Analytical Solitary Wave Solutions Research Articles

Phase trajectories and Chaos theory for dynamical demonstration and explicit propagating wave formation

This paper is subjected to study the nonlinear integrable model which is the (3+1)-dimensional Boussinesq equation which has a lot of applications in engineering and modern sciences. To find and examine the analytical exact solitary wave solutions of (3+1)-dimensional Boussinesq equation, a modified generalized exponential rational functional method is exerted. As a result, waves, singular periodic, hyperbolic, and trigonometric type solutions are obtained. These acquired solutions are more innovative and encouraging to researchers in their endeavor to study physical marvels. To illustrate how some selected exact solutions propagate, the graphical representation in 2D, Contour, and 3D of those solutions is provided with various parametric values. The considered equation is additionally transformed into the planar dynamical structure by applying the Galilean transformation. All potential phase portraits of the dynamical system are investigated using the theory of bifurcation. The Hamiltonian function of the dynamical system of differential equations is established to see that, the system is conservative over time. The presentation of energy levels through graphics provides valuable insights, and it demonstrates that the model has solutions that can be expressed in closed form. The periodic, quasi-periodic, and chaotic behaviors of the 2D, 3D, and time series are also observable once the dynamical system is subjected to an external force. Meanwhile, the sensitivity of the derived solutions is carefully examined for a range of initial conditions.

On the exploration of soliton solutions of the nonlinear Manakov system and its sensitivity analysis

In this study, we investigate the fractional coupled nonlinear Schrödinger equation (FCNLSE) of Manakov type, which has numerous applications in different fields of physics, such as optics and plasma physics. In this study, we employ two different analytical methods, namely the auxiliary equation method (AEM) and the improved F-expansion method, to construct novel analytic exact solitary wave solutions, encompassing trigonometric, hyperbolic, and exponential functions. These solutions include solitary wave solutions dark, bright, combo, singular, rational form solutions, and periodic wave solutions. Additionally, using the characterizations of the Hamiltonian system, the stability property of the discovered soliton wave solutions is examined. We have also illustrated the sensitivity analysis for the modified dynamic structural system using different initial conditions. In order to illustrate various physical properties of wave structures, the well-furnished results are finally demonstrated in various 3D, 2D, contour, and density profiles. This work’s findings highlight the significance of researching diverse nonlinear wave phenomena in physics and nonlinear optics by demonstrating how crucial it is to comprehend the physical meaning and behavior of the examined equation. The procedures used are capable, effective, and succinct enough to make it possible for further research.

Application of modified exp-function method for strain wave equation for finding analytical solutions

The strain wave equation in micro-structured solids, which is significant in solid physics, is offered for consideration in this work. This is accomplished using the modified exp-function method, which is one of the most powerful ways for constructing abundantly distinct, accurate solutions to nonlinear partial differential equations. Wave propagation in microstructured solids is based on the structure of the strain wave equation. As a result, we were able to obtain a variety of new analytical solitary wave solutions like hyperbolic and complex function solutions with free parameters. Additionally, 2D and 3D plots are graphed for each of the solutions found in order to better comprehend their physical interpretation.

Analytical solitary wave solutions of a time-fractional thin-film ferroelectric material equation involving beta-derivative using modified auxiliary equation method

The main objective of current paper is to examine the impacts of fractional parameters on the dynamic response of soliton waves in a nonlinear time-fractional thin-film ferroelectric materials equation (TFFEME). To achieve such a goal, the TFFEME is first rehabilitated into ordinary differential equation using a complex wave transformation. Solitary wave solutions of the governing equation, representing the dynamics of waves in the material and plays a vital rule in many branches of physics and hydrodynamics, are then constructed through applying the modified auxiliary equation method (MAEM). The extracted solutions are demonstrated using definition of the beta derivative to understand their dynamical behavior. The hyperbolic, periodic and trigonometric function solutions are used to derive the analytical solutions for the given model. As a result, dark, bright, periodic and solitary wave solitons are obtained. The fractional impact of the above derivative on the physical phenomena is observed. The 2D and 3D graphs are also shown to confirm the behavior of analytical wave solutions.

Exact Solutions of (2+1) Dimensional Cubic Klein-Gordon (cKG) Equation

In current study, (2+1)-dimensional cubic Klein-Gordon (cKG) equation illustrating dislocation propagation in crystals as well as the behaviour of elementary particles is investigated to establish a variety of new analytic exact solitary wave solutions. Modified exponential expansion method has been implemented to unfold certain wave solutions of considered model. As a result, three sorts of solutions emerge in a fairly systematic manner in the shape of hyperbolic, trigonometric, and rational functions. The kink and periodic wave solitons are acquired and presented geometrically, some 3D plots are simulated and displayed to respond the dynamic behavior of these obtained solutions. In this work we have used symbolic package maxima to obtained our solutions. Our acquired solutions might be most helpful to analyze physical issues that arise from nonlinear complicated dynamical systems.

