Analytical Soliton Research Articles

Analytical Soliton Solutions of the Coupled Radhakrishnan‐Kundu‐Lakshmanan Equation via Three Techniques

The analytical soliton solutions to the coupled Radhakrishnan‐Kundu‐Lakshmanan (RKL) model are greatly important for birefringent fibers without the effect of four‐wave mixing (4WM). A significant number of general and standard analytical soliton solutions to this model have been extracted using three powerful techniques, namely the generalized Kudryashov’s method, the extended tanh method, and the (G′/G)‐expansion method in this article. The schematic profiles of the solitons are sketched using the symbolic mathematical program Mathematica and are presented in two and three dimensions. The reported solutions might be helpful in explaining the RKL equation’s physical significance as well as some other related nonlinear phenomena that appear in engineering and nonlinear sciences.

Analytical soliton solutions to the generalized (3+1)-dimensional shallow water wave equation

In this paper, the soliton solutions and dynamical wave structures for the generalized (3+1)-dimensional shallow water wave (SWW) equation, which is an important physical property in ocean engineering and hydrodynamics, are presented. The generalized exponential rational function (GERF) method is used to investigate the closed-form wave solutions of the generalized SWW equation, which is used to describe the evolutionary dynamics of SWW. We successfully archive a variety of soliton solutions such as exponential solutions, kink wave solutions, non-topological solutions, periodic singular solutions, and topological solutions. These newly established results are also important for understanding the wave-propagation and dynamics of exact solutions of the equation, which is of great significance in physical oceanography and chemical oceanography. Eventually, it is shown that the proposed GERF technique is effective, robust, and straightforward and is also used to solve other types of higher-dimensional nonlinear evolution equations. In our work, we have used Mathematica extensively for such complicated algebraic calculations.

Classes of new analytical soliton solutions to some nonlinear evolution equations

Undoubtedly, all applied sciences are closely corresponding to differential equations, especially partial differential equations. This contribution aims to examine the efficiency of a modified version of the generalized exponential rational function method in solving some well-known nonlinear evolution equations. The investigated models in this article include the Zakharov–Kuznetsov, the cubic Boussinesq, and the modified regularized long-wave equation. In the structure of these solutions, various familiar elementary functions, including exponential, trigonometric, and hyperbolic functions are used. This important feature makes it easy to use these solutions in real applications. Numerical behaviors corresponding to the obtained solutions have been demonstrated through some three-dimensional diagrams in the article. This technique can be also adopted in determining wave soliton solution to other equations with partial derivatives.

The higher-order nonlinear Schrödinger’s dynamical equation with fourth-order dispersion and cubic-quintic nonlinearity via dispersive analytical soliton wave solutions

The higher-order nonlinear Schrödinger’s dynamical equation with fourth-order dispersion and cubic-quintic nonlinearity via dispersive analytical soliton wave solutions

Abundant analytical soliton solutions and different wave profiles to the Kudryashov-Sinelshchikov equation in mathematical physics

The generalized exponential rational function (GERF) method is used in this work to obtain analytic wave solutions to the Kudryashov-Sinelshchikov (KS) equation. The KS equation depicts the occurrence of pressure waves in mixtures of liquid-gas bubbles while accounting for thermal expansion and viscosity. By applying the GERF method to the KS equation, we obtain analytic solutions in terms of trigonometric, hyperbolic, and exponential functions, among others. These solutions include solitary wave solutions, dark-bright soliton solutions, singular soliton solutions, singular bell-shaped solutions, traveling wave solutions, rational form solutions, and periodic wave solutions. We discuss the two-dimensional and three-dimensional graphics of some obtained solutions under the accurate range space by selecting appropriate values for the involved arbitrary parameters to make this research more praiseworthy. The obtained analytic wave solutions specify the GERF method’s dependability, capability, trustworthiness, and efficiency.

The phase shift analysis of the colliding dissipative KdV solitons

In this work, the head-on collisions of the non-stationary dissipative soliton in ultracold neutral plasmas (UNPs) are investigated. The extended Poincare-Lighthill-Kuo (PLK) approach is adopted for reducing the fluid equations of the UNPs to two-counterpropagating damped Korteweg-de Vries (dKdV) equations. The dKdV equation is not an integrable Hamiltonian system, i.e., does not have an exact solution. Thus, one of the main goal of this paper is to find a new general approximate analytical solution to the dKdV equation for investigating the mechanism of the propagation and interaction of the non-stationary dissipative solitons. The residual error is estimated for checking the accuracy of the new obtained solution. The approximate analytical soliton solutions are adopted for deriving the temporal phase shifts after the collision. The impact of physical parameters on the nonstationary dissipative soliton profile and the temporal phase shifts is discussed. The obtained results will contribute to understand the mechanism of propagation and interaction of many nonlinear phenomena in different nonlinear mediums such as ocean, sea, optical fiber, plasma physics, etc.

