Singular Wave Research Articles

Stable abundant computational solitary wave structures of the perturbed time-fractional NLS equation

The conformable fractional derivatives Khater II (Khat. II) method and the Adomian decomposition (AD) approach are used to investigate the analytical and semi-analytical wave solutions to the perturbed time-fractional nonlinear Schrödinger (NLS) model. This model describes the dynamics of optical solitons propagating via nonlinear optical fibers. Several novel solitary wave solutions are constructed in distinct formats such as hyperbolic, trigonometric, rational, dark, brilliant, combined dark-bright, singular, combination singular, and periodic wave solutions. Additionally, the Adomian decomposition technique determines the absolute error associated with analytical and semi-analytical wave solutions. The Hamiltonian system examines the stability of discovered solutions to determine their acceptability for usage in the model’s application.

The Soliton Wave Solutions and Bifurcations of the (2 + 1)-Dimensional Dissipative Long Wave Equation

With the help of the bifurcation theory of dynamical differential system and maple software, we shall devote ourselves to research travelling wave solutions and bifurcations of the (2 + 1)-dimensional dissipative long wave equation. The study of travelling wave solutions for long wave equation derives a planar Hamiltonian system. Based on phase portraits, we obtain exact explicit expressions of some bounded traveling wave solutions and some important singular traveling wave solutions, under different parametric conditions.

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.

Abundant novel nematicon soliton wave solutions in liquid crystals with Kerr law nonlinearity

Nematicons in liquid crystals with Kerr law nonlinearity are studied using three recent analytical techniques. An essential goal of this study is to discover a variety of nonlinear system solution types, such as singular and solitary wave solutions in both bright and dark states. It also seeks to highlight its answers’ uniqueness by comparing and contrasting with previously published results achieved via other approaches. Two-dimensional, three-dimensional, contour, polar, and spherical graphics have been used to demonstrate specific solutions. Their performance shows the effectiveness of imposed methods in solving nonlinear evolution equations. Many different domains, including optics, fluid dynamics, gaseous plasma, and acoustics, rely on molecules’ characteristics in liquid crystals.

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.

Construction of optical solitons of Radhakrishnan–Kundu–Lakshmanan equation in birefringent fibers

Abstract In this article, we are attracted to discover the multiple-optical soiltons in birefringent fibers for Radhakrishnan–Kundu–Lakshmanan equation (RKLE) by applying the Sardar-subequation method (SSM) and the new extended hyperbolic function method (EHFM). We construct the solutions in the form of exponential, trigonometric, and hyperbolic functions solitons solutions like mixed complex solitons and multiple-optical solitons solutions. In addition, singular periodic wave solutions are constructed, and the restraint conditions for the presence of soliton solutions are also defined. Moreover, the physical interpretation of the obtained solutions is disclosed in forms of 3D and 2D plots for different suitable parameters. The attained results indicate that the implemented computational scheme is straight, proficient, and brief and can be applied in more complex phenomena with the associate of representative computations. We have obtained several sorts of new solutions.

Nonparaxial pulse propagation in a planar waveguide with Kerr–like and quintic nonlinearities; computational simulations

This article investigates some novel solitary wave solutions of the cubic-quintic nonlinear Helmholtz equation by implementing two recent analytical (Khater II and generalized exponential methods) techniques. As a result of seeing the nonparaxial effect, this model was developed to show how a pulse propagates in a planar waveguide with Kerr-like and quintic nonlinearities as well as spatial dispersion. When the traditional approximation of a slowly changing envelope fails, this effect takes over. Some novel soliton wave solutions are obtained and formulated in distinct forms such as rational, hyperbolic, and trigonometric forms. The obtained solutions are represented through various figures’ types to show the physical and dynamical behavior of the nonparaxial pulse propagation. The model’s features are demonstrated, such as its intriguing bright, anti-bright, periodic, singular, gray, chirped anti-dark, and dark solitary waves depending upon the nature of nonlinearities. The solutions’ originality is discussed by comparing it with those constructed in recently published articles. All built results are checked their accuracy by putting them back into the original model by employing Mathematica 12.

Fractional derivative-based performance analysis to Caudrey–Dodd–Gibbon–Sawada–Kotera equation

This paper explores the innovative soliton solutions to the fractional Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation with Beta and Atangana–Baleanu (AB) fractional derivatives. The new auxiliary equation method (NAEM) is used to extricate novel analytical solutions to the CDGSK equation. A comparative analysis of the solutions is conducted by implementing these fractional derivatives. The impact of different values of fractional parameter on the behavior of obtained solutions is also explored. The shapes and dynamics of the retrieved solutions are portrayed employing 3D and 2D graphs. These new solitary wave solutions are shown to encompass dark, bright, periodic, and singular wave solutions. Analytical solutions expound that this method is an effective mathematical tool to find exact traveling wave solutions to nonlinear models found in several fields of science and engineering.