A study of nonlinear extended Zakharov–Kuznetsov dynamical equation in (3+1)-dimensions: Abundant closed-form solutions and various dynamical shapes of solitons

In this work, we execute the generalized exponential rational function (GERF) method to construct numerous and a large number of exact analytical solitary wave solutions of the nonlinear extended Zakharov–Kuznetsov (EZK) dynamical equation in (3+1)-dimensions. The implemented method is one of the best, most reliable, and efficient techniques in the present time for determining numerous closed-form wave analytic solutions to NPDEs. We have accomplished a variety of solitary wave solutions related to some arbitrary parameters under various family cases. These solutions take the following forms based on the free parameters chosen: exponential functions form, trigonometric functions form, and hyperbolic functions form. The obtained solutions are dissimilar and entirely new from the previous findings available in the literature. The dynamics of obtained solutions, namely, soltion, singular soliton wave, a periodic wave, bell-shape, anti-bell-shape wave, breather wave, and multisoliton wave solutions by the special-choice of parameters, are shown graphically in 3D, 2D, and corresponding density profiles. The results demonstrate that the employed computational strategy is efficient, direct, concise, and can be executed in various complex phenomena with symbolic computations. Furthermore, it is revealed that the generalized exponential rational function technique can be effectively utilized for several other NPDEs in engineering, sciences, and mathematical physics.

Structure of analytical ion-acoustic solitary wave solutions for the dynamical system of nonlinear wave propagation

Abstract In the present study, the ion-acoustic solitary wave solutions for Kadomtsev–Petviashvili (KP) equation, potential KP equation, and Gardner KP equation are constructed. The nonlinear KP equations are studying the nonlinear process of waves without collisions plasma and having non-isothermal electrons and cold ions. Two-dimensional ion-acoustic solitary waves (IASWs) in magnetized plasma are consisting of electrons and ions. We obtained the ion-acoustic solitary wave solutions same as dark and bright, kink and anti-kink wave solitons. The physical phenomena of various structures for IASWs are represented graphically with symbolic computations. These results are more helpful in the development of soliton dynamics, quantum plasma, dynamic of adiabatic parameters, fluid dynamics, and industrial phenomena.

Abundant stable wave structures for the nonlinear propagation of dislocations in crystals, phase differences across Josephson junctions

This study investigates the analytical solitary wave solutions of the nonlinear Sine–Gordon (SG) system by using two recent computational (Khater II and novel Kudryashov) schemes. The studied model is used as a model for various wave phenomena, including the propagation of dislocations in crystals, phase differences across Josephson junctions, torsion waves in strings and pendula, and waves along lipid membranes. It was already studied in the nineteenth century in connection with the theory of pseudospherical surfaces. The quantum version is used as a simple model for solid–state excitations. The solitonic property of the constructed solutions is also explained through some distinct graphs in two-, three-dimensional, contour, and polar plots. The stability property of the obtained soliton wave solutions is discussed based on the Hamiltonian system’s characterizations.

Transcritical bifurcation of nonlinear electrostatic waves in a superthermal dusty plasma

The phase portraits and bifurcation properties of nonlinear dust ion-acoustic waves are studied in a complex plasma comprising cold dust particles, warm adiabatic ions, and κ-distributed electrons. A characteristic equation is obtained and solved numerically. The existence domain of fixed points and the stability of traveling wave solutions are discussed in the parameter space of Mach number M and dust concentration δ over a wide range of values of κ and ion temperature σ. It is shown that the motion dynamics of homoclinic orbits and nonlinear periodic orbits undergo a transcritical bifurcation at critical points of and , where two fixed points coalesce, and then switch their stabilities. We observe that at these bifurcation values with half-stable fixed points, the system supports a new analytical solitary wave solutions with much fatter tails than regular solitons. Furthermore, the existence domain of transcritical bifurcation parameters and transition between solitary and nonlinear periodic waves has been determined for different values of M and δ. It is found that the Mach number broadens the region prior to the bifurcation point, and enhances the onset of transition between modes, while the dust concentration has an opposite influence.

