Analytical Soliton Solutions Research Articles

Novel soliton solutions and localized structures of KdV equation

A (1 + 1)-dimensional Korteweg-de Vries (KdV) equation is studied. Using the extended Tanh-function method, the analytical soliton solutions are obtained by symbolic calculation, and the localized coherent structures containing the compacton, the peakon, periodic multi-solitons and fractal soliton structures are constructed. Through the analysis of various images, the properties of soliton solutions excited by different local structures are discussed and the fractal structure of soliton solutions are further discussed.

Discovering optical soliton solutions in the Biswas–Milovic equation through five innovative approaches

Purpose:In recent years, depending on the development of models containing highly dispersive nonlinear equations, especially in the field of nonlinear optics, it has been important to obtain the solutions of these equations with analytical methods. This study presents some effective, reliable, and easily applicable solution techniques based on the well-known extended auxiliary equation. Besides, these techniques were applied to the Biswas–Milovic equation. Methodology:First, in order to achieve the ordinary differential equation structure of the Biswas–Milovic equation, the complex wave transform was given and utilized. Then, the algorithms, constraints of each proposed approach, and key points for their implementation were presented. In addition, the relations and transitivities between the methods are also interpreted. Second, analytical optical soliton solutions were obtained by applying the proposed techniques. Lastly, graphical simulations of optical soliton types represented by some solution functions are made. Findings:It has been shown that all of the proposed methods yield effective results in solving the Biswas–Milovic equation and can be used not only for nonlinear evolution equations but also for highly dispersive nonlinear optical equations. It has been observed that the methods have the capacity to be employed easily and do not require tedious calculus and algebraic manipulations. Originality:The work in the article has not been presented before.

Kink soliton dynamics in one-dimensional Bose–Einstein condensate with higher-order nonlinear interactions

We investigated the soliton behavior in one-dimensional Bose–Einstein condensates. Based on the modified Gross–Pitaevskii equation (GPE) model with higher-order nonlinear interaction and through the F-expansion method, we derived the analytical kink soliton solution of the one-dimensional GPE. The typical kink soliton features under the specific experimental setting are identified, and the physical implication of the analytical results is also demonstrated. The derived theoretical results can be used to guide the experimental study of kink soliton in ultracold atomic systems with higher-order nonlinearity.

Analytical soliton solutions of the fractional order dual-mode nonlinear Schrödinger equation with time-space conformable sense by some procedures

Analytical soliton solutions of the fractional order dual-mode nonlinear Schrödinger equation with time-space conformable sense by some procedures

Unraveling the interplay of gravity and surface tension in driving waves on water’s surface

The aim of this paper is to incorporate modern computational and numerical methods to give analytical and approximate soliton solutions for the generalized fractional Hietarinta–type (GFH) model in (2+1) dimensions. Solitary wave solutions for the suggested model, which mathematically shows how gravity and surface tension make waves move again on the water’s surface, are identified and illustrated. The phase velocity of waves with different wavelengths is described by the difference in frequency in fluid dynamics. A new modern numerical technique is used to show how accurate the computational solution is, along with analytical solutions. Hamiltonian system features are also utilized to regulate the solutions’ stability. All findings are double-checked by reinserting them into Mathematica 13.1’s core model.

Exploring the wave solutions of a nonlinear non-local fractional model for ocean waves

In this research, analytical and semi-analytical soliton solutions for the nonlinear fractional (2 + 1)-dimensional integrable Calogero–Bogoyavlenskii–Schiff equation (FCBSE) in the non-local form are obtained using recent computational and numerical methods. The FCBSE is a significant model for investigating various phenomena, such as internal ocean waves, tsunamis, river tidal waves, and magneto-sound waves in plasma. The constructed solution helps in understanding the interaction between a long wave moving along the x-axis and a Riemann wave propagating along the y-axis. Various analytical solutions, such as exponential, trigonometric, and hyperbolic, have been formulated differently for this model, which is a specific derivation of the well-known Korteweg–de Vries equation. Density charts in two and three dimensions are used to visualize the behavior of a single soliton in reality through simulations. The results demonstrate the effectiveness of the employed numerical scheme and various methods to ensure the consistency of computational and approximation answers. Overall, this study demonstrates the potential of recent computational and numerical techniques for solving nonlinear mathematical and physical problems.