Optical solutions and conservation laws of the Chen–Lee–Liu equation with Kudryashov's refractive index via two integrable techniques

The fundamental objective of this study is to find optical solutions to the perturbed Chen–Lee–Liu equation with Kudryashov's arbitrary refractive index model. Therefore, we applied the enhanced Kudryashov's and improved extended tanh-function techniques to the model. We obtained bright soliton, singular soliton, singular periodic wave, Jacobi elliptic function, exponential function, Weierstrass function, and dark soliton solutions. The obtained results are different from the other solutions available in the literature. As a result of the study, we can say that the proposed methods are effective and powerful for securing exact solutions of the nonlinear partial differential equations. Finally, using a new conservation theorem, we were able to establish the conservation laws for the current model.

Traveling Waves for the Generalized Sinh-Gordon Equation with Variable Coefficients

The sinh-Gordon equation is simply the classical wave equation with a nonlinear sinh source term. It arises in diverse scientific applications including differential geometry theory, integrable quantum field theory, fluid dynamics, kink dynamics, and statistical mechanics. It can be used to describe generic properties of string dynamics for strings and multi-strings in constant curvature space. In the present paper, we study a generalized sinh-Gordon equation with variable coefficients with the goal of obtaining analytical traveling wave solutions. Our results show that the traveling waves of the variable coefficient sinh-Gordon equation can be derived from the known solutions of the standard sinh-Gordon equation under a specific selection of a choice of the variable coefficients. These solutions include some real single and multi-solitons, periodic waves, breaking kink waves, singular waves, periodic singular waves, and compactons. These solutions might be valuable when scientists model some real-life phenomena using the sinh-Gordon equation where the balance between dispersion and nonlinearity is perturbed.

Optical wave solutions of perturbed time-fractional nonlinear Schrödinger equation

The conformable fractional derivatives modified Khater (mKhat.) technique and the Adomian decomposition (AD) method is used to examine the perturbed time-fractional nonlinear Schrödinger (NLS) problem's analytical and semi-analytical wave solutions. This model describes the dynamics of optical solitons propagating via nonlinear optical fibers. For this model, we create a variety of different formulae for analytical wave solutions, including hyperbolic, trigonometric, rational, dark, brilliant, combined dark-bright, singular, combined singular, and periodic wave solutions. Additionally, the Adomian decomposition approach is utilized to evaluate the absolute error between analytical and semi-analytical wave solutions. The Hamiltonian system is used to analyze the stability of found solutions to demonstrate their suitability for implementation in the model's application.

Computational Simulations; Abundant Optical Wave Solutions Atangana Conformable Fractional Nonlinear Schrödinger Equation

This research paper explores the Atangana conformable nonlinear fractional Schrödinger equation’s optical soliton wave solutions through three recently introduced computational schemes. The simplest expanded equation, the generalized Kudryashov method, and the sech-tanh expansion approaches are used for describing the structure of optical solitons by nonlinear optical fibers with the modern fractional operator. Several formulas such as hyperbolic, trigonometric, logical, dim, light, moon-bright hybrid, singular, combined singular, and regular wave solutions have been created. The employed methods are effective and worthy of being tested. The features of the Hamiltonian process were used to analyze the stability properties of the solutions obtained.

Travelling wave solutions for the doubly dispersive equation using improved modified extended tanh-function method

In the present paper, the improved modified extended tanh-function method is employed to get new exact traveling wave solutions for the doubly dispersive model. Several types of solutions are obtained using the proposed method. Theses solutions are bright, dark and bright-dark combo soliton solutions. Also, hyperbolic type solutions, singular periodic wave solutions, Jacobi elliptic function solutions, exponential solutions and Weierstrass elliptic doubly periodic solutions are obtained. Some of these solutions are reported in this paper for first time, namely, dark solitons, Jacobi elliptic function solutions and Weierstrass elliptic doubly periodic solutions. Furthermore, 3D graphs of some solutions are introduced for illustrating the physical interpretation.

A variety of exact optical soliton solutions to the generalized (2+1)-dimensional dynamical conformable fractional Schrödinger model

In this work, our main idea is to retrieve a variety of exact optical solitons of the generalized (2+1)-dimensional nonlinear conformable fractional Schrödinger system by employing three novel integration norms. The constructed optical solitons can be divided as dark, singular, combo dark-singular, periodic, mixed periodic, and plane wave solutions. In addition, by selecting different parametric values, the physical representation of some achieved solutions is plotted in 3-dimensional, 2-dimensional, and contour graphs. Computation packages are used to verify all of the secured results by putting them back into the original model. The findings demonstrate that the examined system theoretically contains a large number of optical solitons. The constructed results rendering that the consider methods in this article are effective, concise, straightforward, efficient, and they can be used to handle some more nonlinear partial differential equations in the applied sciences. The achieved outcomes in this article may be valuable in clarifying the true meaning of numerous nonlinear advancement circumstances that develop in various domains of nonlinear sciences. Also, these fresh solutions have many applications in physics and other branches of physical sciences.

