Find Soliton Solutions Research Articles

Extraction of soliton solutions for the fractional Kaup-Boussinesq system: A comparative study

This paper is based on finding soliton solutions to fractional Kaup-Boussinesq (FKB) system. The fractional derivatives such as β-derivative and truncated M-fractional derivative are used in this study. The unified approach, generalized projective riccati equations method (GPREM) and improved tan (φ(ζ)/2)-expansion approaches are efficiently used for obtaining bright soliton, dark soliton, singular soliton, periodic soliton, dark-singular combo soliton and dark-bright combo soliton. The numerical simulations are also carried out by 3D and 2D, graphs of some of the obtained solutions to discuss the fractional effects.

Three different integration schemes for finding soliton solutions in the (1+1)-dimensional Van der Waals gas system

This paper explores three distinct integration schemes applied to the problem of finding soliton solutions in the (1+1)-dimensional Van der Waals gas system. The methods employed include the Enhanced Kudryashov’s method, the Sine–Gordon equation method, and the Projective Riccati equations method. Each of these techniques offers a unique approach to solving the system’s equations and finding soliton solutions. By examining and comparing the results obtained from these different integration schemes, this study aims to provide valuable insights into the behavior and properties of solitons in the Van der Waals gas system.

New optical soliton solutions and dynamical wave formations for a fractionally perturbed Chen-Lee-Liu (CLL) equation with a novel local fractional (NLF) derivative

In this work, we use a novel fractional-order derivative for the fractionally perturbed Chen–Lee–Liu nonlinear equation. The new extended hyperbolic function (EHF) method is applied for obtaining new optical soliton solutions of the mentioned equation. Three-dimensional graphics and projection 3D plots are used for showing the dynamic wave formations of the soliton solutions. Then, we contrast our findings with the earlier existing results for the nonlinearly perturbed CCL. The generated solutions show that the extended hyperbolic function (EHF) method for finding soliton solutions to highly nonlinear equations is productive, suitable, and competent in optical fibers, fractional calculus, and nonlinear sciences.

Nonlinear waves and the Inverse Scattering Transform

Solitons are a class of nonlinear stable, localized waves. They arise widely in physical problems; applications include water waves, plasma physics, Bose–Einstein condensation and nonlinear optics. Such localized water waves can be traced back to research in the 1800s. In fiber optics ‘bright and dark’ solitons were discovered in 1973 by Hasegawa and Tappert. In the 1970s a general theory emerged which allows one to linearize and explicitly find soliton solutions to a class of nonlinear wave equations including physically significant equations such as the Korteweg–deVries, nonlinear Schrödinger (NLS) and sine-Gordon equations. Solitons are connected to eigenvalues/bound states of underlying linear scattering equations. The theory, termed the Inverse Scattering Transform (IST) by Ablowitz, Kaup, Newell, Segur, leads to linearization/solutions to broad classes of nonlinear wave equations. In 2013 the theory was extended to novel classes of nonlocal nonlinear wave equations including PT symmetric and reverse-space–time nonlinear equations. In 2022 the theory was shown to encompass fractional integrable nonlinear systems; the fractional KdV and NLS equations are paradigms.

New solitary wave and computational solitons for Kundu–Eckhaus equation

The goal of this research is to find novel optical solutions to the Kundu–Eckhaus equation, which possess crucial roles in the field of nonlinear optics. A collective variable (CV) strategy is adopted to solve governing equation including the Raman effect and quintic nonlinearity. This method is a suitable to deal with both conservative and non-conservative systems by exposing a set of equations of motion regardless of nonlinearities or dissipative components. The parameters employed in this approach are chirp, temporal position, phase, amplitude, frequency and width, namely, collective variables. The fourth order Runge–Kutta technique is a well-known numerical scheme that aims towards the solution of the resulting system of ordinary differential equations representing the variables involved in the pulse ansatz. This technique presents the evolution of pulse parameters with regard to propagation variables. The graphical profiles at suitable values of pulse parameters are also provided. The unified technique is also applied to find soliton solutions. The obtained solution is a periodic solitary wave, showed graphically. The results developed in this article are found to be new in the literature and the approach utilized, can be applied to solve a variety of nonlinear problems in the mathematical sciences.

