Weierstrass Elliptic Solutions Research Articles

Investigating the noise effect on the CGL model having parabolic law of nonlinearity

The stochastic complex Ginzburg–Landau model with parabolic law nonlinearity and chromatic dispersion is investigated in this research study. In many fields of science, including fluid dynamics and optical fibers, this model is utilized to represent the wave nature. In this study, the improved modified extended tanh approach is used. This method provides a wide variety of solutions, including dark, singular and bright solitons. Furthermore, exponential, singular periodic and Weierstrass elliptic solutions are provided. For a variety of nonlinear partial differential equations, this method offers a practical and effective method for determining exact solutions. The impact of the noise is illustrated graphically using examples of some of the retrieved solutions with various noise strengths.

Investigation of chirped optical solitons perturbation of higher order NLSE via improved modified extended tanh function approach

This context focuses on examining the manner in which localized waves travel through a non-linear medium that adheres to a dual-power law. This medium encompasses a range of disruptions like inter modal dispersion, self frequency shift, and self-steepening effects. This investigation entails employing an improved modified extended tanh function (IMETF) technique to obtain various novel optical soliton solutions which include bright, dark, singular and combo singular-dark solitons. Furthermore, we extract singular periodic, exponential, rational, and Weierstrass elliptic solutions. To showcase their physical characteristics, we provide graphical representations in 3D and 2D of some of the derived solutions.

Soliton solutions to generalized (2+1)-dimensional Hietarinta-type equation and resonant NLSE along with stability analysis

This paper discusses the generalized [Formula: see text]-dimensional Hietarinta-type fourth-order nonlinear equation (GHTE) and Resonant nonlinear Schrödinger equation (RNLSE). We will study the soliton solutions of the GHTE as well as RNLSE by the renowned, efficient and successful method called sub-ODE technique. The solutions are categorized as dark soliton, periodic, rational, positive, hyperbolic and Weierstrass elliptic solutions (WES). Stability analysis of the solutions is performed for both the models. Later on, these results will be displayed graphically by 3D, 2D, contour, density and streamline plots which describe the behavior of the waves.

Derivation new solitons and other solutions for higher order Sasa–Satsuma equation by using the improved modified extended tanh scheme

In this work, we utilize the improved modified extended tanh method to get optical solitons and other exact solutions for (3+1) dimensional Sasa–Satsuma model with third order dispersion that describe the dynamics of ultrashort pulses in the optical fibers. Many exact solutions are raised with the aid of the proposed method. These solutions include bright solitons, dark solitons, singular solitons, periodic wave solutions, hyperbolic wave solutions, Weierstrass elliptic solutions and Jacobi elliptic solutions.

Derivation of optical solitons and other solutions for nonlinear Schrödinger equation using modified extended direct algebraic method

Optical solitons is one of the fastest growing areas of research in the field of nonlinear fiber optics. In the current work, the modified extended direct algebraic method is implemented to obtain optical solitons for the nonlinear Schrödinger’s equation with group velocity dispersion and second order spatiotemporal dispersion that describe the propagation of optical solitons in nonlinear media. Bright soliton solutions, dark soliton solutions and singular soliton solutions are obtained. Also, periodic solutions, exponential solutions, Jacobi elliptic solutions and Weierstrass elliptic solutions are extracted. Under specific parameter values, 2D, 3D and contour graphs are introduced to visualize the model and demonstrate their accurate physical behaviors.

Solitary wave solutions and other solutions for Gilson–Pickering equation by using the modified extended mapping method

In this paper, the modified extended mapping method is applied to obtain solitary wave solutions and other solutions for the Gilson–Pickering equation which describes the wave propagation in plasma physics. Many types of solutions are obtained such as shock wave, singular waves, periodic waves, singular periodic waves, Jacobi elliptic functions, plane waves, hyperbolic waves, Weierstrass elliptic solutions and exponential type solutions. Moreover, 3D graphs of some solutions are introduced for the physical illustration.

Magnetization profiles at the upper critical dimension as solutions of the integer Yamabe problem.