Dynamical behavior of analytical soliton solutions, bifurcation analysis, and quasi-periodic solution to the (2+1)-dimensional Konopelchenko–Dubrovsky (KD) system

Nonlinear evolution equations (NLEEs) are extensively used to establish the elementary propositions of natural circumstances. In this work, we study the Konopelchenko–Dubrovsky (KD) equation which depicts non-linear waves in mathematical physics with weak dispersion. The considered model is investigated using the combination of generalized exponential rational function (GERF) method and dynamical system method. The GERF method is utilized to generate closed-form invariant solutions to the (2+1)-dimensional KD model in terms of trigonometric, hyperbolic, and exponential forms with the assistance of symbolic computations. Moreover, 3D, 2D combined line graph and their contour graphics are displayed to depict the behavior of obtained solitary wave solutions. The model is observed to have multiple soliton profiles, kink-wave profiles, and periodic oscillating nonlinear waves. These generated solutions have never been published in the literature. All the newly generated soliton solutions are checked by putting them back into the associated system with the soft computation via Wolfram Mathematica. Moreover, the system is converted into a planer dynamical system using a certain transformation and the analysis of bifurcation is examined. Furthermore, the quasi-periodic solution is investigated numerically for the perturbed system by inserting definite periodic forces into the considered model. With regard to the parameter of the perturbed model, two-dimensional and three-dimensional phase portraits are plotted.

Obtaining soliton solutions of the nonlinear (4+1)-dimensional Boiti–Leon–Manna–Pempinelli equation via two analytical techniques

This paper tackles the recently introduced (4+1)-dimensional Boiti–Leon–Manna–Pempinelli equation (4D-BLMPE) utilized to model wave phenomena in incompressible fluid and fluid mechanics. Modified extended tanh expansion method (METEM) and the new Kudryashov scheme are implemented to produce analytical soliton solutions for the presented equation. The traveling wave transformation is constructed, and the homogeneous balance principle is utilized to apply the two proposed techniques. Furthermore, the flat-kink, smooth-kink, singular, and periodic singular solutions are successfully extracted. Some produced solutions are illustrated graphically to understand the physical meaning of the presented model. Moreover, for the first time in this study, the effect of model parameters on kink soliton dynamics is examined, and graphical representations are depicted and interpreted.

Non-collisional dynamics of nonautonomous three solitons through tailoring of modulated coefficients and modulation instability gain spectra

In this paper, the generalized nonautonomous nonlinear Schrödinger equation with non-isospectral parameter is investigated from the point of view of soliton management through modulation of dispersion and nonlinearity. Based on the obtained analytic soliton solutions, the effect of modulated coefficients such as dispersion and nonlinearity on the control of soliton transmission is studied. Especially, controllable soliton interactions are investigated through suitably tailoring the inhomogeneous profiles. Additionally, modulation instability (MI) has been analyzed graphically through linear stability analysis (LSA) and MI gain spectrum is analyzed for various forms of modulated coefficients. Through this study, we confirm that the modulated coefficients in the theoretical model under investigation affect the amplitudes, widths and trajectories of the nonautonomous solitons. Thus, one can obtain the optical control system by choosing the different form of modulated coefficients for the specific problem. Results obtained in this paper would be worthy in the field of soliton management for improving the capacity of optical fiber communication system.

New optical soliton solutions to the space-time fractional perturbed Chen-Lee-Liu equation

The key traits of optical fiber and plasma physics are interpreted by the typical space-time fractional nonlinear perturbed Chen-Lee-Liu equation. The equation is regarded in the sense of beta derivative, and a composite fractional wave transformation is used to reshape it into a nonlinear equation of a single variable. This article aims to compute diverse analytical soliton solutions to the Chen-Lee-Liu equation through the potential (G′/G,1/G)-expansion approach. The solutions include periodic soliton, bell-shaped soliton, anti-peakon, V-shaped soliton, compacton, and others that might be supportive to analyze signal transmission in optical fibers and the properties of plasmas. This research formulates some illustrative solitons to the referred equation that will aid in the study of signal transmission and abridge the complexity of the stated sectors. The three- and two-dimensional graphs are portrayed for different values of the fractional-order derivative to illustrate the impact of the fractional derivative. This study shows the effectiveness and reliability of the adopted strategy.

