Hyperbolic Form Research Articles

Investigation of complex hyperbolic and periodic wave structures to a new form of the q-deformed sinh-Gordon equation with fractional temporal evolution

This paper presents the fractional generalized q-deformed sinh-Gordon equation. The fractional effects of the temporal derivative of the proposed model are studied using a conformable derivative. The analytical solutions of the governing model depend on the specified parameters. The resulting equation is studied with two integration architectures: the sine-Gordon expansion method and the modified auxiliary equation method. These strategies extract hyperbolic, trigonometric, and rational form solutions. For appropriate parametric values and different values of fractional parameter α, the acquired findings are displayed via 3D graphics, 2D line plots, and contour plots. The graphical simulations of the constricted solutions depict the existence of bright soliton, dark soliton, and periodic waves. The considered model is useful in describing physical mechanisms that possess broken symmetry and incorporate effects such as amplification or dissipation.

(3+1)-Dimensional Gardner Equation Deformed from (1+1)-Dimensional Gardner Equation and its Conservation Laws

Through the application of the deformation algorithm, a novel (3+1)-dimensional Gardner equation and its associated Lax pair are derived from the (1+1)-dimensional Gardner equation and its conservation laws. As soon as the (3+1)-dimensional Gardner equation is set to be y or z independent, the Gardner equations in (2+1)-dimension are also obtained. To seek the exact solutions for these higher dimensional equations, the traveling wave method and the symmetry theory are introduced. Hence, the implicit expressions of traveling wave solutions to the (3+1)-dimensional and (2+1)-dimensional Gardner equations, the Lie point symmetry and the group invariant solutions to the (3+1)-dimensional Gardner equation are well investigated. In particular, after selecting some specific parameters, both the traveling wave solutions and the symmetry reduction solutions of hyperbolic function form are given.

A NEW PERSPECTIVE ON THE NONLINEAR DATE–JIMBO–KASHIWARA–MIWA EQUATION IN FRACTAL MEDIA

In this paper, we first created a fractal Date–Jimbo–Kashiwara–Miwa (FDJKM) long ripple wave model in a non-smooth boundary or microgravity space recorded. Using fractal semi-inverse skill (FSIS) and fractal traveling wave transformation (FTWT), the fractal variational principle (FVP) was derived, and the strong minimum necessary circumstance was attested with the He Wierstrass function. We have discovered two distinct solitary wave solutions, the square form of the hyperbolic secant function and the hyperbolic secant function form. Then, soliton solutions are cultivated through FVP and the minimum steady state condition. Finally, the influences of non-smooth boundaries on solitons were tackled, and the properties of the solution were demonstrated through three-dimensional contour lines. Fractal dimension can impact waveforms, but cannot affect their vertex values. The presentation of soliton solutions (SWS) using techniques is not only laudable but also noteworthy. The technique employed can also be used to investigate solitary wave solutions of other local fractional calculus partial differential equations.

Traveling Wave Solutions of the Conformable Fractional Klein–Gordon Equation With Power Law Nonlinearity

This article investigates the construction of new traveling wave solutions for the conformable fractional Klein–Gordon equation, which is a well‐known mathematical and physical model that can be used to explain spinless pion and de Broglie waves. In order to accomplish this task, a classic and effective analysis method, namely, the extended tanh–coth method, was utilized. By introducing appropriate transformations, the conformable fractional Klein–Gordon equations are reduced to ordinary differential equations, and then the solutions with hyperbolic function form are obtained. In addition, the effect of fractional parameters on waveform is analyzed by drawing two‐ and three‐dimensional graphics. These results contribute to a deeper understanding of the dynamics of the conformable fractional Klein–Gordon equation. The research in this article also indicates that the extended tanh–coth method is a straightforward and concise technique that has the potential to be applicable to many other conformable fractional partial differential equations that appear in mathematical physics.

