Rational Wave Solutions Research Articles

NEW SOLITON SOLUTIONS IN NANO-FIBERS WITH SPACE-TIME FRACTIONAL DERIVATIVES

The conformable space-time fractional perturbed nonlinear Schrödinger equation (PNLSE) having three different laws of nonlinearity namely, Kerr law, parabolic law, and power law is considered in this paper. The PNLSE is a nonlinear model which arises in nano-fibers. An efficient technique called the exp[Formula: see text]-expansion method is utilized to construct some new soliton solutions of the considered model. As a result, dark, singular, rational and periodic solitary wave solutions are obtained. For the special choice of parameters, the 3D, 2D and contour surfaces of the reported solutions are plotted.

Nonlinear dynamic behaviors of the generalized (3+1)‐dimensional KP equation

AbstractUnder investigation in this article is a generalized (3+1)‐dimensional Kadomtsev–Petviashvili (KP) equation, which is usually used to describe nonlinear phenomena in fluid dynamics. First, multiple waves solutions of this considered equation by means of the linear superposition principle are constructed. Second, the parameters of this considered model control multiple waves solutions are being investigated. We find that the parameters mainly affect the wave shapes, amplitude, and bright/dark nature. Third, we apply a polynomial function to construct rational solutions and rogue wave solutions when the parameters contain in the polynomial function are equal to zero and nonzero, respectively. Finally, the obtained results are demonstrated graphically to illustrate the nonlinear dynamics behaviors of multiple, rational, and rogue wave solutions in fluid mechanics.

A variety of solitons on the oceans exposed by the Kadomtsev Petviashvili-modified equal width equation adopting different techniques

The diverse patterns of waves on the oceans yielded by the Kadomtsev Petviashvili-modified equal width (KP-mEW) equation are highlighted in this paper. Two recent established approaches such as the improved auxiliary equation technique and the enhanced rational (G′/G)-expansion scheme are utilized to construct wave solutions of the proposed governing model. Numerous rational, trigonometric, exponential, and hyperbolic wave solutions bearing many free parameters are successfully acquired in appropriate form. The obtained solutions are plotted in various profiles as three-dimension, two-dimension, and contour to illustrate their physical appearances. The plotting outlines appear in the shapes of singular kink, anti-kink, kink, compacton, anti-compacton, bell, anti-bell, periodic, singular periodic etc. The computational software Maple is used for plotting and checking the validity of the found solutions. This paper claims to be novel for generating new results regarding the earlier results.

Highly dispersive optical solitons in birefringent fibers for complex-Ginzburg–Landau equation with parabolic law of nonlinearity using two integration techniques

Object:This paper aims to deduce optical soliton solutions of the highly dispersive optical solitons with perturbed complex-Ginzburg–Landau equation having parabolic law nonlinear refractive index in birefringent fibers. Methods:The addendum Kudryashov’s method and algebraic sub-equation expansion method are used for solving the given system. Results:Combo soliton solutions, bright soliton solutions and singular soliton solutions as well as rational and periodic wave solutions have been deduced.

Diverse Forms of Breathers and Rogue Wave Solutions for the Complex Cubic Quintic Ginzburg Landau Equation with Intrapulse Raman Scattering

This manuscript consist of diverse forms of lump: lump one stripe, lump two stripe, generalized breathers, Akhmediev breather, multiwave, M-shaped rational and rogue wave solutions for the complex cubic quintic Ginzburg Landau (CQGL) equation with intrapulse Raman scattering (IRS) via appropriate transformations approach. Furthermore, it includes homoclinic, Ma and Kuznetsov-Ma breather and their relating rogue waves and some interactional solutions, including an interactional approach with the help of the double exponential function. We have elaborated the kink cross-rational (KCR) solutions and periodic cross-rational (KCR) solutions with their graphical slots. We have also constituted some of our solutions in distinct dimensions by means of 3D and contours profiles to anticipate the wave propagation. Parameter domains are delineated in which these exact localized soliton solutions exit in the proposed model.

