Singular Periodic Wave Research Articles

Dynamics of breather, multi-wave, interaction and other wave solutions to the new (3+1)-dimensional integrable fourth-order equation for shallow water waves

PurposeThe purpose of this paper is to study the new (3 + 1)-dimensional integrable fourth-order nonlinear equation which is used to model the shallow water waves.Design/methodology/approachBy means of the Cole–Hopf transform, the bilinear form of the studied equation is extracted. Then the ansatz function method combined with the symbolic computation is implemented to construct the breather, multiwave and the interaction wave solutions. In addition, the subequation method tis also used to search for the diverse travelling wave solutions.FindingsThe breather, multiwave and the interaction wave solutions and other wave solutions like the singular periodic wave structure and dark wave structure are obtained. To the author’s knowledge, the solutions obtained are all new and have never been reported before.Originality/valueThe solutions obtained in this work have never appeared in other literature and can be regarded as an extension of the solutions for the new (3 + 1)-dimensional integrable fourth-order nonlinear equation.

Diverse optical solitons to the Radhakrishnan–Kundu–Lakshmanan equation for the light pulses

In this paper, we study the Radhakrishnan–Kundu–Lakshmanan equation (RKLE) that plays a key role in modeling the nonlinear propagation of the light pulses. Two novel methods namely Wang’s Bäcklund transformation-based method and Wang’s direct mapping method are employed to construct the diverse exact optical soliton solutions. Abundant exact solutions like the bright soliton solution, dark soliton solutions, singular soliton solutions, algebraic solitary wave solutions and the singular periodic wave solutions expressed in the form of the rational function type, double-exp function type, Sin-Cos function type, Tan function type, Cot function type, Sinh-Cosh function type, Tanh function type, Coth function type, Sech function type, Csch function type, Sec function type and Csc function type are obtained. The dynamics of the different exact solutions are presented through the 3D plot, contour and 2D curve, and the corresponding physical interpretations are elaborated in detail. It is confirmed that the proposed methods are effective and powerful, and can be adopted to construct the optical solitons of the other PDEs arising in the optical physics.

Further physical study about solution structures for the fractional equal width equation (EW) in shallow water hydrodynamics

In this article, with the help of Matlab R2021a software, the exact solutions of the EW are successfully examined by Sub-Equation method and the unified F-expansion method. The main novelty of this paper lies in the six aspects: (1) Singular periodic wave U-shaped solition is discovered by the sub equation method which did not happen in previous papers [35–37]. (2) More and more intensive multi-periodic M-shape wave solutions are also discovered by the unified F-expansion method which did not happen in previous papers [32,34–37]. (3) The dynamic wave behaviours about the got results and the effects of fractionality α are illustrated by Figs. 1–7. The fractional parameter α masters the appearance of the solitons and makes the involved alphabetic shaped solitions due to the increase in the fractionality number α in Figs. 1–7. (4) The effect about free parameters c,k can be seen from Figs. 8–11 in this paper. If only the increase of wave velocity parameter c or wave number parameter k, the energy trend of the wave is always positive and intensive. This also never happens in previous literatures [34–37]. (5) The Jumarie’s modified Riemann–Liouville and beta fractional and conformable derivatives are tested for accuracy in Figs. 12 and 13. We seem firstly compare the relations and distinctions among different fractional order derivatives in EW model. (6) In our best cognitive, the planar dynamical system about EW model whose phase portraits are also first studied. Furthermore, phase portraits for the space–time fractional equal width equation are plotted, and the corresponding sensitivity and chaotic behaviors are analyzed. The geometrical representations of the EW provide the corporal information to explain the physically phenomenons.

Study on the Nonlinear Dynamics of the (3+1)-Dimensional Jimbo-Miwa Equation in Plasma Physics

The Jimbo-Miwa equation (JME) that describes certain interesting (3+1)-dimensional waves in plasma physics is studied in this work. The Hirota bilinear equation is developed via the Cole-Hopf transform. Then, the symbolic computation, together with the ansatz function schemes, are utilized to seek exact solutions. Some new solutions, such as the multi-wave complexiton solution (MWCS), multi-wave solution (MWS) and periodic lump solution (PLS), are successfully constructed. Additionally, different types of travelling wave solutions (TWS), including the dark, bright-dark and singular periodic wave solutions, are disclosed by employing the sub-equation method. Finally, the physical characteristics and interaction behaviors of the extracted solutions are depicted graphically by assigning appropriate parameters. The obtained outcomes in this paper are more general and newer. Additionally, they reveal that the used methods are concise, direct, and can be employed to study other partial differential equations (PDEs) in physics.

