Hyperbolic Solutions Research Articles

The Propagating Exact Solitary Waves Formation of Generalized Calogero–Bogoyavlenskii–Schiff Equation with Robust Computational Approaches

The generalized Calogero–Bogoyavlenskii–Schiff equation (GCBSE) is examined and analyzed in this paper. It has several applications in plasma physics and soliton theory, where it forecasts the soliton wave propagation profiles. In order to obtain the analytically exact solitons, the model under consideration is a nonlinear partial differential equation that is turned into an ordinary differential equation by using the next traveling wave transformation. The new extended direct algebraic technique and the modified auxiliary equation method are applied to the generalized Calogero–Bogoyavlenskii–Schiff equation to get new solitary wave profiles. As a result, novel and generalized analytical wave solutions are acquired in which singular solutions, mixed singular solutions, mixed complex solitary shock solutions, mixed shock singular solutions, mixed periodic solutions, mixed trigonometric solutions, mixed hyperbolic solutions, and periodic solutions are included with numerous soliton families. The propagation of the acquired soliton solution is graphically presented in contour, two- and three-dimensional visualization by selecting appropriate parametric values. It is graphically demonstrated how wave number impacts the obtained traveling wave structures.

Extraction of soliton solutions for the time–space fractional order nonclassical Sobolev-type equation with unique physical problems

In this paper, we investigate the soliton solutions of the time–space fractional order nonclassical Sobolev-type (TSFNST) equation. Sobolev equations are used in thermodynamics, the flow of fluid through fractured rock, and other fields. There are certain significant partial differential equations having a third-order mixed derivative with regard to time and space that are now being used in real-world applications. The study aims to analyze the TSFNST equation analytically by employing the ϕ6-model expansion and modified G′/G2 expansion techniques respectively. The solution is obtained successfully in the Jacobi elliptic function solutions that will prove to us the hyperbolic, trigonometric, and rational solutions. Additionally, we select some unique physical problems of the solution. Lastly, the 3-dimensional and their corresponding contour plots are drawn by choosing the different values of constants

Impact of Fractional Derivative and Brownian Motion on the Solutions of the Radhakrishnan-Kundu-Lakshmanan Equation

The fractional-stochastic Radhakrishnan-Kundu-Lakshmanan equation (FSRKLE) is considered here. To attain new hyperbolic, elliptic, rational, and trigonometric stochastic-fractional solutions, we use two various methods such as the sine-cosine and the Jacobi elliptic function methods. The solutions acquired are important in understanding some interesting physical phenomena due to the significance of the Radhakrishnan-Kundu-Lakshmanan equation in designing the propagation of solitons through an optical fiber. Furthermore, we graph some of the obtained solutions in 3D to display the influence of fractional derivative and multiplicative noise on these solutions. Finally, we show that when the order of fractional derivative decreases, the surface shrinks, while the multiplicative noise stabilizes the solutions of FSRKLE a round zero.

Modified Exp-Function Method to Find Exact Solutions of Microtubules Nonlinear Dynamics Models

In this paper, we use the modified exp−ψθ-function method to observe some of the solitary wave solutions for the microtubules (MTs). By treating the issues as nonlinear model partial differential equations describing microtubules, we were able to solve the problem. We then found specific solutions to the nonlinear evolution equation (NLEE) covering various parameters that are particularly significant in biophysics and nanobiosciences. In addition to the soliton-like pulse solutions, we also find the rational, trigonometric, hyperbolic, and exponential function characteristic solutions for this equation. The validity of the method we developed and the fact that it provides more solutions are demonstrated by comparison to other methods. We next use the software Mathematica 10 to generate 2D, 3D, and contour plots of the precise findings we observed using the suggested technique and the proper parameter values.

