Solitary Wave Ansatz Research Articles

New perceptions for the bright and dark soliton solutions to the modified nonlinear Schrödinger equation

In this study, we will implement new bright and dark perceptions for the solitary wave solutions to the modified nonlinear Schrödinger equation. The achieved solutions will describe new vision of the following forms. The first form is the rogue wave modes for a derivative nonlinear Schrodinger model with positive linear dispersion which describe the propagation of rogue waves in ocean engineering as well as all similar waves such as dynamics waveguides that have unexpected large displacements. The second form is the waves which occur only in the regime of positive cubic nonlinearity. The third form is the waves that also occur in the regime that coincides exactly with the existence of instabilities of plane waves. The fourth form is the long-wave limit of a breather (a pulsing mode). Two famous different schemas are involved for this purpose. The first schema is the solitary wave ansatz method, while the second schema is the extended simple equation method. The two schemas are implemented in the same vein and parallel to construct new perceptions to the soliton solutions of this model. A comparison between the obtained new perceptions with the old perceptions that were realized before they have been documented.

New impressive analytical optical soliton solutions to the Schrödinger–Poisson dynamical system against its numerical solutions

In our current study, we will derive new diverse enormous impressive analytical optical soliton solutions for the Schrödinger–Poisson dynamical system. The proposed model is applied in gravity field with the corresponding quantum state that produces coupling between different gravity states. Moreover, this model has a significant role in the field of many quantum phenomena. Hereby, we will construct diverse forms of the soliton behaviors that arising from this dynamical system via the solitary wave ansatze method. This technique is one of the ansatze methods that doesn’t surrenders to the homogeneous balance and continuously achieves good results. Moreover, we will construct the numerical solutions that are identical for all achieved exact solutions by using two-dimensional differential transform method (TDDTM). The extracted soliton solutions are new compared with that realized before by other authors who used various techniques. The achieved solutions will give new distinct configurations to soliton behaviors arising from this model and show the fact of charges regular distributions on conductors’ materials surface.

New diverse types of soliton solutions to the Radhakrishnan-Kundu-Lakshmanan equation

<abstract> <p>The main purpose of this study was to produce abundant new types of soliton solutions for the Radhakrishnan-Kundu-Lakshmanan equation that represents unstable optical solitons that emerge from optical propagations through the use of birefringent fibers. These new types of soliton solutions have behaviors that are bright, dark, W-shaped, M-shaped, periodic trigonometric, and hyperbolic and were not realized before by any other method. These new forms have been detected by using four different techniques, which are, the extended simple equation method, the Paul-Painlevé approach method, the Ricatti-Bernoulli-sub ODE, and the solitary wave ansatz method. These new solitons will be arranged to create a soliton catalog with new impressive behaviors and they will contribute to future studies not only for this model but also for the optical propagations through birefringent fiber.</p> </abstract>

Multiple soliton solutions and other solutions of a variable-coefficient Korteweg–de Vries equation

In this paper, we construct multiple soliton solutions of a time-dependent variable-coefficient Korteweg–de Vries equation in the form of bright, dark and singular soliton solutions. Furthermore, the periodic solutions are derived. These will be achieved via the aid of solitary wave ansatz methods. In addition, other exact solutions of a time-dependent variable-coefficient Korteweg–de Vries equation are attained using symmetry reductions. In one special case, the power series method is employed to derive a solution of the aforesaid equation. We also examine the conservation laws of the aforementioned equation using the variational approach. A brief physical representation of the derived conserved vectors will be conversed.

Unexpected configurations for the optical solitons propagation in lossy fiber system with dispersion terms effect

In this work, we will design unexpected configurations for the optical soliton propagation in lossy fiber system in the presence of the dispersion term solitons via two distinct and impressive techniques. The first one is the (G′/G)‐expansion method, while the second is solitary wave ansatze method. The two methods are implemented in the same vein and parallel. The obtained perceptions are new and were not achieved before. The comparison between our achieved visions and that achieved by other authors who used different schemas has been documented.