Diversity of soliton solutions to the (3 + 1)-Dimensional Wazwaz-Benjamin-Bona-Mahony equations arising in mathematical physics

The current study scrutinizes exact singular solitary, kink, anti-kink, bell-shaped, and periodic traveling wave solutions for the newly implemented 3-dimensional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations family using a modified Exp-function approach. In spite of the presence of several trigonometric, hyperbolic, and rational functions, some new exact solitary, hyperbolic, and periodic traveling wave solutions to the considered equations are obtained by putting the modified Exp-function method into practice using the computer program Maple. The arrangement of the functions tanh, sech2, tan, and sec2 expresses the specific solutions produced by the defined procedure. The information was obtained to evaluate how well the suggested method could compute the exact solutions of the WBBM equations that could be applied to nonlinear water model applications in the ocean and coastal engineering. To show the physical composition and properties of the discovered solitons, contour, three-, and two-dimensional graphs are plotted using the computer program Mathematica. The acquired results imply that the proposed method can be applied to find various, enhanced, useful, and compatible solutions for other important nonlinear partial differential equations. All of the solutions given have been qualified by swapping the respective equations with those generated via the computer program Maple. Furthermore, we compared our obtained results with the solutions already put forward in the literature.

A new implementation of a novel analytical method for finding the analytical solutions of the (2+1)-dimensional KP-BBM equation

In this work, we perform a comprehensive analytical study to find the novel exact traveling wave solutions of the (2+1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation. The recently developed (G′G′+G+A) -expansion technique is a capable method for finding the new exact solutions of assorted nonlinear evolution equations. Some new analytical solutions are obtained by utilizing the aforementioned method. The obtained solutions are expressed as trigonometric functions and exponential functions. The extracted exact wave solutions are advanced and fully unique from the earlier literature Moreover, we have presented the contour simulations, 2D and 3D graphical representations of the solution functions and we have observed that the solutions obtained are periodic and solitary wave solutions. We have shown two soliton wave solutions and two singular periodic wave solutions for the particular values of the parameters graphically. As per our knowledge, we must say that the extracted solutions might be significant and essential for new physical phenomenon.

New exact solutions to space–time fractional telegraph equation with conformable derivative

In this paper, we extract variety of new exact traveling wave solutions of space–time fractional nonlinear Telegraph equation for transmission lines by using improved generalized Riccati equation mapping (IGREM) method. The aforementioned equation has been solved for the first time using conformable fractional derivative. The nonlinear Telegraph equation plays crucial role in the mathematical study of voltage and current on an electrical transmission line. In an effort to motivate the researchers, we have used very efficacious technique called IGREM method to find a variety of soliton solutions, in which appropriate conformable fractional wave transformation converts nonlinear partial fractional differential equation into nonlinear ordinary equation which is then solved to find the exact solutions of equation. Hence, we succeed in constructing rich solutions that include periodic wave, singular periodic wave solutions, dark, combined dark–bright and rational solutions. To investigate wave profiles of various solutions, 2D and 3D graphs have been explained. The originality of this work has been proved by comparing the obtained results with previously present results in literature. This method has substantial advantage over other methods that it can be used to solve higher order nonlinear PDEs, and this method allows to solve these equations more conveniently with the aid of mathematical software.

Solitary wave solutions for the originating waves that propagate of the fractional Wazwaz-Benjamin-Bona-Mahony system

In this article, we discuss various solitary wave solutions that are extremely important in applied mathematics. In order to construct accurate solution to nonlinear fractional PDEs, we have employed the Khater technique to nonlinear equation for 3-dimensional fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) using conformable fractional dervatives. We obtain novel sets of solutions by employing this method including both periodic kink and periodic singular kink solution, singular periodic waves solution, dark solitons, bell shaped soliton solution, bell shaped singular solution, bright solution and smooth periodic wave solutions. In order to comprehend the physical principles and significance of the technique, solutions have been graphically represented. The results acquired to demonstrate the efficiency of the computational technique for the WBBM equation, our findings unambiguously demonstrate that the suggested approach is a useful, potent and simple way for experimental result for various kinds of non-integer order differential equations in the engineering and applied sciences.