Abundant closed-form solutions of the (3+1)-dimensional Vakhnenko-Parkes equation describing the dynamics of various solitary waves in ocean engineering

This research article is a study of the (3+1)-dimensional Vakhnenko-Parkes equation that governs the propagation of high-frequency waves in a relaxing medium. To find a variety of new analytic exact solitary wave solutions, we apply three effective schemes, such as the generalized Kudryashov method (GKM), the generalized exponential rational function method (GERFM), and the generalized Riccati equation mapping method (GREMM). Obtained solutions are crucial to understand some existing, complicated physical phenomena. Many solutions are obtained, which are expressed by the exponential, trigonometric, and hyperbolic functions, including tanh, coth, sech, csch, tan, cot, sech, csch, and their combinations. These solutions include periodic-wave solutions, kink waves, combinations of kink and multi solitons, singular solitons, singular-kink structures, and periodic solitons. The 3D and 2D plots are shown by considering the possibility of arbitrary parameters occurring in the solutions better to understand the underlying mechanisms of the stated family. The computational results of this study assure us that the proposed approaches seem useful, efficient, and reliable tools for estimating the solutions attained in the study. Moreover, the method is expected to be frequently applied to examine various types of nonlinear PDEs, which emerge in mathematical physics, ocean engineering, and science.

Variable dust charge generates a special type of nonlinear structures through modified Gardner equations

AbstractNonlinear damped modified Gardner equation is derived from the dust hydrodynamic model in dusty plasma, which is collisional, unmagnetized, and has two temperature ions. An extended version of the reductive perturbation technique (RPT) is used to derive the said equations. Approximate analytical solitary wave solution is obtained for compositional parameters near the critical composition. It is found that various nonlinear structures like super periodic, superhomoclinic, and special types of nonlinear structures are emerging while an investigation is done through the system of first‐order differential equations. The effects of different plasma parameters like collisions and dust charge variables on the wave structures have been noticed. In the circumstances of nebulas, interstellar clouds, planetary magnetospheres, and cometary environments recognized by astrophysicists, Voyager spacecraft observation on the fine structure of Saturn's rings, semiconductor devices, it has been investigated that the dust charge variable modifies the super nonlinear structures significantly.

Evolutionary dynamics of solitary wave profiles and abundant analytical solutions to a (3+1)-dimensional burgers system in ocean physics and hydrodynamics

In the fields of oceanography, hydrodynamics, and marine engineering, many mathematicians and physicists are interested in Burgers-type equations to show the different dynamics of nonlinear wave phenomena, one of which is a (3+1)-dimensional Burgers system that is currently being studied. In this paper, we apply two different analytical methods, namely the generalized Kudryashov (GK) method, and the generalized exponential rational function method, to derive abundant novel analytic exact solitary wave solutions, including multi-wave solitons, multi-wave peakon solitons, kink-wave profiles, stripe solitons, wave-wave interaction profiles, and periodic oscillating wave profiles for a (3+1)-dimensional Burgers system with the assistance of symbolic computation. By employing the generalized Kudryashov method, we obtain some new families of exact solitary wave solutions for the Burgers system. Further, we applied the generalized exponential rational function method to obtain a large number of soliton solutions in the forms of trigonometric and hyperbolic function solutions, exponential rational function solutions, periodic breather-wave soliton solutions, dark and bright solitons, singular periodic oscillating wave soliton solutions, and complex multi-wave solutions under various family cases. Based on soft computing via Wolfram Mathematica, all the newly established solutions are verified by back substituting them into the considered Burgers system. Eventually, the dynamical behaviors of some established results are exhibited graphically through three - and two-dimensional wave profiles via numerical simulation.

New kinds of analytical solitary wave solutions for ionic currents on microtubules equation via two different techniques

New kinds of analytical solitary wave solutions for ionic currents on microtubules equation via two different techniques

New analytical solitary and periodic wave solutions for generalized variable-coefficients modified KdV equation with external-force term presenting atmospheric blocking in oceans

In this study, the generalized modified variable-coefficient KdV equation with external-force term (gvcmKdV) describing atmospheric blocking located in the mid-high latitudes over ocean is studied for integrability property by using consistent Riccati expansion solvability and the necessary integrability conditions between the function coefficients are obtained. Moreover, several new solutions have been constructed for the gvcmKdV. Additionally, the classical direct similarity reduction method is used to reduce the gvcmKdV to a nonlinear ordinary differential equation. Building on the solutions given in the previous literature for the reduced equation, many novel solitary and periodic wave solutions have been obtained for the gvcmKdV.

Two-dimensional ion-acoustic solitary waves obliquely propagating in a relativistic rotating magnetised electron–positron–ion plasma in the presence of external periodic force

The characteristics of ion-acoustic solitary waves in rotating, weakly relativistic, magnetised and collisionless plasma system comprising electron, positron and ion (EPI) under a periodic external force, whose constituents electron and positron obey Boltzmann distribution, are investigated by deriving forced Zakharov–Kuznetsov (FZK) equation. FZK equation is constructed using reductive perturbation technique (RPT) which is based on multiple-scale analysis of dependent variables. The effects of physical parameters such as the ratio of temperature of electron and positron, the strength of external periodic force, positron concentration in electron background, ion temperature and magnetic field on the analytical solitary wave solution of FZK equation are observed. It is seen that the behaviours of ion-acoustic wave are significantly affected due to the presence of positron and the periodic external forces. The results of this work may be applied when the above-mentioned plasma environment is found in laboratory as well as in space.