New unexpected soliton solutions to the generalized (2 + 1) Schrödinger equation with its four-mixing waves

The propagation of solitons in birefringent fiber is one of the phenomena that has an important role in all modern technological means of communication. The generalized [Formula: see text] nonlinear Schrödinger equation (GNLSE) with its four-mixing waves (FMW) is the famous model that describes the propagation of solitons in birefringent fiber perfectly. In fact, the FMW governed effectively the performance of the resultant solitons amplitude. Hereby, we will study this model to extract new unexpected optical soliton solutions to this model via various three techniques. The three famous methods that are a candidate for this purpose are the extended direct algebraic method (EDAM), the extended simple equation method (ESEM) and the solitary wave ansatz method (SWAM). The three techniques are implemented successively for the suggested model to establish the optical solutions of the suggested model successfully. The optical soliton solutions that are achieved by these proposed techniques give surprise expectations that weren’t achieved previously by any other authors who used other techniques.

The higher-order modified Korteweg-de Vries equation: Its soliton, breather and approximate solutions

In this study, the fifth-order modified Korteweg-de Vries (F-MKdV) equation is first addressed using Hirota’s bilinear method. Thereafter, the exact and approximative solutions of the generalized form of the F-MKdV equation are investigated using the modified Kudryashov method, the Riccati equation and its Backlund transformation method, the solitary wave ansatz method, and the homotopy perturbation transform method (HPTM). As a result, solitons, breather, and solitary wave solutions are derived from these methods. In particular, we obtain some new solutions such as the dark soliton, bright soliton, singular soliton, periodic trigonometric, exponential, hyperbolic, and rational solutions. The constraint conditions associated with the resulting solutions are also discussed in detail. The HPTM is employed to construct approximate solutions to the aforementioned generalized model due to its strong nonlinear terms and only a few terms are required to obtain accurate solutions. These findings may help to understand complex nonlinear phenomena.

New optical solitons and modulation instability analysis of generalized coupled nonlinear Schrödinger–KdV system

In this study, the generalized coupled nonlinear Schrödinger–KdV (NLS–KdV) system is investigated to obtain new optical soliton solutions. This system appears as a model for reciprocity between long and short waves in various physical settings. Different kinds of new soliton solutions including dark, bright, combined dark-bright, singular and combined singular soliton solutions are obtained using two effective methods namely, the extended sinh–Gordon equation expansion method and the solitary wave ansatz method. In addition, the modulation instability analysis of the system is presented based on the standard linear-stability analysis. The behaviours of obtained solutions are expressed by 3D graphs.

New impressive ideal optical soliton solutions to the space and time invariant nonlinear Schrödinger equation

• We will achieve new types of ideal optical soliton solution to the space and time invariant nonlinear Schrödinger equation. • The proposed model represents the two layer baroclinic instability and lossless symmetric two streams plasma instability. • Many types of special solutions in different forms as well as kink, anti-kink soliton solutions by using two powerful methods have been extracted. • The obtained results can be considered as benchmark to the results obtained by any numerical methods. In this paper, we will extract new types of ideal optical soliton solution to the space and time invariant nonlinear Schrödinger equation. The proposed model represents the two layer baroclinic instability and lossless symmetric two streams plasma instability. We will extract this Package of these ideal optical soliton solutions for this model by using two distinct methods one of them surrenders to the balance rule and has profile name the solitary wave ansatz method while the other doesn't surrender to the balance rule and has profile name the modified extended mapping method. The two suggested methods have been implemented in the same vein and parallel. Some of our achieved results agree with that realized previously by other authors in some cases and the remains are new.

Accurate impressive optical solitons to the nonlinear refractive index cubic–quartic through birefringent fibers

The main target of this work is implementing new accurate impressive optical solitons for four forms of the nonlinear refractive index cubic–quartic through birefringent fibers which play a vital role in all modern telecommunications process. These four different forms have profile names which are the cubic–quartic in polarization-preserving fibers with its different forms which are the Kerr-law, the quadratic-law, the parabolic-law and the non-local-law NLSE. These new accurate optical solitons for these different forms are extracted in the framework of the solitary wave ansatz method which is powerful technique that achieves accurate results for all problems that are solved in the framework of it. The achieved results will be compared by the previous results which are achieved via other authors.

