Dual-power Nonlinearity Research Articles

Classification of traveling waves to the generalized Camassa–Holm equation with dual-power nonlinearities

In this paper, we mainly devote to investigate the classification of traveling waves to the generalized Camassa–Holm equation with dual-power nonlinearities. Utilizing the celebrated approach of classification of traveling wave solution which was proposed by Jonatan Lenells [Traveling wave solutions of the Camassa-Holm equation. J Diff Equ. 2005;217(2):393–430]. We show a result concerns the regularity of traveling waves and then classify all traveling wave solutions.

Dual-power nonlinear energy sink for targeted energy transfer in ultra-wide range of impulsive energy

Recently, the NES with one third power (OTP) nonlinearity was proposed to attenuate micro-vibrations under low-level initial impulsive energy. However, the targeted energy transfer (TET) region for effective vibration mitigation is narrow. This paper develops a new type of NES with dual-power (DP) nonlinearity by combining the third power (TP) nonlinearity and the OTP one in parallel. Firstly, the dynamic behaviors of the DP NES are investigated, and significant vibration mitigation is observed in two discontinuous TET regions that are separately induced by the OTP and TP nonlinearities. Furthermore, the parameter analysis is conducted to find out a feasible avenue to merge such two discontinuous TET regions. It demonstrates that there exist optimal parameters to achieve a wide and continuous TET region. Finally, the design optimization of the DP NES is carried out with the aid of the genetic algorithm, and the optimal parameters enable the DP NES to trigger TET in an ultra-wide and continuous range of initial impulsive energy. Consequently, the proposed DP NES should be an effective solution to substantially broaden the TET region and attenuate vibration in an ultra-wide range of initial impulsive energy.

Wave-breaking phenomena for the generalized Camassa–Holm equation with dual-power nonlinearities

In this paper, we mainly devote to investigate the generalized Camassa–Holm equation with dual-power nonlinearities. We first establish the local well-posedness by applying the Kato’s semigroup theory. Then, the precise blow-up result is obtained by using the transport equation theory and Moser-type estimates. Moreover, according to the different real-valued intervals in which the dispersive parameter s is located, the sufficient conditions which guarantee the occurrence of wave-breaking of solutions are studied. It is worth noting that we need to overcome the difficulty caused by complicated mixed dual-power nonlinear structure and balance the relationship between the various dispersive parameters to get corresponding convolution estimates.

Pseudo-spectral approach for extracting optical solitons of the complex Ginzburg Landau equation with six nonlinearity forms

In this paper, a compelling pseudo-spectral approach namely split-step Fourier transform (SSFT) is utilized for the first time to report the complex Ginzburg Landau (CGL) equation. In this regard, six nonlinear forms of this notable equation are associated with this model, which are the kerr law nonlinearity, power law nonlinearity, quadratic-cubic nonlinearity, cubic-quintic nonlinearity, dual power nonlinearity, and polynomial law nonlinearity. Since opting for an analytical solution is complicated and only provided for a limited set of soliton solutions, the numerical solution might render a more generalized aspect in addressing versatile solutions for a wide range of arbitrary input pulses. Therefore, the SSFT scheme is efficiently employed to extract a plethora of optical solitons solutions such as kink waves, bright, dark, bright-dark, singular, singular periodic solitons. The obtained results, via MATLAB, are in total agreement with the findings explored form the exact analytical solutions. As a result, this numerical technique may provide a creditable mathematical tool to solve a wide range of other nonlinear evolution partial differential equations in the mathematical physics field.

PT symmetric cubic-quintic nonlinear Schrödinger equation with dual power nonlinearities and its solitonic solutions

We propose a more general nonlocal integrable nonlinear Schrödinger equation with nonlocal cubic and quintic nonlinearities in addition to external potential. We employ similarity transformation and Lax-pair method to achieve integrability of the model and then suitably exploit the Darboux transformation approach to generate more general solitonic solutions which enable us to exhibit all kinds of soliton pairs and breather solutions by fine-tuning spectral parameters. We also witness how one can manoeuver the parameters to suppress or retain the amplitude of the solitons desirably. We also show how the interaction parameter can be manipulated to transfer energy from unstable mode to stable mode in a two soliton solution.

Integrability conditions and solitonic solutions of the nonlinear Schrödinger equation with generalized dual-power nonlinearities, PT-symmetric potentials, and space- and time-dependent coefficients

We consider a generalized nonlinear Schrödinger equation with dual power law nonlinearities, complex potential, and position- and time-dependent strengths of dispersion and nonlinearities. Using a standard similarity transformation, we obtain the integrability conditions and solitonic solutions of this equation by mapping it to its homogeneous version. Using a modified similarity transformation, where a solution of the homogeneous equation, which we denote as a seed, enters also in the transformation operator, a wider range of exact solutions is obtained including cases with complex potentials. We apply these two transformations to obtain two exact solitonic solutions of the homogeneous nonlinear Schrödinger equation, which are derived here for the first time for a general power of the nonlinearities, namely the flat-top soliton and tanh solution. We discuss and derive explicit solutions to the experimentally relevant cases associated with parabolic and PT-symmetric potentials.

Numerical simulation of KdV equation by finite difference method

In this study, the numerical solutions to the KdV equation with dual power nonlinearity by using the finite difference method are obtained. Discretize equation is presented in the form of finite difference operators. The numerical solutions are secured via the analytical solution to the KdV equation with dual power nonlinearity which is present in the literature. Through the Fourier–Von Neumann technique and linear stable, we have seen that the FDM is stable. Accuracy of the method is analyzed via the $${L}_{2}$$ and $${L}_{\infty }$$ norm errors. The numerical, exact approximations and absolute error are presented in tables. We compare the numerical solutions with the exact solutions and this comparison is supported with the graphic plots. Under the choice of suitable values of parameters, the 2D and 3D surfaces for the used analytical solution are plotted.

New exact and approximate stationary solutions of (1+2)D nonlinear Schrödinger equation in dual-power media

In the paper propagation of axially-symmetric (1+2)D beam in nonlinear medium with dual-power nonlinearity is analyzed. The ordinary differential equation for transverse stationary profile of the propagating field is derived and solved using a perturbation technique. The simple analytical formulas for the three lowest order solutions are obtained. They describe fields of algebraic profiles. The zero order solution satisfies exactly the nonlinear Schrödinger equation in (1+2)D case. Higher order solutions are determined by propagation constant and describe fields of different initial amplitude. The accuracy of approximation and stability of the obtained solutions are discussed.

Soliton solution for an inhomogeneous highly dispersive media with a dual-power nonlinearity law

Optical soliton propagation in an inhomogeneous highly dispersive media with a dual-power nonlinearity law is studied. The variable-coefficient nonlinear Schrödinger (NLS) equation describing the propagation in such media includes space-dependent coefficients of fourth-order dispersion, cubic–quintic nonlinearity, and attenuation. By means of a solitary wave ansatz, exact bright soliton solution is found. Importantly, we have found that it is always possible to express the propagation of bright soliton pulses, for any integer value of the exponent in the power law in the form of a power of hyperbolic secant function. All physical parameters in the soliton solution are obtained as a function of the space-dependent model coefficients. Note that, it is always useful and desirable to construct exact analytical solutions for the understanding of most nonlinear physical phenomena.

Optical solitons in highly dispersive media with a dual-power nonlinearity law

We find bright and dark solitary wave solutions for a nonlinear Schrodinger equation in the presence of fourth-order dispersion for an optical medium modelled by a dual-power nonlinearity law. Particular cases are discussed.