Some new soliton solutions to the (3 + 1)-dimensional generalized KdV-ZK equation via enhanced modified extended tanh-expansion approach

In this article, The (3 + 1)-dimensional generalized Korteweg-de-Vries-Zakharov-Kuznetsov equation (gKdV-ZKe) which explains the influence of the magnetic field on the weak nonlinear ion-acoustic waves investigated in the field of plasma conjured up including both cold and hot electrons. GKdV-zk techniques solutions are obtained using the improved modified extended tanh expansion method, which is one of the most efficient algebraic methods for obtaining accurate solution to nonlinear partial differential equations. We aim to show how the analyzed model’s parameter impact soliton behavior by choosing different bright and single soliton forms and by developing various analytical optical soliton solutions for the explored equation. MethodologyIn order to apply the suggested method, we used a complex wave transform to derive the nonlinear ordinary differential form of the analyzed equation. Then, using the method, we were able to obtain the polynomial form, leading to a set of linear equations. The conclusion of solving the linear equations problem, the outcomes of the analyzed model, and the suggested strategy are all included in different solution sets. After choosing the appropriate set from these sets, using the solution functions, and utilizing the wave transformation provided by the approach, we were able to arrive at the optical soliton solutions by providing the central equation. FindingThe proposed method has successfully produced a number of soliton solutions and several analytical optical solutions to such model. The research shows that the parameters of the model may have a variety of effects on the behavior of solitons, categories based on the soliton type. The findings we get in this article can be used to research and compare numerical and experimental data with analytical solving problems in plasma physics. OriginalityThis study differs from others in that it assessed the impact that parameters of the model have on the actions of solitons, despite the fact that the proposed technique was applied for the first time on the topic under investigation and numerous soliton types were created. This study focuses on the influence of model parameters on solitons behavior.

Approximate-analytic optical soliton solutions of a modified-Gerdjikov–Ivanov equation: modulation instability

The Gerdjikov–Ivanov equation (GIE) occupied a remarkable area of research in the literature. In the present work, a modified GIE (MGIE) is considered which is new and was not studied in the literature. Also, the modified-unified method (MUM) is used to obtain approximate analytic solutions (AASs) of MGIE. Up to our knowledge, no AASs for non-integrable complex field equation were found up to now. Thus the AASs found, here, are novel. The UM addresses finding the exact solutions to integrable equations. In this sense as no exact solution for MGIE exists, consequently, it is not integrable. So, here, approximate analytic optical soliton solutions are invoked. The UM stands for expressing the solution of nonlinear evolution equations in polynomial and rational forms in an auxiliary function (AF) with an appropriate auxiliary equation. For finding exact solutions by the UM, the coefficients of the AF, with all powers, are set equal to zero, For a non-integrable equation, only approximate solutions are affordable. In this case, we are led to utilizing the MUM. Herein, non-zero coefficients (residue terms (RTs)) are considered as errors, which are space and time-independent. It is worth mentioning that, this is in contrast to the errors found by the different numerical methods, where they are space and time-dependent. Further, in the present case, the maximum error is controlled via an adequate choice of the parameters in the RTs. These solutions are displayed in graphs. Breather soliton, chirped soliton and M-shape soliton, among others, are observed. Furthermore, modulation instability (MI) is studied and it is found MI triggers when the coefficient of the nonlinear dispersion exceeds a critical value.

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 variety analytical optical soliton solutions for two various models that are emerged from the perturbed nonlinear Schrödinger equation

New diverse variety analytical optical soliton solutions for two various models that are emerged from the perturbed nonlinear Schrödinger equation

Propagation characteristics of bright and mixed solitons based on the variable coefficient (3+1)-dimensional cubic-quintic complex Ginzburg-Landau equation

In the study of telecommunication system, the variable coefficient (3+1)-dimensional cubic-quintic complex Ginzburg-Landau equation is used as the optical solitons transmission model, which not only explains the physical meaning of the existing model with quintic terms, but also has more nonlinear dynamics characteristics of the higher dimensional system than the lower dimensional system. In this paper, the analytical soliton solutions of the (3+1)-dimensional cubic-quintic CGL equations with variable coefficients are obtained by using the modified Hirota method. By selecting certain parameters of the nonlinear coefficients and spectral filtering terms, a special kind of mixed soliton solution is obtained, which has the characteristics of bright soliton, dark soliton and kinked soliton at the same time. Subsequently, the influence of changing the nonlinear, spectral filtering, linear loss parameters and other parameters on the transmission characteristics of solitons is discussed respectively, so as to realize the control of optical solitons, which can not only control the propagation of optical solitons in different forms, but also can realize the adjustment of the amplitude and pulse width of the pulse and control the propagation direction and energy of the pulse for the mixed solitons of a particular form. The research results of high dimensional CGL system in this paper can be applied to nonlinear optical system, ultra-fast optical digital logic system and other different experiments and application fields.