Stable abundant computational solitary wave structures of the perturbed time-fractional NLS equation

The conformable fractional derivatives Khater II (Khat. II) method and the Adomian decomposition (AD) approach are used to investigate the analytical and semi-analytical wave solutions to the perturbed time-fractional nonlinear Schrödinger (NLS) model. This model describes the dynamics of optical solitons propagating via nonlinear optical fibers. Several novel solitary wave solutions are constructed in distinct formats such as hyperbolic, trigonometric, rational, dark, brilliant, combined dark-bright, singular, combination singular, and periodic wave solutions. Additionally, the Adomian decomposition technique determines the absolute error associated with analytical and semi-analytical wave solutions. The Hamiltonian system examines the stability of discovered solutions to determine their acceptability for usage in the model’s application.

Optical wave solutions of perturbed time-fractional nonlinear Schrödinger equation

The conformable fractional derivatives modified Khater (mKhat.) technique and the Adomian decomposition (AD) method is used to examine the perturbed time-fractional nonlinear Schrödinger (NLS) problem's analytical and semi-analytical wave solutions. This model describes the dynamics of optical solitons propagating via nonlinear optical fibers. For this model, we create a variety of different formulae for analytical wave solutions, including hyperbolic, trigonometric, rational, dark, brilliant, combined dark-bright, singular, combined singular, and periodic wave solutions. Additionally, the Adomian decomposition approach is utilized to evaluate the absolute error between analytical and semi-analytical wave solutions. The Hamiltonian system is used to analyze the stability of found solutions to demonstrate their suitability for implementation in the model's application.

Spatially discrete Hirota equation: Rational and breather solution, gauge equivalence, and continuous limit

It is well known that the nonlinear Schrödinger equation, the Hirota equation, and their integrable discrete versions are very important integrable equations. These integrable equations not only have deep integrability theory, but also have wide physical applications. In this paper, we focus on an integrable spatially discrete Hirota equation. The new Lax pair, Darboux transformation, rational wave solution, rogue wave solution, breather solutions and gauge equivalent structure of the spatially discrete Hirota equation are investigated. The continuous limit of the rogue wave solution and breather solution of the discrete Hirota equation yields the counterparts of the Hirota equation, which shows that the spatially discrete Hirota equation is a very good model for the numerical analysis for considering the Cauchy problem with a general initial date of the Hirota equation. Besides, we prove that this spatially discrete Hirota equation is gauge equivalent to a discrete integrable generalized Heisenberg spin model under the discrete gauge transformation. In this performance, we see that utilizing the new Lax pair, an essential properties of integrability, of the spatially discrete Hirota equation plays a significant role in the investigation of continuous limit of Lax pair of discrete integrable generalized Heisenberg spin model.

Gradient-optimized physics-informed neural networks (GOPINNs): a deep learning method for solving the complex modified KdV equation

Recently, the physics-informed neural networks (PINNs) have received more and more attention because of their ability to solve nonlinear partial differential equations via only a small amount of data to quickly obtain data-driven solutions with high accuracy. However, despite their remarkable promise in the early stage, their unbalanced back-propagation gradient calculation leads to drastic oscillations in the gradient value during model training, which is prone to unstable prediction accuracy. Based on this, we develop a gradient optimization algorithm, which proposes a new neural network structure and balances the interaction between different terms in the loss function during model training by means of gradient statistics, so that the newly proposed network architecture is more robust to gradient fluctuations. In this paper, we take the complex modified KdV equation as an example and use the gradient-optimized PINNs (GOPINNs) deep learning method to obtain data-driven rational wave solution and soliton molecules solution. Numerical results show that the GOPINNs method effectively smooths the gradient fluctuations and reproduces the dynamic behavior of these data-driven solutions better than the original PINNs method. In summary, our work provides new insights for optimizing the learning performance of neural networks and improves the prediction accuracy by a factor of 10 to 30 when solving the complex modified KdV equation.

Rational vector rogue waves for the [formula omitted]-component Hirota equation with non-zero backgrounds

In this paper, the dimensionless n-component Hirota (alias the n-Hirota) equation is investigated, which describes the wave propagations of n ultrashort optical fields in a fiber. Starting from the modified Darboux transform and its Lax pair with initial non-zero plane-wave conditions, we find the novel multi-parametric families of rational vector rogue wave (RW) solutions for the n-Hirota equation. Furthermore, some weak and strong interactions of rational vector RWs are exhibited for the n-Hirota equation with n=2,3,4,5,6 in detail. In particular, we also deduce the rational vector W-shaped dark and bright solitons of the n-component complex mKdV equation, whose representative wave structures are illustrated for n=2,3,4,5. Finally, the effect of a small non-integrable deformation of the 3-Hirota equation is explored numerically on the excitation of vector RWs in terms of the Fourier spectral method. These obtained rational vector RW and W-shaped soliton solutions will be useful to further explore the related nonlinear wave phenomena in the sense of multi-component physical systems with non-zero backgrounds.