New wave solutions, exact and numerical approximations to the nonlinear Klein–Gordon equation

This study investigates the nonlinear Klein–Gordon equation (KGE). We successfully construct some new topological kink-type, non-topological, singular solitons, periodic waves and singular periodic wave solutions to this nonlinear model by using the extended ShGEEM, rational sine-cosine extended (ERSC), and sinh-cosh (ERSCh) methods. In addition, a numerical method for solving the KGE is described in this paper. We use a combination of two numerical techniques called fictitious time integration method and the group preserving scheme (GPS). Fictitious time integration method converts the main equation into a new problem then the GPS is used to gain the numerical solutions. Few experiments are provided to successfully demonstrate the correctness of the approach.

Nonlinear Dynamic Behaviors of the (3+1)-Dimensional B-Type Kadomtsev—Petviashvili Equation in Fluid Mechanics

This paper provides an investigation on nonlinear dynamic behaviors of the (3+1)-dimensional B-type Kadomtsev—Petviashvili equation, which is used to model the propagation of weakly dispersive waves in a fluid. With the help of the Cole—Hopf transform, the Hirota bilinear equation is established, then the symbolic computation with the ansatz function schemes is employed to search for the diverse exact solutions. Some new results such as the multi-wave complexiton, multi-wave, and periodic lump solutions are found. Furthermore, the abundant traveling wave solutions such as the dark wave, bright-dark wave, and singular periodic wave solutions are also constructed by applying the sub-equation method. Finally, the nonlinear dynamic behaviors of the solutions are presented through the 3-D plots, 2-D contours, and 2-D curves and their corresponding physical characteristics are also elaborated. To our knowledge, the obtained solutions in this work are all new, which are not reported elsewhere. The methods applied in this study can be used to investigate the other PDEs arising in physics.

Optical solitons to the Perturbed Gerdjikov-Ivanov equation with quantic nonlinearity

In this article, we examine the Perturbed Gerdjikov-Ivanov equation by using the Jacobi elliptic function expansion method. This results in obtaining distinct solutions including dark, bright, singular solitons, periodic waves, singular periodic waves, and Jacobi elliptic function solutions. The 2- and 3-dimensional graphs of the reported solutions are presented. The reported results may be useful in explaining the physical features of the studied equation.

Optical Solitons for a Concatenation Model by Trial Equation Approach

This paper addresses the newly proposed concatenation model by the usage of trial equation approach. The concatenation is a chain model that is a combination of the nonlinear Schrodinger’s equation, Lakshmanan–Porsezian–Daniel model as well as the Sasa–Satsuma equation. The recovered solutions are displayed in terms of dark solitons, singular solitons, cnoidal waves and singular periodic waves. The trial equation approach enables to recover a wide spectrum of solutions to the governing model. The numerical schemes give a visual perspective to the solutions derived analytically.

Study of explicit travelling wave solutions of nonlinear evolution equations

The Bernoulli Sub-ODE approach is used in this study to look for comprehensive travelling wave solutions to the nonlinear evolution equations (NLEEs). The analysis in the present paper shows the existence of travelling waves for the time-regularized long-wave (TRLW) equation, the modified Korteweg–de Vries –Zakharov–Kuznetsov (mKdV–ZK) equation, and the (2+1)-dimensional Zoomeron equation. The outcomes demonstrate the richness of explicit solutions of the studied models. As a result, precise solitary wave solutions to the studied problems, such as kink waves, singular kink waves, dark soliton, and periodic waves are found. The phase plane is briefly examined after the determination of the Hamiltonian function. Using Maple 13, we validated the accuracy of the obtained solutions by reintroducing them into the original equation. We will demonstrate how the amplitudes and wave profiles are impacted by free parameters. In this article, we firmly establish that the wave amplitude varies as the free parameters change. It is demonstrated that the technique is efficient and applicable to several different NLEEs in mathematical physics.

On the multiple explicit exact solutions to the double‐chain DNA dynamical system

The nonlinear dynamics of deoxyribonucleic acid (DNA) is the subject of this research. For an organism to flourish and reproduce, it needs the genetic information that is contained in its DNA. In the DNA molecule, the hydrogen bonds between the base pairs of the two polynucleotide chains are represented by an elastic membrane (or a series of linear springs) that connects the model's two long elastic homogeneous strands (or rods). Using two integration norms, namely, the new auxiliary equation method (NAEM) and the extended sinh‐Gordon equation expansion method (ShGEEM), we obtain a variety of nonlinear dynamical solitary wave structures in various shapes, including hyperbolic, trigonometric, and exponential function solutions, as well as some unique solitary wave solutions, including bell‐shaped, kink, singular, combined, and multiple solitons. Using the logarithmic transformation assistant, we examine a variety of wave structures and recover singular periodic wave solutions with unknown parameters, along with the validity constraints. According to the results, the governing model contains an unusually large number of theoretically exact solitary wave solutions. The physical characterization of some of the presented results can be visualized in 3D and 2D profiles by selecting the appropriate parameter values.