Dark and bright hump solitons in the realm of the quintic Benney-Lin equation governing a liquid film

<p>This study explored and examined soliton solutions for the Quintic Benney-Lin equation (QBLE), which describes the dynamic of liquid films, using the Riccati modified extended simple equation method (RMESEM). The proposed approach, which is designed for nonlinear partial differential equations (NPDEs), effectively generates a large number of soliton solutions for the given QBLE, which basically captures the fundamental dynamics of the system. The rational, hyperbolic, rational-hyperbolic, trigonometric, and exponential forms of the scientifically specified soliton solutions are the main determinants of the hump solitons. We used 2D, 3D, and contour visualizations to offer accurate representations of the researched soliton phenomena associated with these solutions. These representations revealed the existence of dark and bright hump solitons in the framework of the QBLE and offer a thorough way to examine the model's behavioral characteristics in the liquid film by analyzing the QBLE model's soliton dynamics. Moreover, applying the suggested approach advances our knowledge of the unique features of the other similar NPDEs and the underlying dynamics.</p>

Wave nature of Rosensweig instability

The explicit analytical solution of Rosensweig instability spikes’ shapes obtained by Navier–Stokes (NS) equation in diverse magnetic field H vertical to the flat free surface of ferrofluids are systematically studied experimentally and theoretically. After carefully analyzing and solving the NS equation in elliptic form, the force balanced surface equations of spikes in Rosensweig instability are expressed as cosine wave in perturbated magnetic field and hyperbolic tangent in large magnetic field, whose results both reveal the wave-like nature of Rosensweig instability. The results of hyperbolic tangent form are perfectly fitted to the experimental results in this paper, which indicates that the analytical solution is basically correct. Using the forementioned theoretical results, the total energy of the spike distribution pattern is calculated. By analyzing the energy components under different magnetic field intensities H, the hexagon–square transition of Rosensweig instability is systematically discussed and explained in an explicit way.

Calculation of the results of basic arithmetic operations with quaternions defined in the interval form

The goal of the work. Creation of an EXCEL-oriented calculator for determining the results of basic arithmetic operations with quaternions, which are given by hyperbolic interval numbers. The subject of research is the set of quaternions and arithmetic operations defined on this set. Research method: Algorithmic and numerical analysis of procedures for performing basic arithmetic operations with quaternions given by hyperbolic interval numbers. The obtained results. The definition of a quaternion, the coefficients of which are interval numbers, is given. It was obtained the ratios for determining in the interval form the following characteristics: quaternion norm, quaternion modulus, quaternion vector part modulus, quaternion argument, quaternion sign was obtained. It was obtained the ratios for determining in the interval form the following basic arithmetic operations: sum of quaternions, difference of quaternions, multiplication of a constant value by a quaternion, scalar product of quaternions, product of vector parts of quaternions, product of quaternions, vector product of quaternions, left and right division of quaternions. The ratio for calculating the determinants of the second order, the elements of which are defined in the interval form, is obtained. The ratio for the approximate calculation of determinants of arbitrary order, the elements of which are defined in the interval form, is obtained. It is shown that the operations of multiplication (division) of interval numbers and raising them to an integer power are expedient to perform when they have a hyperbolic form. It is advisable to perform the addition (subtraction) operation with interval numbers given in the classic form or in the CENTER-RADIUS system. It is better to use the last form in the case of determining the coefficients of the quaternions based on the results of technological calculations. Screenshots of the formulas for determining the vector product of quaternions are given, provided that the coefficients at their orths are represented by interval numbers.

The residual symmetry, Bäcklund transformations, CRE integrability and interaction solutions: (2+1)-dimensional Chaffee–Infante equation

In this paper, we consider the (2+1)-dimensional Chaffee–Infante equation, which occurs in the fields of fluid dynamics, high-energy physics, electronic science etc. We build Bäcklund transformations and residual symmetries in nonlocal structure using the Painlevé truncated expansion approach. We use a prolonged system to localize these symmetries and establish the associated one-parameter Lie transformation group. In this transformation group, we deliver new exact solution profiles via the combination of various simple (seed and tangent hyperbolic form) exact solution structures. In this manner, we acquire an infinite amount of exact solution forms methodically. Furthermore, we demonstrate that the model may be integrated in terms of consistent Riccati expansion. Using the Maple symbolic program, we derive the exact solution forms of solitary-wave and soliton-cnoidal interaction. Through 3D and 2D illustrations, we observe the dynamic analysis of the acquired solution forms.