Enormous soliton solutions to a (2+1)-dimensional Heisenberg ferromagnetic spin chain equation

In our current study, we will extract enormous, impressive soliton solutions to a (2+1)-dimensional Heisenberg ferromagnetic spin chains with bilinear and anisotropic interactions in the semi classical limit. The suggested model has its wide importance for many magnetism phenomena. Specially, it is very momentous in modern magnetism theory in which the nonlinear long-range ferromagnetic ordering magnetism can be explained. Moreover, it describes the characteristic of magnetism to many insulating crystals as well as interaction spins. The ferromagnetism plays fundamental role in modern technology and industry and it is principal for many electrical and electromechanical devices such as electromagnets, electric motors, generators etc. We will document various types of the soliton solutions by using four various techniques to extract the soliton configuration behaviors arising from the magnet dynamics model. The techniques that are used to derive these types of soliton solutions are the extended direct algebraic method, the extended simple equation method, the solitary wave ansatz method and the Paul-Painleve approach method. The expected achieved soliton solutions will offer a rich podium to study the nonlinear spin dynamics in magnetic materials. The achieved soliton solutions are new compared with that achieved previously by other authors who used different techniques.

On some novel solution solutions to the generalized Schrödinger-Boussinesq equations for the interaction between complex short wave and real long wave envelope

This paper explores some novel solutions to the generalized Schrödinger-Boussinesq (gSBq) equations, which describe the interaction between complex short wave and real long wave envelope. In order to derive some novel complex hyperbolic and complex trigonometric function solutions, the sine-Gordon equation method (sGEM) is applied to the gSBq equations. Novel complex hyperbolic and trigonometric function solutions are expressed by the dark, bright, combo dark-bright, W-shaped, M-shaped, singular, combo singular, and periodic wave solutions. The accuracy of the explored solitons is examined under the back substitution to the corresponding equations via the symbolic computation software Maple. It is found from the back substitution outcomes that all soliton solutions satisfy the original equations. The proper significance of the explored outcomes is demonstrated by the three-dimensional (3D) and two-dimensional (2D) graphs, which are presented under the selection of particular values of the free parameters. All the combo-soliton (W-shaped, M-shaped, and periodic wave) solutions are found to be new for the interaction between complex short wave and real long wave envelope in laser physics that show the novelty of the study. Moreover, the applied method provides an efficient tool for exploring novel soliton solutions, and it overcomes the complexities of the solitary wave ansatz method.

New solitary wave solutions of the Sawada-Kotera equation and its bidirectional form

In this paper, the solitary wave ansatz method along with Hirota’s bilinear form are used to study the well-known Sawada-Kotera equation and its bidirectional form. Numerical simulations related to some of these solutions are given in the article. These figures provide us with a better understanding of visualizing the dynamic behavior of solutions. For the latter equation, we have found an indefinite quadratic bilinear Hirota solution with three homogenous terms and a constant. The techniques used in this paper can be adopted to solve other nonlinear models.

Exact solutions of the space time-fractional Klein-Gordon equation with cubic nonlinearities using some methods

Recently, finding exact solutions of nonlinear fractional differential equations has attracted great interest. In this work, the space time-fractional Klein-Gordon equation with cubic nonlinearities is examined. Firstly, suitable exact soliton solutions are formally extacted by using the solitary wave ansatz method. Some solutions are also illustrated by the computer simulations. Besides, the modified Kudryashov method is used to construct exact solutions of this equation.