Stability of structure-aware Taylor methods for tents

Structure-aware Taylor (SAT) methods are a class of timestepping schemes designed for propagating linear hyperbolic solutions within a tent-shaped spacetime region. Tents are useful to design explicit time marching schemes on unstructured advancing fronts with built-in locally variable timestepping for arbitrary spatial and temporal discretization orders. The main result of this paper is that an s s -stage SAT timestepping within a tent is weakly stable under the time step constraint Δ t ≤ C h 1 + 1 / s \Delta t \leq Ch^{1+1/s} , where Δ t \Delta t is the time step size and h h is the spatial mesh size. Improved stability properties are also presented for high-order SAT time discretizations coupled with low-order spatial polynomials. A numerical verification of the sharpness of proven estimates is also included.

Asymptotically hyperbolic Einstein constraint equations with apparent horizon boundary and the Penrose inequality for perturbations of Schwarzschild-AdS *

We prove the existence of asymptotically hyperbolic solutions to the vacuum Einstein constraint equations with a marginally outer trapped boundary of positive mean curvature, using the constant mean curvature conformal method. As an application of this result, we verify the Penrose inequality for certain perturbations of Schwarzschild Anti-de Sitter black hole initial data.

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.

Stationary solutions and dynamical evolution of the nonlocal Gross–Pitaevskii equation in spin-2 Bose–Einstein condensates

In this paper, we investigate the nonlocal Gross–Pitaevskii equation which describes the phenomena of Bose–Einstein condensates under the mean field approximation. Using mean-field theory and adopting separation of variables method, the five-component Gross–Pitaevskii spin–orbit-coupled spin-2 Bose–Einstein condensate model can be reduced to five-component ordinary differential equation, then twelve families of solutions are obtained by the combination of some Jacobi elliptic functions. In particular, when the modulus of Jacobi elliptic function tends to one, a series of bright soliton and dark soliton and their combined solution are obtained. These periodic and hyperbolic soliton solutions are analyzed in detail, the nature and images will be of practical application to provide theoretical guidance for spin systems.

Explicit Soliton Structure Formation for the Riemann Wave Equation and a Sensitive Demonstration

The motive of the study was to explore the nonlinear Riemann wave equation, which describes the tsunami and tidal waves in the sea and homogeneous and stationary media. This study establishes the framework for the analytical solutions to the Riemann wave equation using the new extended direct algebraic method. As a result, the soliton patterns of the Riemann wave equation have been successfully illustrated, with exact solutions offered by the plane solution, trigonometry solution, mixed hyperbolic solution, mixed periodic and periodic solutions, shock solution, mixed singular solution, mixed trigonometric solution, mixed shock single solution, complex soliton shock solution, singular solution, and shock wave solutions. Graphical visualization is provided of the results with suitable values of the involved parameters by Mathematica. It was visualized that the velocity of the soliton and the wave number controls the behavior of the soliton. We are confident that our research will assist physicists in predicting new notions in mathematical physics.

Solitary Wave Solutions of the Fractional-Stochastic Quantum Zakharov–Kuznetsov Equation Arises in Quantum Magneto Plasma

In this paper, we consider the (3 + 1)-dimensional fractional-stochastic quantum Zakharov–Kuznetsov equation (FSQZKE) with M-truncated derivative. To find novel trigonometric, hyperbolic, elliptic, and rational fractional solutions, two techniques are used: the Jacobi elliptic function approach and the modified F-expansion method. We also expand on a few earlier findings. The extended quantum Zakharov–Kuznetsov has practical applications in dealing with quantum electronpositron–ion magnetoplasmas, warm ions, and hot isothermal electrons in the presence of uniform magnetic fields, which makes the solutions obtained useful in analyzing a number of intriguing physical phenomena. We plot our data in MATLAB and display various 3D and 2D graphical representations to explain how the stochastic term and fractional derivative influence the exact solutions of the FSEQZKE.

