Exact Soliton Solutions Research Articles

New optical soliton solutions to the coupled fractional Lakshmanan–Porsezian–Daniel equations with Kerr law of nonlinearity

This article studies the optical soliton solutions of the coupled fractional Lakshmanan–Porsezian–Daniel equations with kerr’s law nonlinearity based on the complete discriminant system of cubic polynomials. By means of traveling wave transformation, the fractional Lakshmanan–Porsezian–Daniel equations are simplified into two ordinary differential equations. Then, according to the classification of the roots of cubic polynomials, the optical soliton solutions of the fractional model with Kerr’s law nonlinearity are traversed. Then new traveling wave solutions are obtained, including the rational solutions, triangle function solutions, implicit function solutions and Jacobian elliptic function solutions, and the corresponding solutions are shown in two-dimensional and three-dimensional figures. Compared with previous related research, the exact optical soliton solutions of the LPD equation system for nonlinear Kerr’s law presented in this study are more completely and clearly classified, and researchers are easier to obtain these solutions.

Exact traveling wave solutions and soliton solutions of conformable M-fractional modified nonlinear Schrödinger model

The traveling wave solutions and soliton solutions of the conformable M-fractional modified nonlinear Schrödinger equation have been devised in this work by employing the generalized projective Riccati equations technique. The proposed method is straightforward and concise to obtain exact soliton solutions of nonlinear partial differential equations in various scientific discipline. This approach has been applied successfully to a wide variety of physical and mathematical problems. In many cases, the method has been able to provide new insights into the behavior of these systems and has led to the discovery of new phenomena. Several types of soliton solutions, including hyperbolic traveling wave solution, kink-shape soliton solution, singular soliton solution and dark soliton-type traveling wave solution, have been derived from the governing equation by using this effective technique. The solutions achieved by the governing model contain significant and dynamic justifications for certain real-world physical phenomena. Some acquired soliton solutions are displayed in 3D plots, corresponding contour plots and density plots to illustrate the physical dynamics of the governing model. Furthermore, the resulting outcomes manifest that the proposed method is highly efficient for revealing wave solutions for a wide range of nonlinear partial differential equations that emerge in optics.

An investigation of the variable coefficients modified KdV equation arising in arterial mechanics by using two expansion techniques

This article investigates the exact soliton solutions to the mKdV equation with time-dependent coefficients using the adapted sine-Gordon and modified simple equation approaches. The variations of the achieved solutions are demonstrated by plotting them with the corresponding solutions of the constant coefficients mKdV equation. Solitary waves with discrete wave profiles are developed from the resultant solution functions when specific values are assigned in place of the variable coefficients. The compatible two- and three-dimensional configurations show that the wave profiles are strongly affected by the variable wave velocity and its associated variable parameters. The analysis and interpretation emphasize the importance of the nonlinear evolution equations with time-varying coefficients, and it might open the purview to a notable domain of research.

Optical solitons and exact solutions of the (2+1) dimensional conformal time fractional Kundu-Mukherjee-Naskar equation via novel extended techniques

The current work uses a (2+1) dimensional conformal time fractional Kundu-Mukherjee-Naskar (KMN) model to investigate optical soliton transmission across an optical fiber that maintains polarization. Three constructive techniques, namely, the extended power series solution, the new generalized method, and the extended sinh-Gordon expansion method are utilized to find the exact soliton solutions of this model. The invariant analysis has been performed on the (2+1) dimensional time fractional KMN model by using the conformal time fractional derivative. The symmetries obtained using conformal fractional derivative are compared with the symmetries obtained for integer order KMN model because symmetries using Riemann Liouville fractional derivative turned out to be trivial. The given system of fractional PDEs has been reduced by using differential invariants obtained from various linear combinations of vector fields associated with the infinitesimal generator of symmetry transformations. These reduced systems of equations are then investigated for their exact solutions.

Various exact optical soliton solutions for time fractional Schrodinger equation with second-order spatiotemporal and group velocity dispersion coefficients

Various exact optical soliton solutions for time fractional Schrodinger equation with second-order spatiotemporal and group velocity dispersion coefficients

Analytical and numerical solutions with bifurcation analysis for the nonlinear evolution equation in (2+1)-dimensions

This paper is focused on the nonlinear evolution equation in (2+1)-dimensions which is found in different engineering and scientific areas. Many sets of exact soliton solutions of the nonlinear evolution equation in (2+1)-dimensions are presented via two analytical methods such as the Kudryashov method and the G′G-expansion method. Numerical solutions based on the finite difference method are introduced. The bifurcation analysis of solitary wave solutions is exhibited. Our analytical and numerical results are compared and illustrate them through some tables and graphical figures.

