3 are known. Here we work with the case n = 5 and we classify the global centres of a four parameter family of systems (1). In particular we illustrate how to study the local phase portraits of the singular points whose linear part is identically zero using only vertical blow ups.

(Invited) Quiescent solitary waves in dual-core systems with separated Bragg grating and cubic–quintic nonlinearity

The existence and stability of zero-velocity solitons in a dual-core optical fiber, where one core possesses cubic–quintic nonlinearity and the other core is linear with a Bragg grating, are investigated. The frequency spectrum for the model consists of three bandgaps, of which only the central band gap is a genuine one. There are no soliton solutions in the central gap. On the other hand, the upper and lower gaps are filled with two distinct and disjoint families of soliton solutions, termed Type-1 and Type-2 solitons, that differ in their shapes and phase characteristics. When the relative group velocity in the linear core is zero, implicit analytical solutions for Type-1 and Type-2 quiescent solitons are found. For nonzero values of the relative group velocity, soliton solutions can only be found using numerical techniques. The stability of solitons has been analyzed systematically by means of numerical stability analysis. It is found that Type-2 solitons are always unstable. In the case of Type-1 solitons, stability regions are identified in the upper and lower bandgaps. The effects and interplay of the relative group velocity and the Bragg grating-induced linear coupling coefficient on the stability of the solitons are also analyzed.

Application of Odd Harmonic Resonances of a Single Crystal to Generation and Reception of Superharmonic Waves for Sensitive Monitoring of Heat-Treated Materials.

In nonlinear ultrasonic testing, the quadratic and more recently cubic nonlinearity parameters are frequently measured as a quantitative indicator of damaged material state. Application of higher-order harmonics can improve the sensitivity of detection and monitoring for damages and microstructures due to their higher values of nonlinearity parameters. The excitation and reception of higher-order harmonics, so-called superharmonics, which use the third to fifth harmonics arising from nonlinear wave propagation, is not sufficiently investigated and applied. The purpose of this communication is to develop a highly sensitive superharmonic nondestructive technique that efficiently generates and receives third- and fifth-order harmonics using the odd harmonic resonances of a single piezoelectric crystal. The method focuses on the measurement of fifth harmonic generation and reception, and the calculation of the relative quintic nonlinearity parameter (δ'). The method also addresses the issue of source nonlinearity that may be contained in the measured fifth harmonic amplitude. The measurement results of δ' for a series of precipitation heat-treated samples clearly show a much better sensitivity than the results of the cubic nonlinearity parameter (γ'). The proposed method enables a highly sensitive and true pulse-echo mode nonlinear ultrasound testing.

Stabilizing solitons of the cubic–quintic nonlinear Schrödinger equation by frequency-dependent linear gain-loss and delayed Raman response

We demonstrate transmission stabilization against radiation emission by frequency-dependent linear gain-loss and perturbation-induced frequency shifting for solitons of the cubic–quintic nonlinear Schrödinger (CQNLS) equation. We consider soliton propagation in a nonlinear optical waveguide with focusing cubic nonlinearity, defocusing quintic nonlinearity, and dissipative perturbations due to weak frequency-dependent linear gain-loss, cubic loss, and delayed Raman response. The frequency shifting is induced by delayed Raman response. Our perturbation analysis and numerical simulations with the perturbed CQNLS equation show that transmission stabilization with CQNLS solitons is indeed possible, and in this way provide the first demonstration of the stabilization method for solitons of a nonintegrable nonlinear wave model. Moreover, we find that transmission stabilization with energetic CQNLS solitons is realized with significantly smaller frequency shifts and pulse distortion compared with stabilization with energetic solitons of the cubic nonlinear Schrödinger equation. Therefore, our study also demonstrates that stabilization of energetic solitons by the method is significantly improved by the presence of defocusing quintic nonlinearity.