Soliton solutions to the electric signals in telegraph lines on the basis of the tunnel diode

Soliton solutions (SSs) of the electric signals in telegraph lines on the basis of the tunnel diode (ESTLBTD) have been examined using the modified G′/G-expansion method (MG′/GEM). We create general SSs that define rational, hyperbolic and trigonometric solutions. The 3D, 2D and contour plots of the gained SSs are represented to explain the ESTLBTD in the optical fibers. Equating our attained answers and that gotten in beforehand written examination papers offerings the novelty of our study. The gained SSs might show imperative part in nonlinear science and engineering (NLSE) areas. It is remarkable to notice that the MG′/GEM is easy, well-matched and influential scientific device to determine SSs for nonlinear engineering models (NLEMs). The study procedures could also be implemented to develop SSs for other NLEMs in mathematical physics, applied mathematics, and engineering.

Derivation of new optical solitons for Biswas-Milovic equation with dual-power law nonlinearity using improved modified extended tanh-function method

Cubic-quartic optical solitons and other solutions for Biswas-Milovic equation with dual-power law nonlinearity are investigated using improved modified extended tanh-function method. Bright soliton solutions, dark soliton solutions, singular soliton solutions, singular periodic solutions, Jacobi elliptic solutions and hyperbolic solutions are extracted. Furthermore, some selected solutions are described graphically to demonstrate the physical nature of obtained solutions.

Dynamical solutions and quadratic resonance of nonlinear perturbed Schrödinger equation

This work investigates the perturbed nonlinear Schrödinger equation using the modified (G′G2)-expansion method. The obtained results are generalized and classified into classes of trigonometric, hyperbolic, and rational solutions. The kinematics of soliton and kink profiles are very helpful to understand the propagation of electromagnetic waves inside nonlinear optical fibers. The proposed modified method is unique, straightforward, concise, and effective in the sense that it gives more traveling wave solutions. The findings of this study can strengthen a system's nonlinear dynamic behavior and show how practical the methodology used to attempt to replicate has been. Wolfram Mathematica 11 is used for mathematical simplification and MATLAB is used for graphical simulation.

Exploration of New Optical Solitons in Magneto-Optical Waveguide with Coupled System of Nonlinear Biswas–Milovic Equation via Kudryashov’s Law Using Extended F-Expansion Method

Optical soliton solutions in a magneto-optical waveguide and other exact solutions are investigated for the coupled system of the nonlinear Biswas–Milovic equation with Kudryashov’s law using the extended F-expansion method. Various types of solutions are extracted, such as dark soliton solutions, singular soliton solutions, a dark–singular combo soliton, singular combo soliton solutions, Jacobi elliptic solutions, periodic solutions, combo periodic solutions, hyperbolic solutions, rational solutions, exponential solutions and Weierstrass solutions. The obtained different types of wave solutions help in obtaining nonlinear optical fibers in the future. Furthermore, some selected solutions are described graphically to demonstrate the physical nature of the obtained solutions. The results show that the current method gives effectual and direct mathematical tools for resolving the nonlinear problems in the field of nonlinear wave equations.

The weakly non-linear waves propagation for Kelvin-Helmholtz instability in the magnetohydrodynamics flow impelled by fractional theory.

The weakly nonlinear wave propagation that occurs in the presence of magnetic fields, in which energy is concentrated in a narrow band of wave-numbers in a dispersive and dissipative fluid. The main objective of this paper is to analyze the dimensional elliptic nonlinear Schrodinger equation under the influence of three different fractional operators. The generalized fractional soliton solutions and propagation of magnetohydrodynamics fluid in sort of solition will be visualized. The Conformable, and M-truncated fractional operator applied to classical evolution Schrodinger equation. In order to get the analytical closed form solution, one of the generalized approach new extended direct algebraic method is utilized. The fractional nonlinear elliptic Schrodinger equation is developed in three different fractional sense. The similarity transformation technique converted the controlling fractional system to ordinary differential equations. The fractional analytical solutions such as, plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, mixed trigonometric solution, trigonometric solution, shock solution, mixed shock singular solution, mixed singular solution, complex solitary shock solution, singular solution and shock wave solutions are obtained. The graphical 2-D and 3-D representation of the results is shown to express the propagation of fluid with the magnetic field by assuming the appropriate values of the involved parameters. The graphical performance of the obtained solution at various settings of parametric values and fractional order reveals new perspectives and fascinating model phenomena. The attained outcomes have significant applications and have opened up innovative development areas for research across numerous scientific fields.