Obtaining optical soliton solutions of the cubic–quartic Fokas–Lenells equation via three different analytical methods

In this study, we have focused on finding soliton solutions of the cubic–quartic Fokas–Lenells equation, which models the nonlinear pulse transmission through optical fiber, is a pretty new and updated model. The main motivation of this study is to produce new solutions with previously unused methods for data transmission models in fiber optic cables together with the developing technology, which has been used very frequently today. We have used three different efficient analytical methods, namely, the Sinh–Gordon expansion method, enhanced modified extended tanh expansion method and unified Riccati equation expansion method. By modifying our proposed method and using it effectively, we have obtained much more optical solutions. We have supported the results with the 3D surface, 2D and contour plots of the soliton solutions, such as singular, periodic, periodic-singular, dark and M-shaped solitons. The originality of the method we propose is that it has never been used before. With the innovation of the method, we have obtained M-shaped solitons, which are very rarely obtainable in soliton waves. On the other hand, enhanced modified extended tanh expansion method have provided us more extra solutions. As the last and most important part, in order to examine physical behaviors, we have investigated the effect of the coefficient of the fourth order dispersion parameter on wave propagation by presenting the graphical representation.

Multi-peak soliton solutions of the generalized breaking soliton equation

In this article, we apply the generalized exponential rational function method to find soliton solutions of generalized (3+1)-dimensional breaking soliton equation, which describes two-dimensional interchange of Riemann wave disseminating alongside y-axis with a long wave disseminating alongside x-axis. As a result, we obtain many soliton solutions to the equation by choosing appropriate parameters. We demonstrate a variety of travelling waves using three-dimensional, two-dimensional and contour plots. We observe that solutions involving parameters can drastically influence the behaviour of solitary waves. Such solutions are applicable to enrich the dynamical study of nonlinear waves in fluid dynamics and other fields of research.

On the interaction phenomena to the nonlinear generalized Hietarinta-type equation

In this paper, we describe the nonlinear behavior of a generalized fourth-order Hietarinta-type equation for dispersive waves in (2 + 1) dimension. The various wave formations are retrieved by using Hirota’s bilinear method (HBM) and various test function perspectives. The Hirota method is a widely used and robust mathematical tool for finding soliton solutions of nonlinear partial differential equations (NLPDEs) in a variety of disciplines like mathematical physics, nonlinear dynamics, oceanography, engineering sciences, and others requires bilinearization of nonlinear PDEs. The different wave structures in the forms of new breather, lump-periodic, rogue waves, and two-wave solutions are recovered. In addition, the physical behavior of the acquired solutions is illustrated in three-dimensional, two-dimensional, density, and contour profiles by the assistance of suitable parameters. Based on the obtained results, we can assert that the employed methodology is straightforward, dynamic, highly efficient, and will serve as a valuable tool for discussing complex issues in a diversity of domains specifically ocean and coastal engineering. We have also made an important first step in understanding the structure and physical behavior of complex structures with our findings here. We believe this research is timely and relevant to a wide range of engineering modelers. The results obtained are useful for comprehending the fundamental scenarios of nonlinear sciences.

A novel and efficient method for obtaining Hirota’s bilinear form for the nonlinear evolution equation in (n+1) dimensions

Bilinearization of nonlinear partial differential equations (PDEs) is essential in the Hirota method, which is a widely used and robust mathematical tool for finding soliton solutions of nonlinear PDEs in a variety of fields, including nonlinear dynamics, mathematical physics, and engineering sciences. We present a novel systematic computational approach for determining the bilinear form of a class of nonlinear PDEs in this article. It can be easily implemented in symbolic system software like Mathematica, Matlab, and Maple because of its simplicity. The proven results are obtained by using a developed method in Mathematica and applying a logarithmic transformation to the dependent variable. Finally, the findings validate the implemented technique’s competence, productivity, and dependability. The approach is a useful, authentic, and simple mathematical tool for calculating multiple soliton solutions to nonlinear evolution equations encountered in nonlinear sciences, plasma physics, ocean engineering, applied mathematics, and fluid dynamics. • This paper introduces a new systematic computational approach for constructing bilinear forms for nonlinear PDEs. • The given approach can handle a class of (n+1)-dimensional nonlinear PDEs. • Various examples of well-known nonlinear evolution equations up to the (4+1)-dimension are shown. • The results of various nonlinear equations are provided using the proposed method.

Soliton solution of Generalized Zakharov-Kuznetsov and ZakharovKuznetsov-Benjamin-Bona-Mahoney equations with conformable temporal evolution

In this paper, we proposethe method of functional variable for finding soliton solutions of two practical problems arising in electronics, namely, the conformable time-conformable Generalized Zakharov-Kuznetsov equation (GZKE) and the conformable time-conformable Generalized Zakharov-Kuznetsov-Benjamin-BonaMahoney equation (GZK-BBM). The soliton solutions are expressed by two types of functions which are hyperbolic and trigonometric functions. Implemented method is more effective, powerful and straightforward to construct the soliton solutions for nonlinear conformable time-conformable partial differential equations.