Chirped periodic waves and solitary waves for a generalized derivative resonant nonlinear Schrödinger equation with cubic–quintic nonlinearity

Chirped periodic waves and solitary waves for a generalized derivative resonant nonlinear Schrödinger equation with cubic–quintic nonlinearity

Peculiar optical solitons and modulated waves patterns in anti-cubic nonlinear media with cubic–quintic nonlinearity

Peculiar optical solitons and modulated waves patterns in anti-cubic nonlinear media with cubic–quintic nonlinearity

Spatiotemporal instabilities in [formula omitted]-symmetric periodic structures with non-Kerr fiber Bragg gratings

The influence of cubic–quintic (CQ) nonlinearity on the modulation instability (MI) in fiber Bragg grating structures in the conventional and three PT-symmetry types such as unbroken, singularity and broken are investigated. The MI gain was obtained through standard linear stability analysis. The effect of cubic nonlinearity combined with focusing and defocusing quintic nonlinearity on MI for both anomalous and normal GVD regimes, as well as the top and bottom edges of the photonic bandgap. We find that the MI gain varies dramatically because of the consequences of cubic–quintic nonlinearity. Finally, we show that even in the presence of PT-symmetry, effective control of the MI can be brought to light by adjusting the spatial instability parameter, cubic nonlinearity, focusing and defocusing quintic nonlinearity. The investigation yields a variety of spectrum types, including symmetry, asymmetry, monotonically increasing gain, sideband spectrum, broad spectrum, and so on. For the few physical variables, we observe an unusual spectrum that is more discrete in nature rather than continuous. After introducing the PT-symmetry, the emergence of solitons as in nonlinear periodic structure is also explored.

Optical solitons with an extended [formula omitted]-dimensional nonlinear conformable Schrödinger equation including cubic–quintic nonlinearity

In this paper, we study the extended (3+1)-dimensional nonlinear conformable Schrödinger equation with cubic–quintic nonlinearity. We use three different methods to obtain exact solutions of this equation: the G′/G expansion method, the extended hyperbolic method, and Nucci’s reduction method. We show that these methods are effective in finding solitary wave solutions, periodic wave solutions, and rational solutions of the equation. Besides, a first integral of the considered equation is derived by the Nucci reduction technique. Our results demonstrate the applicability of these methods in finding exact solutions to nonlinear PDEs, especially in cases where other methods are not effective.

Group classification and exact solutions of fractional differential equation with quintic non-Kerr nonlinearity term

Group classification and exact solutions of fractional differential equation with quintic non-Kerr nonlinearity term

Discrete solitons in competitive zigzag waveguide arrays with cubic-quintic nonlinearity

In this paper, we study one-dimensional discrete solitons in zigzag waveguide arrays with competitive cubic-quintic nonlinearity and competitive linear mixing between the nearest-neighbor (NN) and next-nearest-neighbor (NNN) couplings. The competitive nonlinearity features a cubic self-focusing associated with a quintic self-defocusing nonlinearities. The competitive linear mixing between the NN and NNN couplings is induced by making the two coefficients opposite of each other, which is assumed to be induced by the embedding synthetic gauge phase within the coupling constants. The combination of these two types of competition, linear mixing and nonlinearity can create four types of soliton: multipeak bell-shaped solitons, multipeak flat-top solitons, staggered bell-shaped solitons, and staggered flat-top solitons. The stability and dynamics of these types of solitons are verified systematically through the paper. The total power and the types of competition between the linear mixing play important roles in tuning these solitons.

Analytical study of Brinkman–Bénard convection in a bidisperse porous medium: Linear and weakly nonlinear study

Linear and weakly nonlinear stability analyses of Brinkman–Bénard convection of a Newtonian fluid saturating a bidisperse porous medium (BDPM) are made. Local-thermal-non-equilibrium (LTNE) is assumed between the fluid and the porous spheres (micro-pores) that make up the macro porous medium. Two coupled linear momentum equations are considered one each for the macro- and micro-pores. Results of mono-disperse porous medium (MDPM) with solid spheres are recovered as a limiting case of the present study. Further, in the case of both types of porous media considered the results of Darcy–Bénard and Brinkman–Bénard convection are extracted under suitable limiting procedures. To keep the work analytical, we reduce the intractable hexa-modal Lorenz model with four quadratic nonlinearities into the tractable mono-modal Stuart–Landau equation (SLE) with cubic and quintic nonlinearities. Subcritical instability is discounted in the study thereby suggesting that cubic SLE and cubic–quintic SLE both expound similar results qualitatively. The concept of a BDPM is shown to be meaningful only when the pores are not large, and when they are very small, then the MDPM assumption applies. Similar observation can be made when the ratio of permeabilities is large. The presence of micro-pores does not alter the size of the convective cell significantly at the onset. The present study reiterates the findings of several earlier works.