Coupled superposed solutions in nonlinear nonlocal equations

We obtain novel periodic as well as hyperbolic solutions of an Ablowitz–Musslimani variant of the coupled nonlocal, nonlinear Schrödinger equation (NLS) as well as a coupled nonlocal modified Korteweg–de Vries (mKdV) equation which can be re-expressed as a linear superposition of the sum or the difference of two hyperbolic or two periodic kink or pulse solutions. Besides, we also discuss some of the other solutions admitted by these coupled equations. These results demonstrate that the notion of the superposed solutions extends to the coupled nonlocal nonlinear equations as well.

The exact solutions of the fractional-stochastic Fokas-Lenells equation in optical fiber communication

<abstract><p>The fractional-stochastic Fokas-Lenells equation (FSFLE) in the Stratonovich sense is taken into account here. The modified mapping method is used to generate new trigonometric, hyperbolic, elliptic and rational stochastic fractional solutions. Because the Fokas-Lenells equation has many implementations in telecommunication modes, complex system theory, quantum field theory, and quantum mechanics, the obtained solutions can be employed to describe a wide range of exciting physical phenomena. We plot several 2D and 3D diagrams to demonstrate how multiplicative noise and fractional derivatives affect the analytical solutions of the FSFLE. Also, we show how multiplicative noise at zero stabilizes FSFLE solutions.</p></abstract>

Solitary wave solutions to Gardner equation using improved tan$ \left(\frac{\Omega(\Upsilon)}{2}\right) $-expansion method

<abstract><p>In this study, the improved tan$ \left(\frac{\Omega(\Upsilon)}{2}\right) $-expansion method is used to construct a variety of precise soliton and other solitary wave solutions of the Gardner equation. Gardner equation is extensively utilized in plasma physics, quantum field theory, solid-state physics and fluid dynamics. It is the simplest model for the description of water waves with dual power law nonlinearity. Hyperbolic, exponential, rational and trigonometric traveling wave solutions are obtained. The retrieved solutions include kink solitons, bright solitons, dark-bright solitons and periodic wave solutions. The efficacy of this method is determined by the comparison of the newly obtained results with already reported results.</p></abstract>

Some new chiral and perturbed chiral solitary waves in quantum Hall effect

In this article, we present a new form of solutions to the Chiral/perturbed Chiral nonlinear Schrödinger equation, using the unified solver method. This solver introduces the closed formula in an explicit way. In each problem, different families of elliptic functions solutions are found under suitable conditions. Moreover, some cases of hyperbolic solutions are obtained. The acquired solutions may be applicable for explaining various complex phenomena in the quantum Hall effect. The presented study displays that the proposed solver is simple, powerful and sturdy. For more details about the physical dynamical representation of the presented solutions, we introduce some graphs of some selected solutions using Matlab software. Finally, the presented solver can be used to solve many other models in real life problems.

Solutions to the (4+1)-Dimensional Time-Fractional Fokas Equation with M-Truncated Derivative

In this paper, we consider the (4+1)-dimensional fractional Fokas equation (FFE) with an M-truncated derivative. The extended tanh–coth method and the Jacobi elliptic function method are utilized to attain new hyperbolic, trigonometric, elliptic, and rational fractional solutions. In addition, we generalize some previous results. The acquired solutions are beneficial in analyzing definite intriguing physical phenomena because the FFE equation is crucial for explaining various phenomena in optics, fluid mechanics and ocean engineering. To demonstrate how the M-truncated derivative affects the analytical solutions of the FFE, we simulate our figures in MATLAB and show several 2D and 3D graphs.