New vision for the soliton solutions to the complex Hirota-dynamical model

In this paper, the nonlinear complex Hirota-dynamical model NLCHM in which the third derivative term represents the self-interaction in the high-frequency subsystem is established. This model plays a vital role in plasma physics because there are agreements between the self-interaction in the high-frequency and the well- known self-focusing effect in plasma. Many soliton solutions to this equation model have been achieved perfectly using the solitary wave ansatz method (SWAM). Furthermore, in the same vein and related subject the extended simple equation method (ESEM) has been applied perfectly to achieve new perception of soliton solutions to this model. A good isomorphic between the achieved results and that achieved previous by other authors have been listed.

Optical solitons for weakly nonlocal Schrödinger equation with parabolic law nonlinearity and external potential

This study discusses the soliton and other solutions for a weakly nonlocal Schrödinger equation of the fifth order involving nonlinearity of the parabolic law and an external potential. We use solitary wave ansatz of different forms to obtain bright, dark, singular, periodic soliton and exponential solutions. The results obtained can be useful in understanding and interpreting the wave dynamics of the proposed model.

Bright and dark soliton solutions for the complex Kundu-Eckhaus equation

The main target of this article is to realize bright and dark soliton solutions to the complex Kundu-Eckhaus equation which represents a general form of integrable system that is gauge-equivalent to the mixed nonlinear Schrödinger equation. This soliton solution which represents the propagation of the ultra-short femtosecond pulses and the rogue wave in an optical fiber achieved for the first time in the framework of the solitary wave ansatz method. Furthermore, in this connection at the same time and parallel the extended simple equation method has been successfully applied to achieve new impressive solitary wave solutions to this model. A comparison between the obtained results and that satisfied in previous work is established.

New analytical wave structures for the (3 + 1)-dimensional Kadomtsev-Petviashvili and the generalized Boussinesq models and their applications

Different types of soliton wave solutions for the (3 + 1)-dimensional Kadomtsev-Petviashvili and the generalized Boussinesq equations are investigated via the solitary wave ansatz method. These solutions are classified into three categories, namely solitary wave, shock wave, and singular wave solutions. The corresponding integrability criteria, termed as constraint conditions, obviously arise from the study. Moreover, the influences of the free parameters and interaction properties in these solutions are discussed graphically for physical interests and possible applications.

Investigation of dynamics of nematicons in liquid crystals by extended sinh-Gordon equation expansion method

This paper secures new nematicons in liquid crystals from its governing equation by extended sinh-Gordon equation expansion method with the aid of symbolic computational package Maple. The two laws of nonlinearity, namely Kerr and parabolic laws are studied. As outcomes, some new explicit complex hyperbolic and complex trigonometric function solutions are derived regarding dark, bright, combined dark–bright, singular, combined singular optical, periodic wave, dipole soliton solutions and others. To show the real physical meaning of the derived results, three dimensional graphs with contours and two dimensional graphs are prepared under the choice of proper values of the arbitrary parameters. The derived solutions have been checked back into their corresponding equations with the symbolic computational package Maple. All the combined solutions are found to be new, which may play an important role in the context of liquid crystals. The applied method is found to provide a powerful tool for extracting exact solitary wave solutions and overcome the difficulties of the solitary wave ansatz method.

Soliton solutions and modulation instability analysis of the coupled Zakharov-Kuznetsov equation

The bright, dark and kink soliton solutions of the coupled Zakharov-Kuznetsov (ZK) equation are obtained by using the solitary wave ansatz method, variational approximation (VA), variational iteration method (VIM) and Adomian's decomposition method (ADM). The bright and dark soliton solutions are multiple soliton solutions at time t = 0 which reduce into stationary soliton through single soliton for sufficiently large time t . The approximate solutions by the VA, VIM and ADM are compared with the exact solution obtained by the ansatz method. The VIM gives better approximate solution of the coupled ZK equation than the VA. The absolute error and convergence analysis of the approximate solutions by the VIM and ADM show that the approximate solutions converge to the exact solution. The modulation instability is used to discuss the stability of the steady state solution of the coupled ZK equation and it demonstrates that the nonlinear term in the equation decides the modulational stability of the soliton solution.