Abundant analytical soliton solutions and Evolutionary behaviors of various wave profiles to the Chaffee–Infante equation with gas diffusion in a homogeneous medium

The generalized exponential rational function method is used in this work to obtain a variety of rich families of analytical soliton solutions and to exemplify the dynamics of solitonic structures of the (2+1)-dimensional ChaffeeInfante (CI) equation. This CI equation is a well-known reaction–diffusion equation that can describe the physical process of mass transport and particle diffusion and has been widely used in fluid dynamics, electromagnetic wave fields, high-energy physics, fluid mechanics, coastal engineering, and ion-acoustic waves in plasma physics, optical fibers, and other disciplines. The GERF method is successfully used to derive closed-form analytic solutions to the CI equation, including rational, exponential, trigonometric, and hyperbolic function solutions. The CI equation has indeed been observed to have bright and dark solitons, singular and combined singular soliton profiles, periodic oscillating nonlinear waves, and kink-wave profiles. Furthermore, three-dimensional graphical representations are presented to depict the dynamical behavior of the achieved solutions. We can better understand the dynamical properties and structures of these solutions by using these three-dimensional postures. This technique could be used to solve a wide variety of higher-dimensional nonlinear equations confronted in mathematical physics and applied sciences.

High-dimensional vector solitons for a variable-coefficient partially nonlocal coupled Gross–Pitaevskii equation in a harmonic potential

A (3+1)-dimensional variable-coefficient partially nonlocal coupled Gross–Pitaevskii equation trapped in a harmonic potential becomes a focus of this paper. A counterpart of this variable-coefficient coupled equation is found as a (2+1)-dimensional constant-coefficient single nonlinear Schrödinger equation via the reduction procedure. By solutions of constant-coefficient single equation via the Hirota method, and from this counterpart, analytical high-dimensional vector soliton solutions with the Hermite–Gaussian envelope of the variable-coefficient coupled equation are deduced. Expanded behaviors of high-dimensional vector solitons emerge in the exponential diffraction decreasing system.

Analytical Traveling Wave and Soliton Solutions of the $$(2+1)$$ Dimensional Generalized Burgers–Huxley Equation

Analytical Traveling Wave and Soliton Solutions of the $$(2+1)$$ Dimensional Generalized Burgers–Huxley Equation

Analytical new soliton wave solutions of the nonlinear conformable time-fractional coupled Whitham–Broer–Kaup equations

The study of nonlinear physical and abstract systems is greatly important in order to determine the behavior of the solutions for Fractional Partial Differential Equations (FPDEs). In this paper, we study the analytical wave solutions of the time-fractional coupled Whitham–Broer–Kaup (WBK) equations under the meaning of conformal fractional derivative. These solutions are derived using the modified extended tanh-function method. Accordingly, different new forms of the solutions are obtained. In order to understand its behavior under varying parameters, we give the visual representations of all the solutions. Finally, the graphs are discussed and a conclusion is given.

Exact analysis and elastic interaction of multi-soliton for a two-dimensional Gross-Pitaevskii equation in the Bose-Einstein condensation

IntroductionThe Gross-Pitaevskii equation is a class of the nonlinear Schrödinger equation, whose exact solution, especially soliton solution, is proposed for understanding and studying Bose-Einstein condensate and some nonlinear phenomena occurring in the intersection field of Bose-Einstein condensate with some other fields. It is an important subject to investigate their exact solutions. ObjectivesWe give multi-soliton of a two-dimensional Gross-Pitaevskii system which contains the time-varying trapping potential with a few interactions of multi-soliton. Through analytical and graphical analysis, we obtain one-, two- and three-soliton which are affected by the strength of atomic interaction. The asymptotic expression of two-soliton embodies the properties of solitons. We can give some interactions of solitons of different structures including parabolic soliton, line-soliton and dromion-like structure. MethodsBy constructing an appropriate Hirota bilinear form, the multi-soliton solution of the system is obtained. The soliton elastic interaction is analyzed via asymptotic analysis. ResultsThe results in this paper theoretically provide the analytical bright soliton solution in the two-dimensional Bose-Einstein condensation model and their interesting interaction. To our best knowledge, the discussion and results in this work are new and important in different fields. ConclusionsThe study enriches the existing nonlinear phenomena of the Gross-Pitaevskii model in Bose-Einstein condensation, and prove that the Hirota bilinear method and asymptotic analysis method are powerful and effective techniques in physical sciences and engineering for analyzing nonlinear mathematical-physical equations and their solutions. These provide a valuable basis and reference for the controllability of bright soliton phenomenon in experiments for high-dimensional Bose-Einstein condensation.