Diversity of exact solutions and solitary waves with the influence of damping effect in ferrites materials

The main concentration of this article is to investigate a series of abundant new soliton solutions to the ferrites materials. The Kraenkel-Manna-Merle (KMM) system is used as a governing model that expresses the nonlinear ultra-short wave pulse motions in ferrite’s materials having an external field with zero-conductivity. We extract the solutions in different forms like, Jacobi’s elliptic, hyperbolic, periodic, rational function solutions including a class of solitary wave solutions such that dark, singular, complex combo solitons, and mixed complex soliton solutions. Recently developed integration tool known as Φ6-model expansion method is applied to analyze the governing model. Moreover, the constraints conditions are explicitly presented for the resulting solutions and singular periodic wave solutions are recovered. Furthermore, for explaining the solutions in physical phenomena, the three dimensional and two dimensional graphs are plotted under the selection of appropriate parameters. The accomplished results show that the applied computational system is direct, productive, reliable and can be carried out in more complicated phenomena. The findings indicate that the system has very rich soliton structures theoretically.

Exact analytical wave solutions for space-time variable-order fractional modified equal width equation

The variable-order evolution equation is an impressive mathematical model that explain complex dynamical problems efficiently and accurately. This latest research investigates the modified equal width nonlinear space–time variable-order fractional differential equation and fractional derivative operator in the sense of Caputo. Using transformation, an ordinary differential equation is obtained from the variable-order differential equation. For accuracy, the space–time variable-order fractional modified equal width equation is solved by the modified (G'/G) -expansion method. This model describes the simulation for one-dimensional wave transmission in nonlinear media with dispersion processes. As a result, new traveling wave solutions are developed for various values of parameters. The obtained solutions have several applications in a recent area of research in mathematical physics. The obtained graphical solutions are in the form of singular solitons, kink solitons and periodic solitons waves which demonstrate that the physical signifance and dynamical behaviors of fractional variable-order differential equations and the proposed method are more effective, powerful, and easy in solving problems arising in mathematical physics.

BIFURCATIONS AND EXACT SOLUTIONS OF THE OPTICAL SOLITON MODEL IN METAMATERIALS DOMINATED BY ANTI-CUBIC NONLINEARITY

Optical soliton model in metamaterials, dominanted by anti-cubic nonlinearity, is investigated by the method of dynamical systems. By using travelling wave transformation, the model can be converted into a singular integrable travelling wave system. Then we discuss the dynamical behavior of the associated regular system. Further, all bounded exact solutions of the model can be calculated because of its integrability. Finally, twenty exact explicit parametric representations are derived.

DEGENERATION OF LUMP-TYPE LOCALIZED WAVES IN THE (2+1)-DIMENSIONAL ITO EQUATION

<abstract> The degeneration of lump-type localized waves in the (2+1)-dimen-sional Ito equation is investigated through the parallel relationship of wave numbers. These lump-type localized waves can degenerate into three different kinds of localized wave solutions: singular lump-type localized wave, periodic variable amplitude localized wave, rogue wave. In the process of propagation, the lump-type localized waves keep the same waveform structure and amplitude. However, the periodic variable amplitude localized wave demonstrates three different kinds of waveform structures, which presents an interesting emit-absorb interaction phenomenon. By an emitting and absorbing interaction, the localized wave realizes the energy exchange from one localized wave to another, and keeps the original waveform structure. Rogue wave is a rational growing-and-decaying localized wave which is localized in both space and time. </abstract>

Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations

In this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fractional order partial differential equation into an integer order ordinary differential equation. Using the novel (G′/G2)-expansion method, several exact solutions with extra free parameters are found in the form of hyperbolic, trigonometric, and rational function solutions. When parameters are given appropriate values along with distinct values of fractional order α travelling wave solutions such as singular periodic waves, singular kink wave soliton solutions are formed which are forms of soliton solutions. Also, the solutions obtained by the proposed method depend on the value of the arbitrary parameters H. Previous results are re-derived when parameters are given special values. Furthermore, for numerical presentations in the form of 3D and 2D graphics, the commercial software Mathematica 10 is incorporated. The method is accurately depicted, and it provides extra general exact solutions.

Cubic–quartic optical soliton perturbation and modulation instability analysis in polarization-controlled fibers for Fokas–Lenells equation

The objective of this study is to investigate miscellaneous wave structures for perturbed Fokas–Lenells equation (FLE) with cubic-quartic dispersion in polarization-preserving fibers. Based on the improved projective Riccati equations method, various types of soliton solutions such as bright soliton, combo dark–bright soliton, singular soliton and combo singular soliton are constructed. Additionally, a set of periodic singular waves are also retrieved. The dynamical behaviors of some obtained solutions are depicted to provide a key to understanding the physics of the model. The modulation instability of the FLE is reported by employing the linear stability analysis which shows that all solutions are stable.