We study the connection between the magnetization profiles of models described by a scalar field with marginal interaction term in a bounded domain and the solutions of the so-called Yamabe problem in the same domain, which amounts to finding a metric having constant curvature. Taking the slab as a reference domain, we first study the magnetization profiles at the upper critical dimensions d=3, 4, 6 for different (scale-invariant) boundary conditions. By studying the saddle-point equations for the magnetization, we find general formulas in terms of Weierstrass elliptic functions, extending exact results known in literature and finding ones for the case of percolation. The zeros and poles of the Weierstrass elliptic solutions can be put in direct connection with the boundary conditions. We then show that, for any dimension d, the magnetization profiles are solution of the corresponding integer Yamabe equation at the same d and with the same boundary conditions. The magnetization profiles in the specific case of the four-dimensional Ising model with fixed boundary conditions are compared with Monte Carlo simulations, finding good agreement. These results explicitly confirm at the upper critical dimension recent results presented in Gori and Trombettoni [J. Stat. Mech: Theory Exp. (2020) 0632101742-546810.1088/1742-5468/ab7f32].

Solitons dynamics in optical metamaterial with quadratic–cubic nonlinearity using modified extended direct algebraic method

In this article, the modified direct algebraic method is applied for the perturbed nonlinear Schrödinger equation (NLSE) describing the dynamics of optical solitons in metamaterials, in the presence of quadratic–cubic nonlinearity. Various types of solutions are extracted such as bright solitons, singular solitons, dark solitons, singular periodic wave solutions, periodic wave solutions, plane wave solutions, Jacobi elliptic wave solutions, exponential wave solutions, hyperbolic wave solutions and Weierstrass elliptic solutions. Moreover, for the physical illustration of the obtained solutions, 3D graphs are presented.

New Meromorphic Solutions of the Cubic Nonlinear Schrödinger Equation by Using the Complex Method

We show abundant simply periodic solutions, trigonometric solutions, hyperbolic function solutions and Weierstrass elliptic solutions of the reduction of the cubic nonlinear Schrodinger equation by the complex method with Painleve analysis, and some solutions appear to be new. At last, we give some computer simulations to illustrate our main results.

Two Integrable Classes of Emden–Fowler Equations with Applications in Astrophysics and Cosmology

Abstract We show that some Emden–Fowler (EF) equations encountered in astrophysics and cosmology belong to two EF integrable classes of the type d 2 z / d χ 2 = A χ − λ − 2 z n ${\mathrm{d}^{2}}z/\mathrm{d}{\chi^{2}}=A{\chi^{-\lambda-2}}{z^{n}}$ for λ = ( n − 1 ) / 2 $\lambda=(n-1)/2$ (class 1), and λ = n + 1 $\lambda=n+1$ (class 2). We find their corresponding invariants which reduce them to first-order nonlinear ordinary differential equations. Using particular solutions of such EF equations, the two classes are set in the autonomous nonlinear oscillator the form d 2 ν / d t 2 + a d ν / d t + b ( ν − ν n ) = 0 ${\mathrm{d}^{2}}\nu/\mathrm{d}{t^{2}}+a\mathrm{d}\nu/\mathrm{d}t+b(\nu-{\nu^{n}})=0$ , where the coefficients a , b $a,b$ depend only on λ , n $\lambda,n$ . For both classes, we write closed-form solutions in parametric form. The illustrative examples from astrophysics and general relativity correspond to two n = 2 cases from class 1 and 2, and one n = 5 case from class 1, all of them yielding Weierstrass elliptic solutions. It is also noticed that when n = 2, the EF equations can be studied using the Painlevé reduction method, since they are a particular case of equations of the type d 2 z / d χ 2 = F ( χ ) z 2 ${\mathrm{d}^{2}}z/\mathrm{d}{\chi^{2}}=F(\chi){z^{2}}$ , where F ( χ ) $F(\chi)$ is the Kustaanheimo-Qvist function.

Weierstrass elliptic solutions to a Zakharov equation in plasmas with power law nonlinearity

Weierstrass elliptic solutions to a Zakharov equation in plasmas with power law nonlinearity

Note on exact solutions for a class of nonlinear diffusion equations

An effective characterization is given for a class of generalized nonlinear diffusion equations with power law dependent terms. Further, a new auxiliary equation ansatz is derived. Consequently, new exact traveling wave trigonometric function, solitary-like and Weierstrass elliptic solutions to a subclass are obtained by means of an auxiliary equation method and a generalized Riccati equation expansion method.

Weierstrass Elliptic Function Solutions to Nonlinear Evolution Equations

This paper is based on the relations between projection Riccati equations and Weierstrass elliptic equation, combined with the Groebner bases in the symbolic computation. Then the novel method for constructing the Weierstrass elliptic solutions to the nonlinear evolution equations is given by using the above relations.