Multi-wave, homoclinic breather, M-shaped rational and other solitary wave solutions for coupled-Higgs equation

Multi-wave, homoclinic breather, M-shaped rational and other solitary wave solutions for coupled-Higgs equation

Rational Wave Solutions and Dynamics Properties of the Generalized ( 2 + 1 )-Dimensional Calogero-Bogoyavlenskii-Schiff Equation by Using Bilinear Method

Through symbolic computation with Maple, fifty-seven sets of rational wave solutions to the generalized Calogero-Bogoyavlenskii-Schiff equation are presented by employing the generalized bilinear operator when the parameter p = 2 . Via the three-dimensional plots and contour plots with the help of Maple, the dynamics of these solutions are described very well. These solutions have greatly enriched the exact solutions of the generalized Calogero-Bogoyavlenskii-Schiff equation on the existing literature. The result will be widely used to describe many nonlinear scientific phenomena.

Solving localized wave solutions of the derivative nonlinear Schrödinger equation using an improved PINN method

The solving of the derivative nonlinear Schrodinger equation (DNLS) has attracted considerable attention in theoretical analysis and physical applications. Based on the physics-informed neural network (PINN) which has been put forward to uncover dynamical behaviors of nonlinear partial different equation from spatiotemporal data directly, an improved PINN method with neuron-wise locally adaptive activation function is presented to derive localized wave solutions of the DNLS in complex space. In order to compare the performance of above two methods, we reveal the dynamical behaviors and error analysis for localized wave solutions which include one-rational soliton solution, genuine rational soliton solutions and rogue wave solution of the DNLS by employing two methods, also exhibit vivid diagrams and detailed analysis. The numerical results demonstrate the improved method has faster convergence and better simulation effect. On the bases of the improved method, the effects for different numbers of initial points sampled, residual collocation points sampled, network layers, neurons per hidden layer on the second order genuine rational soliton solution dynamics of the DNLS are considered, and the relevant analysis when the locally adaptive activation function chooses different initial values of scalable parameters are also exhibited in the simulation of the two-order rogue wave solution.

Parity-time-symmetric rational vector rogue waves of the n-component nonlinear Schrödinger equation.

Extreme events are investigated in the integrable n-component nonlinear Schrödinger (NLS) equation with focusing nonlinearity. We report novel multi-parametric families of rational vector rogue wave (RW) solutions featuring the parity-time ( PT) symmetry, which are characterized by non-identical boundary conditions for the components that are consistent with the degeneracy of n branches of Benjamin-Feir instability. Explicit examples of PT-symmetric rational vector RWs are presented. Subject to the specific choice of the parameters, high-amplitude RWs are generated. The effect of a small non-integrable deformation of the 3-NLS equation on the excitation of vector RWs is discussed. The reported results can be useful for the design of experiments for observation of high-amplitude RWs in multi-component nonlinear physical systems.

On the Convergence of LHAM and its Application to Fractional Generalised Boussinesq Equations for Closed Form Solutions

By fractional generalised Boussinesq equations we mean equations of the form \begin{equation} \Delta\equiv D_{t}^{2\alpha}-[\mathcal{N}(u)]_{xx}-u_{xxxx}=0, \: 0<\alpha\le1,\label{main}\nonumber \end{equation} where $\mathcal{N}(u)$ is a differentiable function and $\mathcal{N}_{uu}\ne0$ (to ensure nonlinearity). In this paper we lay emphasis on the cubic Boussinesq and Boussinesq-like equations of fractional order and we apply the Laplace homotopy analysis method (LHAM) for their rational and solitary wave solutions respectively. It is true that nonlinear fractional differential equations are often difficult to solve for their {\em exact} solutions and this single reason has prompted researchers over the years to come up with different methods and approach for their {\em analytic approximate} solutions. Most of these methods require huge computations which are sometimes complicated and a very good knowledge of computer aided softwares (CAS) are usually needed. To bridge this gap, we propose a method that requires no linearization, perturbation or any particularly restrictive assumption that can be easily used to solve strongly nonlinear fractional differential equations by hand and simple computer computations with a very quick run time. For the closed form solution, we set $\alpha =1$ for each of the solutions and our results coincides with those of others in the literature.