On ground state (in-)stability in multi-dimensional cubic-quintic Schrödinger equations

We consider the nonlinear Schrödinger equation with a focusing cubic term and a defocusing quintic nonlinearity in dimensions two and three. The main interest of this article is the problem of orbital (in-)stability of ground state solitary waves. We recall the notions of energy minimizing versus action minimizing ground states and prove that, in general, the two must be considered as nonequivalent. We numerically investigate the orbital stability of least action ground states in the radially symmetric case, confirming existing conjectures or leading to new ones.

Pure–quartic optical solitons and modulational instability analysis with cubic–quintic nonlinearity

We analyze the propagation dynamics of intense light pulses in an optical medium exhibiting cubic–quintic nonlinearity and pure fourth-order dispersion. We show that this physical system may support a rich variety of periodic wave solutions in the presence of all physical processes. Interestingly, in the long-wave limit, two different types of quartic dark solitons with equal amplitudes and wave numbers but different widths are identified for the first time. The obtained dark soliton solutions are shown to be independent of any free parameters and their characteristics are uniquely determined by system parameters. The numerical examples are presented for illustrating the soliton evolution dynamics in the waveguiding system. The formation of these quartic dark solitons occurs in normal pure fourth-order dispersion when the quintic and cubic nonlinearities are competitive. The modulational instability of a continuous wave is also analyzed under the combined effects of quartic dispersion and cubic–quintic nonlinearities. Additionally, the role of quintic nonlinearity and fourth-order dispersion effects on the modulational instability has been discussed.

A study of torsional vibration attenuation in closed-loop crankshaft model using absorbers with cubic and quintic smooth nonlinearities

A study of torsional vibration attenuation in closed-loop crankshaft model using absorbers with cubic and quintic smooth nonlinearities

Novel solutions of the generalized mixed nonlinear Schrödinger equation with nonzero boundary condition

We present a determinant representation of generalized Darboux transformation for a generalized mixed nonlinear Schrödinger equation, and obtain several novel solutions with nonzero boundary condition. A complete classification of first-order solutions with a nonzero boundary condition is considered, and several second-order solutions, including some interesting structures, are discussed. Furthermore, by selecting special parameters in the equation, several novel kinds of solutions for the equation are displayed. Finally, we discuss the effects of parameter $$\beta $$ , representing quintic nonlinearity term, on the breather solutions.

Hamiltonian form of extended cubic-quintic nonlinear Schrödinger equation in a nonlinear Klein-Gordon model.

We derive an extended cubic-quintic nonlinear Schrödinger equationwith Hamiltonian structure in a nonlinear Klein-Gordon model with cubic-quintic nonlinearity. We use the nonlinear dispersion relation to properly take into account the input of high-order nonlinear effects in the Hamiltonian perturbation approach to nonlinear modulation. We demonstrate that changing the balance between the cubic and quintic nonlinearities has a significant effect on the stability of unmodulated wave packets to long-wave modulations.

Soliton formation and dynamics in the quintic nonlinear media with [formula omitted]-invariant harmonic-Gaussian potential

In this paper, we introduce a PT-symmetric harmonic-hyperbolic-Gaussian (HHG) potential in the nonlinear Schrödinger (NLS) equation with quintic nonlinearity. Firstly, we present the critical thresholdless PT-phase symmetry breaking in the linear regime. Secondly, we give both exact and numerical single-/double-hump solitons of the quintic NLS equations with the PT-symmetric HHG potential, and show their stabilities by studying the spectrum stability analysis and direct wave propagations. Finally, we investigate the interactions between the single-/double-hump solitons and the exotic hyperbolic secant function, as well as the adiabatic excitations of the solitons with the operation of an adiabatic switch function. All these obtained results provide a theoretical guidance for the applications of the PT-symmetric theory into diverse areas of physics.