The solitary wave solutions to the Klein–Gordon–Zakharov equations by extended rational methods

In this paper, using the extended rational sine–cosine and rational sinh–cosh methods, we find new soliton solutions for the Klein–Gordon–Zakharov equations. The extended rational sine–cosine and rational sinh–cosh methods are prospering in finding soliton solutions of the Klein–Gordon–Zakharov equations. By means of these methods, we found some young solitons of the above mentioned equation. The conclusions we receive are dark, bright, and periodic. In addition, in order to imagine the underlying dynamics of the obtained soliton solutions, 2D and 3D plots are drawn.

On the soliton solutions for an intrinsic fractional discrete nonlinear electrical transmission line

This paper focuses on finding soliton solutions for an intrinsic fractional discrete nonlinear electrical transmission lattice. Our investigation is based on the fact that for a realistic system, the electrical characteristics of a capacitor (and an inductor via skin effect) should include a fractional-order time derivative. In this respect for the model under consideration, we derive a fractional nonlinear partial differential equation for the voltage dynamics by applying the Kirchhoff’s laws. It is realized that the behavior of new soliton solutions obtained is influenced by the fractional-order time derivative as well as the coupling values. The fractional order also modifies the propagation velocity of the voltage wave notwithstanding their structure and tends to set up localized structure for low coupling parameter values. However, for a high value of the coupling parameter, the fractional order is less seen on the shapes of the new solitary solutions that are analytically derived. Several methods such as the Kudryashov method, the $$(G'/G)$$ -expansion method, the Jacobi elliptical functions method and the Weierstrass elliptic function expansion method led us to derive these solitary solutions while using the modified Riemann–Liouville derivatives in addition to the fractional complex transform. An insight into the overall dynamics of our network is provided through the analysis of the phase portraits.

Highly Dispersive Optical Solitons of an Equation with Arbitrary Refractive Index

A nonlinear fourth-order differential equation with arbitrary refractive index for description of the pulse propagation in an optical fiber is considered. The Cauchy problem for this equation cannot be solved by the inverse scattering transform and we look for solutions of the equation using the traveling wave reduction. We present a novel method for finding soliton solutions of nonlinear evolution equations. The essence of this method is based on the hypothesis about the possible type of an auxiliary equation with an already known solution. This new auxiliary equation is used as a basic equation to look for soliton solutions of the original equation. We have found three forms of soliton solutions of the equation at some constraints on parameters of the equation.

Преобразование собственных значений задачи Захарова–Шабата при столкновении солитонов

Background and Objectives: The Zakharov–Shabat spectral problem allows to find soliton solutions of the nonlinear Schrodinger equation. Solving the Zakharov–Shabat problem gives both a discrete set of eigenvalues λj and a continuous one. Each discrete eigenvalue corresponds to an individual soliton with the real part Re(λj) providing the soliton velocity and the imaginary part Im(λj) determining the soliton amplitude. Solitons can be used in optical communication lines to compensate both non-linearity and dispersion. However, a direct use of solitons in return-to-zero signal encoding is inhibited. The interaction between solitions leads to the loss of transmitted data. The problem of soliton interaction can be solved using eigenvalues. The latter do not change when the solitons obey the nonlinear Schrodinger equation. Eigenvalue communication was realized recently using electronic signal processing. To increase the transmission speed the all-optical method for controlling eigenvalues should be developed. The presented research is useful to develop optical methods for the transformation of the eigenvalues. The purpose of the current paper is twofold. First, we intend to clarify the issue of whether the dispersion perturbation can not only split a bound soliton state but join solitons into a short oscillating period breather. The second goal of the paper is to describe the complicated dynamics and mutual interaction of complex eigenvalues of the Zakharov–Shabat spectral problem. Materials and Methods: Pulse propagation in single-mode optical fibers with a variable core diameter can be described using the nonlinear Schrödinger equation (NLSE) which coefficients depends on the evolution coordinate. The NLSE with the variable dispersion coefficient was considered. The dispersion coefficient was described using a hyperbolic tangent function. The NLSE and the Zakharov– Shabat spectral problem were solved using the split-step method and the layer-peeling method, respectively. Results: The results of numerical analysis of the modification of soliton pulses under the effect of variable dispersion coefficient are presented. The main attention is paid to the process of transformation of eigenvalues of the Zakharov–Shabat problem. Collision of two in-phase solitons, which are characterized by two complex eigenvalues is considered. When the coefficients of the nonlinear Schrodinger equation change, the collision of the solitons becomes inelastic. The inelastic collision is characterized by the change of the eigenvalues. It is shown that the variation of the coefficients of the NLSE allows to control both real and imaginary parts of the eigenvalues. Two scenarios for the change of the eigenvalues were identified. The first scenario is characterized by preserving the zero real part of the eigenvalues. The second one is characterized by the equality of their imaginary parts. The transformation of eigenvalues is most effective at the distance where the field spectrum possesses a two-lobe shape. Variation of the NLSE coefficient can introduce splitting or joining of colliding soliton pulses. Conclusion: The presented results show that the eigenvalues can be changed only with a small variation of the NLSE coefficients. On the one hand, a change in the eigenvalues under the effect of inelastic soliton collision is an undesirable effect since the inelastic collision of solitons will lead to unaccounted modulation in soliton optical communication links. On the other hand, the dependence of the eigenvalues on the parameters of the colliding solitons allows to modulate the eigenvalues using all-fiber optical devices. Currently, the modulation of the eigenvalues is organized using electronic devices. Therefore, the transmission of information is limited to nanosecond pulses. For picosecond pulse communication, the development of all-optical modulation methods is required. The presented results will be useful in the development of methods for controlling optical solitons and soliton states of the Bose–Einstein condensate.