Analytical Solution Of Time Fractional Nonlinear Schrodinger Equation By Homotopy Analysis Method

In this research paper, we have applied Homotopy Analysis Approach to the “time fractional nonlinear Schrodinger equation” to find its analytical periodic and solitary wave solution. Presence of convergence control parameter in this method guarantee the solution of time fractional differential equation in the form of rapidly convergent series. Obtained analytical solution has been compared and found in good agreement. This work demonstrates reliability and potential of HAM to study the time fractional partial differential equation

Novel analytical cnoidal and solitary wave solutions of the Extended Kawahara equation

In this work, some novel analytic traveling wave solutions including the cnoidal and solitary wave solutions of the planar Extended Kawahara equation are deduced. Four different analytical methods (the Jacobian elliptic function, Weierrtrass elliptic function, the traditional tanh method and the sech-square) are devoted for solving this equation. By means of the Jacobian elliptic function ansatz, the cnoidal and soliary wave solutions are obtained. Also, new cnoidal wave solutions are derived via a new hypothesis in the form of the Weierrtrass elliptic function. Moreover, the standard tanh method is utilized to get a new set of solitary wave solutions for the evolution equation. Over and above, the hyperbolic ansatz method (a new ansatz in the form of squre-sech) is employed to get a new set of solitary wave solutions for the evolution equation. Furthermore, all obtained solutions of the planar Extended Kawahara equation cover the traveling wave solutions of the planar modified Kawahara equation. These solutions maybe useful to many researchers interested in studying the propagation of nonlinear waves in nonlinear dispersion mediums like plasma physics, optical fibers, fluid mechanics, and many different branches of science.

Dynamical properties of dust-ion-acoustic wave solutions in a nonextensive collisional dusty plasma

Dynamical properties of dust-ion-acoustic waves (DIAWs) are analysed in a collisional nonextensive dusty plasma composing of mobile ions, q-nonextensive electrons and stationary dust grains with slight collisions between dusts and ions. Reductive perturbation technique (RPT) is practiced to acquire the damped Korteweg-de Vries (DKdV) equation. In absence of collision between dusts and ions, the damping term vanishes and the DKdV equation reduces to the KdV equation. Bifurcation analysis of the dynamical system obtained from the KdV equation is carried out. Analytical solitary wave solution and numerical periodic wave solution of the KdV equation are presented. Approximate analytical DIAW solutions for the DKdV equation are furnished using the novel -expansion technique. As an outcome, novel approximate analytical solutions of DIAWs are obtained for the DKdV equation. The application of our work is significant to explore nonextensive plasma environment, such as, ionosphere of Earth.

Analytical solitary wave solution of dust ion acoustic waves in nonextensive plasma in the framework of damped forced Korteweg–de Vries–Burgers equation

Dust ion acoustic waves (DIAWs) are investigated analytically in the framework of damped forced Korteweg–de Vries–Burgers (DFKdVB) equation in an unmagnetized collisional dusty plasma consisting of negatively charged stationary dust grains, positively charged ions and q-nonextensive distributed electrons. Reductive perturbation technique is used to obtain the DFKdVB equation for DIAWs. The effects of different physical parameters such as the entropic index (q), ratio of unperturbed number densities of electrons and ions $$(\mu )$$ , dust ion collision frequency $$(\nu _{\mathrm{id}0})$$ , kinematic viscosity ( $$\eta _0$$ ), the speed of the traveling wave $$(M_0)$$ , strength $$(f_0)$$ and frequency $$(\omega )$$ of the external periodic force on the analytical DIAWs are shown. The results of the present paper will be useful in laboratory and space plasmas.

Analytical methods: Nonlinear longitudinal wave equation in a magnetoelectro-elastic circular rod, foam drainage and modified Degasperis–Procesi models arising in nonlinear water wave models

We form the analytical solitary wave solutions with the execution of generalized direct algebraic technique on three well-known nonlinear wave models, namely, called foam drainage, longitudinal magnetoelectro-elastic circular rod and modified Degasperis–Procesi equations. The derived solutions are hyperbolic functions in which some are plotted graphically on meticulous values to the parameters which provides the basic knowledge to understand physical significant of these three wave models. The obtained solutions show the efficiency and precision of our scheme. These derived new results prove that our novel technique is awfully effective and can be productive as a instrument solving for sundry other nonlinear evolution equations.