Riemann–Hilbert problem for the Fokas–Lenells equation in the presence of high-order discrete spectrum with non-vanishing boundary conditions

We extend the Riemann–Hilbert (RH) method to study the Fokas–Lenells (FL) equation with nonzero boundary conditions at infinity and successfully find its multiple soliton solutions with one high-order pole and N high-order poles. The mathematical structures of the FL equation are constructed, including global conservation laws and local conservation laws. Then, the conditions (analytic, symmetric, and asymptotic properties) needed to construct the RH problem are obtained by analyzing the spectral problem. The reflection coefficient r(z) with two cases appearing in the RH problem is considered, including one high-order pole and N high-order poles. In order to overcome the difficulty of establishing the residue expressions corresponding to high-order poles, we introduce the generalized residue formula. Finally, the expression of exact soliton solutions with reflectionless potential is further derived by a closed algebraic system.

Lie group theory, stability analysis with dispersion property, new soliton solutions and conserved quantities of 3D generalized nonlinear wave equation in liquid containing gas bubbles with applications in mechanics of fluids, biomedical sciences and cell biology

Nonlinear wave equations emerge naturally in optics as well as fluid dynamics. It is noteworthy that, an essential class of special solutions that satisfy the underlying equations are observed to be localized waves, which are frequently referred to as solitons or solitary waves. Thus, in the context of small amplitude alongside shallow-water waves, such solutions were discovered in the nineteenth century. Therefore, we present in this paper, the analytical investigations accomplished on a three dimensional generalized nonlinear wave equation in a fluid accommodating gas fizzes with applications. This equation was developed in the field within which liquid in conjunction with gas fizzes exists to describe the proliferation of feebly-nonlinear-waves. The underlying equation is transformed into a nonlinear structured ordinary differential equation (NLODE) by Lie group theory. Direct integration of the resulting NLODE produced periodic, trigonometric bright soliton together with singular soliton solutions. Moreover, some general exact soliton solutions of the equation under study are secured via the simplest equation technique (SET) in the structure of various Jacobi elliptic functions. Thus, we secure diverse periodic solitons of the equation under consideration. In addition, the dynamics of the results are depicted using suitable graphs which were also discussed. Furthermore, We conduct stability analysis on the model under study and outline the significance of our results in fluid dynamics, biomedical sciences and biological cells. Conclusively, we constructed conservation laws of the aforementioned equation by invoking Ibragimov’s theorem for conserved quantities via its formal Lagrangian structure.

Exact soliton solutions of a (2+1)-dimensional time-modulated nonlinear Schrödinger equation with cubic-quintic nonlinearity

In the present paper, based on the wave ansatz method, exact bright and dark solitons of a (2+1)-dimensional time variable coefficients nonlinear Schrödinger equation with cubic-quintic nonlinearity term are presented. This model can be used to describe the propagation of optical pulse in inhomogeneous media. These obtained solitons exhibit nonautonomous properties due to the amplitude and velocity of them are time-dependent coefficients. At this time, Tangent and Secant function solutions are also reported.

On the dynamics of exact solutions to a (3+1)-dimensional YTSF equation emerging in shallow sea waves: Lie symmetry analysis and generalized Kudryashov method

This paper focuses on obtaining a large number of exact solutions to the highly nonlinear (3+1)-dimensional Yu–Toda–Sassa–Fukuyama (YTSF) equation, which is important in fluid dynamics, plasma physics, nonlinear sciences, and weakly dispersive media, among other things. The primary goal of this research is to obtain a variety of exact soliton solutions and different dynamical wave-forms of the YTSF equation using two different techniques: Lie symmetry and generalized Kudryashov (GK) methods. These techniques reduce the considered partial differential equation (PDE) into ordinary differential equations (ODEs). Hereafter, a variety of exact solutions are derived with the help of the associated subalgebras and wave transformation by the Lie symmetry technique and GK technique, respectively. The obtained exact solutions involve various arbitrary constants, parameters, and arbitrary functionals that enhance the dynamic representation of the derived solutions in the research area. We use numerical simulations to generate 3-dimensional graphics and contour plots to determine the dynamical behavior of the obtained exact solutions. The physical interpretation of interaction soliton solutions demonstrates a rich diversity of attained solutions, such as solitons, kink waves, kink wave interaction with solitons, and others.