Dispersive wave propagation of the nonlinear Sasa-Satsuma dynamical system with computational and analytical soliton solutions

The Sasa-Satsuma equation on a continuous background describes a nonlinear fiber system with higher-order effects including the third-order dispersion and Kerr dispersion. The Sasa-Satsuma equations describe the simultaneous propagation of two ultrashort pulses in the birefringent or two-mode fiber with the third-order dispersion, self-steepening, and stimulated Raman in scattering effects, and govern the propagation of ultra-fast pulses in optical fiber transmission systems. We consider the Sasa-Satsuma equation, which is one of the integrable extensions of the nonlinear Schrödinger equations. We find the functional integral and the Lagrangian of this model. We derived the computational and analytical soliton solutions of the nonlinear Sasa-Satsuma dynamical system. We discuss the stability analysis for our solutions.

Painlevé analysis, auto-Bäcklund transformation and analytic solutions for modified KdV equation with variable coefficients describing dust acoustic solitary structures in magnetized dusty plasmas

In this paper, variable coefficients mKdV equation is examined by using Painlevé analysis and auto-Bäcklund transformation method. The proposed equation is an important equation in magnetized dusty plasmas. The Painlevé analysis is used to determine the integrability whereas an auto-Bäcklund transformation technique is being explored to derive unique family of analytical solutions for variable coefficients mKdV equation. New kink–antikink and periodic-kink- type soliton solutions have been determined successfully for the considered equation. This paper shows that auto-Bäcklund transformation method is effective, direct and easy to use, and used to determine the analytic soliton solutions of various nonlinear evolution equations in the field of science and engineering. The results are plotted graphically to signify the potency and applicability of this proposed scheme for solving the above considered equation. The obtained results are in the form of soliton-like solutions, solitary wave solutions, exponential and trigonometric function solutions. Therefore, these solutions help us to understand the potential and physical behaviors of the proposed equation.

On the approximate and analytical solutions to the fifth-order Duffing oscillator and its physical applications

ABSTRACT Novel approximate and analytical solutions to the fifth-order Duffing equation (FODE) are reported. These solutions are expressed in terms of the Jacobi elliptic and trigonometric functions. As a possible realization in the nature of the FODE, we study some nonlinear differential equations (NLDEs) that describe many physical problems. The numerical solution to the FODE is compared with the theoretical results. Moreover, the proposed method shows a new vision, which could be of great interest in the solution of the family of higher-order nonlinear Schrödinger equation such as the cubic-quintic NLSE (CQNLSE) which is used for interpreting the high Langmuir fields energy in magnetoplasmas. This method extends to find the solution of the derivative NLSE (DNLSE), which is used to describe the weakly modulated Alfvén waves propagation in magnetoplasmas. Furthermore, the equation of motion for the general pendulum could be solved using the techniques under consideration. All mentioned equations could be reduced to the FODE via a suitable transformation. Finally, this study is very interesting in describing several natural phenomena and solving many physical NLDEs that were difficult to solve them before. Moreover, the analytical bright soliton solution to the higher-order NLSE with incorporating cubic-quintic nonlinearity is obtained.

Nondegenerate solitons for coupled higher-order nonlinear Schrödinger equations in optical fibers

Nondegenerate solitons have been demonstrated in recent years that they potentially allow a further increase of information transmission rates in optical communication due to their stable double-hump structures. In this paper, the femtosecond nondegenerate solitons in optical fibers which are described by the coupled higher-order nonlinear Schrödinger equations, are investigated. Analytical nondegenerate soliton solutions are constructed by the developing Hirota method. And the constraints for stable double-hump structures are put forward. Furthermore, soliton molecules and asymmetric solitons are also found as the new types of nondegenerate solitons. The interactions between two nondegenerate solitons are studied in their propagation.