Overstatement in Communicative Demonstrativeness: Pragmatic and Linguistic Aspects

Overstatement plays the leading role organizing the phenomenon of communicative demonstrativeness by highlighting and translating meaningful information about the producer of the utterance. The purpose of the study is to identify and describe the properties and means of overstatement actualization. An interdisciplinary approach and the method of psycholinguistic analysis of the communicants’ speech output are used. The material for the analysis was the statements of the participants from the 2013-2022 television interviews. It is established that at present overstatement, the property of communicative demonstrativeness, is widely used as a means of the speakers’ attractive verbal behavior, caused by the need to obtain recognition of the personality identity that the speech producer presents from other people and its consolidation in the consciousness of the recipients. It is emphasized that the semantic basis of overstatement is the expression of the worldview possessed by a manipulative linguistic personality. Demonstrative overstatement, used to highlight a significant message, may have involuntary, intentional and periodic types and perform expressive, conative, aesthetic, pragmatic, denotative and contact functions. The study shows that overstatement in communicative demonstrativeness is realized in speech strategies of bragging, dramatization, false complaint, compliment, praise, condemnation, and humiliation as a way of self-promotion, over-evaluation and double over-evaluation, consisting of the adverb «absolutely» and an adjective. The linguistic means of actualization of overstatement are revealed: hyperbole, hyperbolic form of tropes (metaphors, epithets, comparisons), phraseological units, superlative adjectives, emotionally marked verbs, semantic and grammatical superlatives conveying the maximum degree of quality, intensifiers «very», «quite», exclamation sentences, shortened quotations, modified sayings and imperative suggestions. The results obtained can be used in the course of semiotics and psycholinguistics.

New wave behaviors of the Fokas-Lenells model using three integration techniques.

In this investigation, we apply the improved Kudryashov, the novel Kudryashov, and the unified methods to demonstrate new wave behaviors of the Fokas-Lenells nonlinear waveform arising in birefringent fibers. Through the application of these techniques, we obtain numerous previously unreported novel dynamic optical soliton solutions in mixed hyperbolic, trigonometric, and rational forms of the governing model. These solutions encompass periodic waves with W-shaped profiles, gradually increasing amplitudes, rapidly increasing amplitudes, double-periodic waves, and breather waves with symmetrical or asymmetrical amplitudes. Singular solitons with single and multiple breather waves are also derived. Based on these findings, we can say that our implemented methods are more reliable and useful when retrieving optical soliton results for complicated nonlinear systems. Various potential features of the derived solutions are presented graphically.

Stochastic soliton solutions of conformable nonlinear stochastic systems processed with multiplicative noise

Stochastic equations are powerful mathematical tools used to study systems having random effects. The current manuscript studies the two distinct systems: the stochastic conformable Zakharov system (SCZs) and stochastic conformable Hirota-Macari system (SCHMs) with multiplicative noises. Both systems are investigated using the conformable derivative in terms of stratonovich formulation. These systems are chosen for study due to their relevance and significance in understanding the behavior of complex physical phenomena influenced by stochastic effect. The novelty lies in the applications of the unified solver method, which is a succinct, direct and efficient technique for obtaining solutions in rational, trigonometric and hyperbolic forms. To demonstrate the effect of multiplicative noise and conformable derivative on the solutions of SCZs and SCHMs, 3D and 2D graphs are illustrated by using Mathematica 11. Our findings highlight the sustaining role of the multiplicative noise, which stabilizes the solutions of systems around zero. Overall, this study deepens the understanding of the interaction between stochastic effects and conformable derivatives in the context of these complex systems.