The dispersion relation of a dark soliton

The energy-velocity relation of a dark soliton is usually derived by its exact solution, which has been used to explain the kinetic motion of the dark soliton widely in many-body physical systems. We perform a variational method to re-derive the dispersion relation, with the consideration that the number of particles of the dark soliton could be conserved. The re-derived dispersion relation is completely different from that given by the exact dark soliton solution. The validity of these two dispersion relations is tested by observing the motion of the dark soliton when we drive impurity atoms that coupled with the soliton. The results suggest that the dispersion relation given by the exact solution usually works better than the one with particle number conservation. This motivates us to reveal that density waves (carrying particle transport) are generated during the acceleration process of a dark soliton, in addition to the previously known sound waves (only carrying energy transport). We further show that the density wave emissions of dark solitons can be inhibited by increasing the impurity atom number, which is trapped by the dark soliton through nonlinear coupling. The discussion is meaningful for investigating and understanding the kinetic motion of dark solitons in many different circumstances.

Investigation of exact soliton solutions of nematicons in liquid crystals according to nonlinearity conditions

In this work, new traveling wave solutions are generated for the system that models nematicons in liquid crystals using the [Formula: see text]-expansion method. In the equation system, nonlinearity is taken into account for Kerr law and Power law. Also, the existence of exact solutions under restriction conditions is guaranteed. We suggest that the solutions produced are of a different type than the solutions in the literature. Figures representing the intensity of the produced traveling wave solutions are presented. In addition, the simulation of the solitary wave is made for different values of the parameter that affects the inclination angle of the molecules in the nematicons mechanism in liquid crystals. How classical solitary wave behavior translates into triangular wave behavior is discussed. We believe this paper will provide an important perspective on the problems encountered in various application areas such as fluid dynamics, chemical engineering, chaos and complex networks in terms of investigating different mechanisms by taking into account nonlinearity factors.

The plethora of exact and explicit soliton solutions of the hyperbolic local (4+1)-dimensional BLMP model via GERF method

In this article, some novel results to the hyperbolic local (4+1)-dim Boiti-Leon-Manna-Pempinelli (BLMP) equation signifies the wave propagation over an incompressible fluid. In comparison to deep water, more essential wave capacity conditions are required for the linearization of the wave structure and strong nonlinear aspects are evident in shallow water. The explicit solitary wave solutions of the hyperbolic local (4+1)-dim BLMP equation are secured in this study using the generalized exponential rational function (GERF) approach. The resulting solutions are also validated by the use of computational tools like Maple. The dynamic characteristics of selected obtained solutions are depicted in 3D along their projections and as 2D graphs for various choices of arbitrary parameters. The outcomes demonstrate the effectiveness of the GERF approach for determining the sws of the (4+1)-dim BLMP equation.

Soliton solutions of Zakhrov equation in ionized plasma using new extended direct algebraic method

In this paper, we study the well known Zakhrov equation (ZE) which states to the Langmuir waves propagation in the ionized plasma. The exact soliton solutions of ZE are extracted using new extended direct algebraic method (EDAM). The obtained solutions are given in the form of trigonometric, rational and hyperbolic functions. Moreover, some of the obtained solutions are illustrated graphically to establish the physical nature of the selected solutions. The existence of these exact solutions is guaranteed by the constraints on the used parameters. It is noted that the new EDAM is easy and efficient technique and it can be useful to many other non-linear partial differential equations (PDEs).

Exact soliton solutions to the time-fractional Kudryashov model via an efficient analytical approach

Exact soliton solutions to the time-fractional Kudryashov model via an efficient analytical approach

Different forms of optical soliton solutions to the Kudryashov’s quintuple self-phase modulation with dual-form of generalized nonlocal nonlinearity

In this paper, the exact soliton solutions and other exact solutions for nonlinear Schrodinger’s equation having Kudryashov’s quintuple power law of refractive index together with dual form of generalized nonlocal nonlinearity are studied. By noticing that the system is a non-integrable one, the diverse of solitary wave solutions by using a generalized trial equation scheme are reached. In particular, four forms of the solution functions including soliton, bright soliton, singular soliton, and periodic wave solutions are investigated. To achieve this, an illustrative example of the Schrodinger’s equation to demonstrate the feasibility and reliability of the used procedure in this study is provided. The effect of the free parameters on the behavior a few obtained solutions for some nonlinear exact solutions was also analyzed due to the nature of nonlinearities. The dynamic properties of the obtained results are shown and analyzed by some density, two and three-dimensional images. We believe that our results would pave a way for future research generating optical memories based on the optical solitons. The constraint conditions to the existence of constructed solutions are also provided in this article.