Soliton solutions of the generalized Davey-Stewartson equation with full nonlinearities via three integrating schemes

This paper considers the generalized Davey-Stewartson equation that is used to investigate the dynamics of wave propagation in water of finite depth under the effects of gravity force and surface tension. The model is considered in the presence of full nonlinearity. The main objective of this paper is to extract soliton solutions of the generalized Davey-Stewartson equation. Three state-of-the-art integration schemes, namely exp(-Φ(ξ))-expansion method, the first integral method and the Sine-Gordon expansion method have been employed for obtaining the desired soliton solutions. The proposed methods successfully attain different structures of explicit solutions such as bright, dark, singular, rational and periodic solitary wave solutions. All the newly found solutions are discussed with their existence criteria. The 2D and 3D portraits are also shown for some of the reported solutions.

Optical dromions for perturbed nonlinear Schrödinger equation with cubic quintic septic media

In this paper, we obtain optical solitons and other solitary wave solutions for (2+1)-dimensional nonlinear Schrödinger equation (NLSE) with the aid of extended modified auxiliary equation mapping method, together with cubic-quintic-septic nonlinearity. We obtain bright, dark, singular, rational, periodic and many other solitary wave solutions for our governing model with constraint conditions.

Rational Localized Waves and Their Absorb-Emit Interactions in the (2 + 1)-Dimensional Hirota–Satsuma–Ito Equation

In this paper, we investigate the (2 + 1)-dimensional Hirota–Satsuma–Ito (HSI) shallow water wave model. By introducing a small perturbation parameter ϵ, an extended (2 + 1)-dimensional HSI equation is derived. Further, based on the Hirota bilinear form and the Hermitian quadratic form, we construct the rational localized wave solution and discuss its dynamical properties. It is shown that the oblique and skew characteristics of rational localized wave motion depend closely on the translation parameter ϵ. Finally, we discuss two different interactions between a rational localized wave and a line soliton through theoretic analysis and numerical simulation: one is an absorb-emit interaction, and the other one is an emit-absorb interaction. The results show that the delay effect between the encountering and parting time of two localized waves leads to two different kinds of interactions.

Construction of traveling waves patterns of [formula omitted]-dimensional modified Zakharov-Kuznetsov equation in plasma physics

In this research, we examine the modified model of (1+n)-dimensional Zakharov-Kuznetsov (ZK) equation, which will be used to analyze the nature of weakly nonlinear traveling waves in the existence of a constant magnetic area in a plasma comprising in cold ions and hot isothermal electrons. The modified Zakharov-Kuznetsov (mZK) equation will have solutions describing the traveling solitary waves, using the extended (G′G2)-expansion method and extended direct algebraic method gives way to the mZK equation regulating the transmission of ion dynamics for nonlinear traveling waves in a plasma. The sufficient conditions for the stability and existence of the traveling wave solutions are reported. Semi-dark, rational, and singular solitary wave solutions are computed. Graphical interpretations of certain practical solutions for specific values of parameters have also been available. The research findings reported throughout this evaluation are fresh and from which this model is employed to analyze waves in numerous plasmas, could be valuable and important. Subsequently, there are concluding remarks mentioned.

On Solutions of an Extended Nonlocal Nonlinear Schrödinger Equation in Plasmas

The parity-time symmetric nonlocal nonlinear Schrödinger equation with self-consistent sources (PTNNLSESCS) is used to describe the interaction between an high-frequency electrostatic wave and an ion-acoustic wave in plasmas. In this paper, the soliton solutions, rational soliton solutions and rogue wave solutions are derived for the PTNNLSESCS via the generalized Darboux transformation. We find that the soliton solutions can exhibit the elastic interactions of different type of solutions such as antidark-antidark, dark-antidark, and dark-dark soliton pairs on a continuous wave background. Also, we discuss the degenerate case in which only one antidark or dark soliton remains. The rogue wave solution is derived in some specially chosen situations.