Discovery of New Exact Wave Solutions to the M-Fractional Complex Three Coupled Maccari’s System by Sardar Sub-Equation Scheme

In this paper, we succeed at discovering the new exact wave solutions to the truncated M-fractional complex three coupled Maccari’s system by utilizing the Sardar sub-equation scheme. The obtained solutions are in the form of trigonometric and hyperbolic forms. These solutions have many applications in nonlinear optics, fiber optics, deep water-waves, plasma physics, mathematical physics, fluid mechanics, hydrodynamics and engineering, where the propagation of nonlinear waves is important. Achieved solutions are verified with the use of Mathematica software. Some of the achieved solutions are also described graphically by 2-dimensional, 3-dimensional and contour plots with the help of Maple software. The gained solutions are helpful for the further development of a concerned model. Finally, this technique is simple, fruitful and reliable to handle nonlinear fractional partial differential equations (NLFPDEs).

Control of distributed-parameter systems using normal forms: an introduction

Abstract This paper gives an overview of the control of distributed-parameter systems using normal forms. Considering linear controllable PDE-ODE systems of hyperbolic type, two methods derive tracking controllers by mapping the system into a form that is advantageous for the control design, analogous to the finite-dimensional case. A flatness-based controller makes use of the hyperbolic controller canonical form that follows from a parametrization of the system’s solutions. A backstepping design exploits the strict-feedback form of the system to recursively stabilize and transform the subsystems.

Bouncing universe in modified Gauss–Bonnet gravity

In this paper, a bouncing cosmological scenario is studied in the background of a flat FLRW model with a specific parametrized hyperbolic form of scale factor a in terms of t, where λ is taken as the model parameter. This model is discussed in f(R,G) formalism having structured as f(R,G)=R+F(G), where R is Ricci scalar and G is the Gauss–Bonnet invariant. The proposed functional form of the Hubble parameter is considered in such a way that it satisfies the bouncing criteria of the model, which is free from the initial singularity. The physical consequences of the model are discussed. In this model, one can see that the EoS parameter crosses the quintom line ω=−1 in the neighbourhood of bouncing point t=0, which is also a criteria for a successful bouncing cosmological model. Finally, we examine the stability of the model and find that all the essential features of bouncing model are satisfied successfully.

Design and Computational Analysis of an Axisymmetric Contoured Supersonic Propulsion Nozzle

The design of supersonic axisymmetric nozzles to a required exit Mach number necessitates applying the Method of Characteristics (MoC) that reduces the partial differential equations system of hyperbolic form governing such internal flow-fields to two ordinary differential equations known as the characteristic and the compatibility equations. These relationships are then transformed into finite difference equations and solved using an appropriate predictor-corrector algorithm.The present study applies the MoC to the design of a contoured nozzle that accelerates the combustion gases to supersonic velocities. The selected configuration has a throat constituted of two circular arcs with the one downstream attached to a 2nd-order polynomial profile at the attachment point. The solution, in terms of the static pressure along the centerline and the wall is obtained. It was found that the flow expands gradually without any instability or possibility of shock waves. A gradual expansion represented by a decrease in pressure is noticed immediately downstream of the throat, the flow tending to expand gradually from the throat to the exit. Along the wall, the expansion is important immediately downstream of the throat while it tends to stabilize downstream to reach the ambient pressure at the outlet. In terms of performance, the thrust coefficient developed shows the great expansion process within the divergent supersonic section. The configuration is also found to develop a great specific thrust.

Revisiting (2+1)-dimensional Burgers’ dynamical equations: analytical approach and Reynolds number examination

Classical Burgers’ equation is an indispensable dynamical evolution equation that is autonomously devised by Burgers and Harry Bateman in 1915 and 1948, respectively. This important model is featured through a nonlinear partial differential equation (NPDE). Furthermore, the model plays a crucial role in many areas of mathematical physics, including, for instance, fluid dynamics, traffic flow, nonlinear acoustics, turbulence phenomena, and linking convection and diffusion processes to state a few. Thus, in the present study, an efficient analytical approach by the name ‘generalized Riccati equation approach’ is adopted to securitize the class of (2+1)-dimensional Burgers’ equations by revealing yet another set of analytical structures to the governing single and vector-coupled Burgers’ equations. In fact, the besieged method of the solution has been proven to divulge various sets of hyperbolic, periodic, and other forms of exact solutions. Moreover, the method first begins by transforming the targeted NPDE to a nonlinear ordinary differential equation (NODE), and subsequently to a set of an algebraic system of equations; where the algebraic system is then solved simultaneously to obtain the solution possibilities. Lastly, certain graphical illustrations in 2- and 3-dimensional plots are set to be depicted - featuring the evolutional nature of the resulting structures, and thereafter, analyze the influence of the Reynolds number Ra on the respective wave profiles.