Further results about the non-traveling wave exact solutions of nonlinear Burgers equation with variable coefficients

Nonlinear Burgers equation with variable coefficients (BEVC) which involves mathematical physics in plasma and fluid dynamics and so on. Investigating the exact solutions of BEVC to show the different dynamics of wave phenomena becomes a hot topic both on many mathematicians and physicists. In this paper, by (G′G2)-expansion and the Jacobian elliptic functions (JEFs) two different methods, we found that various forms for exact wave solutions of BEVC. To our best knowledge, we found a variety of new solutions that have not been studied in previous articles such as u13,u14,u511,u512,u513,u514. The most important thing is that there are double Jacobian elliptic functions ideas in finding solution process, which has not been seen before in seeking for nonlinear BEVC. These new exact soliton solutions contain variable coefficients derived in the form of trigonometric function, rational function, and Jacobian elliptic function, hyperbolic function. The obtained results showed many different types such as annihilation, parabolic kink, curved shaped kink, tine shaped, shock solitons and so on. Furthermore, the above obtained solutions for Burgers equation with variable coefficients are different from Mohanty et al. (2022), Zayed and Abdelaziz (2010).

Bright, dark and kink exact soliton solutions for perturbed Gerdjikov–Ivanov equation with full nonlinearity

In this paper, the perturbed Gerdjikov-Ivanov equation (PGI) which describes the dynamics of the soliton in optical fiber is investigated. Using traveling wave transformation, the nonlinear PGI equation is transformed into two nonlinear ordinary differential equations. Consequently, these equations are reduced to a first -order ordinary differential equation that has a well-known exact solution. The exact solutions for the PGI equation will be introduced including Weierstrass, Jacobi elliptic, and hyperbolic functions. Various types of optical soliton solutions were raised such as bright, dark, and kink soliton solution. The obtained optical soliton solutions are presented graphically to clarify some of its physical parameters.

Optical solitons in birefringent fibers with the generalized coupled space–time fractional non-linear Schrödinger equations

The nonlinear Schrödinger (NLS) equation is an ideal model for describing optical soliton transmission. This paper first introduces an integer-order generalized coupled NLS equation describing optical solitons in birefringence fibers. Secondly, the semi-inverse and fractional variational method is used to extend the integer‐order model to the space–time fractional order. Moreover, various nonlinear forms of fractional NLS equations are discussed, including the Kerr, power, parabolic, dual-power, and log law. The exact soliton solutions, such as bright, dark, and singular solitons, are given. Finally, the behavior of the solution is shown by three-dimensional figures with different fractional orders, which reveals the propagation characteristics of optical solitons in birefringence fibers described by the generalized coupled space–time fractional NLS equation.

A Parametric Method Optimised for the Solution of the (21)-Dimensional Nonlinear Schrödinger Equation+

We investigate the numerical solution of the nonlinear Schrödinger equation in two spatial dimensions and one temporal dimension. We develop a parametric Runge–Kutta method with four of their coefficients considered as free parameters, and we provide the full process of constructing the method and the explicit formulas of all other coefficients. Consequently, we produce an adaptable method with four degrees of freedom, which permit further optimisation. In fact, with this methodology, we produce a family of methods, each of which can be tailored to a specific problem. We then optimise the new parametric method to obtain an optimal Runge–Kutta method that performs efficiently for the nonlinear Schrödinger equation. We perform a stability analysis, and utilise an exact dark soliton solution to measure the global error and mass error of the new method with and without the use of finite difference schemes for the spatial semi-discretisation. We also compare the efficiency of the new method and other numerical integrators, in terms of accuracy versus computational cost, revealing the superiority of the new method. The proposed methodology is general and can be applied to a variety of problems, without being limited to linear problems or problems with oscillatory/periodic solutions.