RETRACTED: On soliton solutions of a modified nonlinear Schrödinger’s equation of third-order governing in optical fibers

Recently, the development of novel methods for finding solutions to partial differential equations has made great progress in various fields of science. These equations are an important tool in describing many phenomena that occur around us. The research work aims to study a third-order nonlinear Schrödinger’s equation using the modified generalized exponential rational function method, and the extended sinh-Gordon equation expansion method. The common idea between the two methods is that it is first necessary to reduce the equation with partial derivatives to a form of the equation with ordinary derivatives using a new wave definition. Moreover, the retrieved solutions are obtained in a closed form of elementary functions which are important in practical applications. We successfully construct some soliton, singular soliton and singular periodic wave solutions to the nonlinear complex model. Further, numerical simulations are also included via several figures. A significant advantage of both techniques is the ability to introduce a wide range of types of analytical solutions in terms of well-known elementary functions. It is obtained that these methods are also applicable in solving other equations in nonlinear science.

Optical solitons and stability analysis for the new (3+1)-dimensional nonlinear Schrödinger equation

This paper explores the new [Formula: see text]-dimensional nonlinear Schrödinger equation which is used to model the propagation of ultra-short optical pulses in highly-nonlinear media. This equation is newly derived based on the extended [Formula: see text]-dimensional zero curvature equation. An effective technique, namely, the extended sinh-Gordon equation expansion method is applied to find optical soliton solutions and other solutions for this model. As a result, dark, bright, combined dark–bright, singular, combined singular soliton solutions, and singular periodic wave solutions are obtained. The stability of the model is investigated by using the modulation instability analysis which guarantees that the model is stable and all solutions are stable and exact. Physical explanations of the obtained solutions are presented by using 3D and 2D plots. The reported outcomes are useful in the empirical application of fiber optics.

On abundant wave structures of the unsteady korteweg-de vries equation arising in shallow water

The aim of this paper is to study the unsteady korteweg-de vries equation that plays an important role in describing the shallow water. Two analytical techniques namely the Sardar-subequation method and the energy balance method are employed to seek the abundant traveling wave solutions for the first time. By these two methods, plenty of traveling wave solutions such as the bright solitary wave solutions, dark solitary wave solutions, singular periodic wave solutions and perfect periodic wave solution that expressed in terms of the generalized hyperbolic functions, generalized trigonometric functions and the cosine function are obtained. Finally, the dynamic behaviors of the solutions are described through the 3D plot and 2D curve. The results in this paper demonstrate that the proposed methods are powerful and effective to construct the traveling wave solutions of the nonlinear evolution equations in ocean engineering and science.

Optical solitons with Kudryashov’s quintuple power law nonlinearity having nonlinear chromatic dispersion using modified extended direct algebraic method

In this article, nonlinear Schrödinger’s equation with Kudryashov’s quintuple power law of refractive index is considered. The modified extended direct algebraic algorithm is implemented to extract solitons and other types of solutions for this model. Bright solitons, dark solitons and singular solitons are extracted with their constraints for the validity of these solitons. Additionally, different structures of solutions are listed such as, periodic wave solutions, singular periodic wave solutions, hyperbolic wave solutions, exponential wave solutions, rational-type solution, Jacobi elliptic wave solutions and Weierstrass elliptic wave solutions. Moreover, some selected solutions are illustrated graphically.

Investigation of new solitons and other solutions to the modified nonlinear Schrödinger equation in ocean engineering

This article deals with the new solitons and other solutions to the modified nonlinear Schrödinger equation (MNLSE) that represents the propagation of rogue waves in deep water. The solutions are achieved in the shape of trigonometric and hyperbolic functions as well as solitary wave solutions by the mean of extended sinh-Gordon equation expansion and (G′G2)-expansion function methods. Furthermore, the singular periodic wave solutions are obtained and the constraint conditions for the existence of soliton solutions are also listed. We also plot 3D graphs for a better explanation of the achieved solutions by choosing the appropriate values of the parameters involved in solutions. We can assert from the obtained results that the applied techniques are simple, vibrant, and quite well, and will be helpful tool for addressing more highly nonlinear issues in various of fields, especially in ocean and coastal engineering. Furthermore, our findings are first step toward understanding the structure and physical behavior of complicated structures. We anticipate that our results will be highly valuable in better understanding the waves that occur in the ocean. We feel that this work is timely and will be of interest to a wide spectrum of experts working on ocean engineering models.

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.