Bilinearization and Soliton Solutions of a Supersymmetric Multicomponent Coupled Dispersionless Integrable System

We present a supersymmetric generalization of a multicomponent coupled dispersionless (SUSY-MCD) integrable system. Expanding the superfields of the SUSY-MCD system, we obtain the component fermionic and bosonic held equations. We propose a bilinear form of the SUSY-MCD system and find soliton solutions of it.

Soliton Solutions of a Generalization of the Coupled Volterra System

The possibility of finding soliton solutions of a nonintegrable generalization of the coupled Volterra system is studied. This generalization is a system of two equations each of which includes terms that take into account the spatial dependence. At the first stage, the continual limit of the generalization is studied. At the second stage, soliton solutions for the continual limit are sought. At the third, final, step, soliton solutions of the nonintegrable generalization are sought.

The role of the free surface on interfacial solitary waves

We investigated theoretically and experimentally internal solitary waves (ISWs) in a two-layer fluid system with a top free surface. Laboratory experiments are performed by lock-release, under Boussinesq and non-Boussinesq conditions. Experimental results are compared with those obtained by the analytical solution of the Korteweg–de Vries (KdV) weakly nonlinear equation and by the strongly nonlinear Miyata-Choi-Camassa (MCC) model. We analyze the initial conditions which allow to find soliton solutions for both rigid-lid (-RL) and free-surface (-FS) boundary conditions. For the MCC-FS model, we employ a new mathematical procedure to derive the ISW-induced free surface displacement. The density structure strongly affects the elevation of the free surface predicted by the MCC-FS model. The free surface maximum displacement increases mostly with the density difference, assuming non-negligible values also for smaller interfacial amplitudes. Larger displacements occur for thinner upper layer thickness. The MCC-FS model gives the best prediction in terms of both internal waves geometric/kinematic features and surface displacements. The existence of a free surface allows the ISW to transfer part of its energy to the free surface: the wave celerity assumes lower values with respect to ISW speed resulting from the MCC-RL model. For ISWs with a very large amplitude, this behavior tends to fade, and the MCC-RL and the MCC-FS model predict approximately the same celerity and interfacial geometric features. For small-amplitude waves also, the predictions of the KdV-RL equation are consistent with experimental results. Thus, ISWs with an intermediate amplitude should be modeled taking into account a free top surface as the boundary condition.

Nonlocal modified KdV equations and their soliton solutions by Hirota Method

We study the nonlocal modified Korteweg–de Vries (mKdV) equations obtained from AKNS scheme by Ablowitz–Musslimani type nonlocal reductions. We first find soliton solutions of the coupled mKdV system by using the Hirota direct method. Then by using the Ablowitz–Musslimani reduction formulas, we find one-, two-, and three-soliton solutions of nonlocal mKdV and nonlocal complex mKdV equations. The soliton solutions of these equations are of two types. We give one-soliton solutions of both types and present only first type of two- and three-soliton solutions. We illustrate our solutions by plotting their graphs for particular values of the parameters.

Nonlocal nonlinear Schrödinger equations and their soliton solutions

We study standard and nonlocal nonlinear Schrödinger (NLS) equations obtained from the coupled NLS system of equations (Ablowitz-Kaup-Newell-Segur (AKNS) equations) by using standard and nonlocal reductions, respectively. By using the Hirota bilinear method, we first find soliton solutions of the coupled NLS system of equations; then using the reduction formulas, we find the soliton solutions of the standard and nonlocal NLS equations. We give examples for particular values of the parameters and plot the function |q(t, x)|2 for the standard and nonlocal NLS equations.

Soliton solutions and eigenfunctions of linearized operator for a higher-order nonlinear Schrödinger equation

The Inverse Scattering Transform (IST) method is applied to find soliton solutions for a higher-order nonlinear Schrödinger (NLS) equation. Eigenfunctions of linearized operator which have a central role in soliton perturbation theory are explicitly found.