The simulation of non-stationary gas pressure in a pipeline with pumping and injection

In this work, based on the fundamental research of I. A. Charny on the unsteady movement of a real fluid in pipes, an equation that describes the pressure dynamics in a complex section of a gas pipeline was compiled. The use of the Dirac impulse function made it possible to describe the dynamics of unsteady pressure in a single equation in the case of point pumping and injection at given points. Linearization of the model by averaging the velocity of the gas made it possible to bring the equation to a hyperbolic form. If we neglect the forces of inertia compared to the forces of resistance, then the mathematical model will represent a partial differential equation of the second order parabolic type. The dynamics of pressure at specific points of pumping and injection under boundary conditions of the second kind (rates are given) was obtained using the finite cosine Fourier transform. The working formula that allows you to determine the pressure at any point at a given point in time is a trigonometric series. The series are rapidly convergent, so there is no difficulty in working with them. Particular cases are considered (without pumping and injection). The fulfillment of the boundary conditions is checked. It is easy to rebuild the working formula for pumping and / or injection at several points in a given area. Calculations of the mass flow and average velocity are made. In practice, the pressure is usually set at the start and end points of the area under study. In this case, it becomes possible to go to the costs (to derivatives) at the ends of the section under consideration.

On logarithmic transformation-based approaches for retrieving traveling wave solutions in nonlinear optics

Finding solutions in the solitary wave-form for nonlinear Schrödinger’s equations with dispersive and dispersion terms is challenging but would yield interesting results. The basic idea of this paper is to use the idea of logarithmic transformation-based approaches in combination with symbolic structures of exponential functions. The article discusses a nonlinear equation that serves as a universal version of Schrödinger’s equations and has numerous applications in the field of nonlinear optics. We obtain a diverse set of solutions involving trigonometric, hyperbolic, and rational forms. These forms have a broad application spectrum in fields such as plasma physics, nonlinear optics, optical fibers, and nonlinear sciences. Due to the presence of various arbitrarily chosen constants, these solutions exhibit extensive and rich dynamical behavior. In this direction, several numerical simulations corresponding to these results are presented in the article. One notable aspect to consider is that the methodologies utilized in this paper have not been previously employed to solve the model being studied. Furthermore, these techniques may be readily adaptable for solving a diverse range of other nonlinear partial differential equations.

On Preheating after Inflation in Scalar‐Tensor Theories of Gravity

AbstractIn this work, the preheating mechanism is investigated after inflation in general scalar‐tensor theories of gravity. In the present scenario, two‐field scenario is considered and then derived the potential of exponential and hyperbolic tangent forms. The evolution of the background system is studied when the back reaction on the background field is neglected. The particle production due to parametric resonances in both models is examined. It is found that in Minkowski space the stage of parametric resonances can be described by the Mathieu equation. Finally, parametric resonances in the models are sufficiently broad possible for the exponential growth of the number of particles are demonstrated. The case in which the expansion of space is taken into account is also discussed.

Exact Solutions of the Bloch Equations to the Asymmetric Hyperbolic Cosine Pulse with Chirped Frequency

In this research study, we derive the exact solutions of the Bloch equations describing the dynamics of a two-level atom with dephasing. In the two-level atom, a strong laser pump couples a ground state to an upper excited state with a time-dependent Rabi-frequency. The exact solutions are given for the atomic population inversion and the real and imaginary parts of the coherence while the input pulse is an asymmetric hyperbolic cosine form. Additionally, the system is under a chirped detuning. The method of solving the Bloch equations analytically is a very tedious part of the research, and as far as we know, there are few exact solutions available in this field. Hence, our solutions might be of great interest to various research areas, including nuclear magnetic resonance, where analytical solutions to the Bloch equations play a major role in the study of the information on the state of the medium as determined by the NMR signals.