Analytical solitons for the space-time conformable differential equations using two efficient techniques

Exact solutions to nonlinear differential equations play an undeniable role in various branches of science. These solutions are often used as reliable tools in describing the various quantitative and qualitative features of nonlinear phenomena observed in many fields of mathematical physics and nonlinear sciences. In this paper, the generalized exponential rational function method and the extended sinh-Gordon equation expansion method are applied to obtain approximate analytical solutions to the space-time conformable coupled Cahn–Allen equation, the space-time conformable coupled Burgers equation, and the space-time conformable Fokas equation. Novel approximate exact solutions are obtained. The conformable derivative is considered to obtain the approximate analytical solutions under constraint conditions. Numerical simulations obtained by the proposed methods indicate that the approaches are very effective. Both techniques employed in this paper have the potential to be used in solving other models in mathematics and physics.

New solutions to the fractional perturbed Chen–Lee–Liu equation with a new local fractional derivative

In this research a new fractional-order derivative is defined and applied to the fractional perturbed Chen–Lee–Liu nonlinear equation. Analytical soliton solutions are obtained by the modified exp ⁡ ( − Ω ( ξ ) ) -expansion function method (MEFM). Three new groups of bright soliton solutions, dark soliton solutions, periodic tan-type singular soliton solutions and rational singular soliton solutions are constructed with specific contains over the parameters of the nonlinear fractional perturbed Chen–Lee–Liu equation. Solutions are illustrated in 3D graphs, contour plots and 2D plots. Next, we compare our results with the results obtained previously for the nonlinear perturbed CCL. It is noticed that the solutions here introduced present some complex propagation features that appears for different time values in the 2D space graphs, these complex propagation phenomena are not observed in the integer-order nonlinear perturbed CCL equation or in the fractional case for other type of fractional derivatives previously introduced in the literature The fractional-order derivative here introduced will be very useful for the description of some nonlinear dissipation, nonlinear dispersion and other complex effects in optical wave phenomena.

Analytical soliton solutions and modulation instability for a generalized (3 + 1)-dimensional coupled variable-coefficient nonlinear Schrödinger equations in nonlinear optics

In this work, the soliton solutions of a (3 + 1)-dimensional coupled nonlinear Schrödinger system with time-dependent coefficients arising in optical fiber has been studied analytically. A complex traveling wave ansatz is used to transform the proposed coupled partial differential equations into ordinary differential equations (ODEs). Then, we solved the obtained nonlinear ODEs for the envelope function in the traveling wave parameter. The obtained waves envelope have the forms of bright and dark soliton-like solutions which are considered as new solutions of the studied system. As a kind of completing the analysis, the modulation instability is discussed based on the linear stability analysis.

Dark soliton dynamics for quantum systems with higher-order dispersion and higher-order nonlinearity

In this work, we investigated one-dimensional and two-dimensional quantum systems with higher-order dispersions and higher-order nonlinear interactions. Based on the high-order nonlinear Schrödinger equation (NLSE) and via the [Formula: see text]-expansion method, we derived the analytical dark soliton solution for the one-dimensional system first. By applying the self-similar method and using the results of the one-dimensional case, the analytical dark soliton solution of the system in the two-dimensional case was derived. The dynamic evolution pattern of the two-dimensional dark soliton is pictorially demonstrated. The theoretical results of our work can be used to guide the detection and experimental study of dark soliton in a two-dimensional quantum system, using high-order dispersion and higher-order nonlinear interactions.

Collapse arrest in the space-fractional Schrödinger equation with an optical lattice* *Project supported by the National Natural Science Foundation of China (Grant No. 11947122), the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2019A1515110935), the Research Start-up Foundation of Dongguan University of Technology, the Guangdong Science and Technology Planning Program (Grant No. 2017A010102019), the Guangdong Province Natural Science Foundation

The soliton solution and collapse arrest are investigated in the one-dimensional space-fractional Schrödinger equation with Kerr nonlinearity and optical lattice. The approximate analytical soliton solutions are obtained based on the variational approach, which provides reasonable accuracy. Linear-stability analysis shows that all the solitons are linearly stable. No collapses are found when the Lévy index 1 < α ≤ 2. For α = 1, the collapse is arrested by the lattice potential when the amplitude of perturbations is small enough. It is numerically proved that the energy criterion of collapse suppression in the two-dimensional traditional Schrödinger equation still holds in the one-dimensional fractional Schrödinger equation. The physical mechanism for collapse prohibition is also given.