Pure-quartic solitons in presence of weak nonlocality

In this paper, we study the characteristics of pure-quartic optical solitons in a nonlinear medium having a weakly nonlocal response. We propose an nonlinear Schrödinger equation incorporating pure fourth-order diffraction, Kerr nonlinearity and weak nonlocality to describe the transmission of solitons in the system. We obtain analytic pure-quartic soliton solutions to this model in a variety of shapes and forms, unlike the case of the model without taking into account the weak nonlocal contribution. Besides, we find that the obtained localized structures have both bright and dark type waveforms in the transmission system. Interestingly, the functional forms of these solitons are expressed in terms of sech and tanh functions which do not possess oscillating tails, markedly different from the conventional pure-quartic solitons in the absence of weak nonlocality. These structures exist for both positive and negative quartic diffraction coefficient whereas pure-quartic solitons in the absence of nonlocality occur only for negative fourth-order dispersion. The numerical examples are presented for illustrating the soliton evolution dynamics in the optical waveguide.

Dynamics of quantum solitons in Lee-Huang-Yang spin-orbit-coupled Bose-Einstein condensates

We present the numerical results of the structure and dynamics of the self-bound ground state arising solely because of the presence of beyond mean field quantum fluctuation in spin-orbit coupled binary Bose-Einstein condensates in one dimension. Depending upon spin-obit (SO) and Rabi couplings, we observe that the ground state exhibits either quantum-bright (plane) or quantum-stripe soliton nature. We find an analytical soliton solution for non-zero SO coupling that matches quite well with the numerical results. Further, we investigate the dynamical stability of these solitons by adopting three protocols, such as (i) adding initial velocity to each component, (ii) quenching the SO and Rabi coupling parameters at initial and finite time, and (iii) allowing collision between the two spin-components by giving equal and opposite direction velocity to them. Many interesting dynamical features of the solitons, like, multi-fragmented, repelling, and breathing in time and space-time, are observed. For given Rabi coupling frequency, the breathing frequency of the soliton increases upon the increase in SO coupling, attaining a maximum at the critical SO coupling where the phase transition from the bright to stripe soliton occurs. We observe that the maximum breathing frequency exhibits power law dependence on the Rabi coupling frequency with an exponent $\sim 0.16$.

Exact analytical soliton solutions of N-component coupled nonlinear Schrödinger equationswith arbitrary nonlinear parameters.

Exact analytical soliton solutions play an important role in soliton fields. Soliton solutions were obtained with some special constraints on the nonlinear parameters in nonlinear coupled systems, but they usually do not hold in real physical systems. We successfully release all usual constrain conditions on nonlinear parameters for exact analytical vector soliton solutions in N-component coupled nonlinear Schrödinger equations. The exact soliton solutions and their existence condition are given explicitly. Applications of these results are discussed in several present experimental parameter regimes. The results would motivate experiments to observe more novel vector solitons in nonlinear optical fibers, Bose-Einstein condensates, and other nonlinear coupled systems.

Study of optical solitons for Kudryashov’s Quintuple power-law with dual form of nonlinearity using two modified techniques

The main purpose of this article is to explore the analytic soliton solutions of a nonlinear Schrodinger equation (NLSE), including Kudryashova’s quintuple power-law with dual form nonlinearity by presenting two efficient analytical techniques. The present equation is a significant model which deals with a variety of nonlinear phenomenons in the fields of optical fibers. Modified G′G2 -expansion technique and exp(−φ(ρ)) technique explore kink-singular, bright, dark and cupson-singular form of soliton solutions. The present techniques are simple and efficient because these techniques produced a rational, trigonometric solution of the nonlinear partial differential equation. The above two techniques are novel and have not been applied yet on present model.

On the investigation of optical soliton solutions of cubic–quartic Fokas–Lenells and Schrödinger–Hirota equations

Purpose:When it comes to third and higher-order dispersion, the Schrödinger–Hirota equation is one of the main models developed outside the classical NLSE management models for optical soliton transmission. The cubic–quartic Fokas–Lenells equation is also one of the recently developed equations, which has importance in the field of telecommunications regarding optical soliton transmission in the absence of chromatic dispersion. In this study, in order to examine the optical solitons, the Schrödinger–Hirota equation in the presence of the chromatic dispersion and the cubic–quartic Fokas–Lenells equation discarding the chromatic dispersion were investigated. For this intent, by obtaining certain soliton types using the unified Riccati equation expansion method (UREEM), optical soliton solutions were obtained for both models and graphical representations and comments were made. Methodology:By developing appropriate computer algorithms and applying UREEM in the following ways, symbolic calculation software was made and analytical optical soliton solutions were obtained. Findings:Through computer algebra software, we plotted the obtained results via 3D, 2D views and we also illustrated the investigation of wave behavior caused by parameter change on 2D graphics. Originality:Different soliton behavior under the parameters effect of the Schrödinger–Hirota equation having chromatic dispersion and the cubic–quartic Fokas-Lenells equation is investigated and the obtained results are reported.