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In this paper, the integrable (4 + 1)-dimensional Fokas equation is investigated. Exploiting a set of non-singular local multipliers, we present a set of local conservation laws for the equation. The nonlocally related partial differential equation (PDE) systems are found. Nine nonlocally related systems are discussed reveal thirty five interesting closed form solutions of the equation. These solutions contain different types of wave solutions, double soliton, multi-solitons, kink and periodic wave solutions. Some of the resulting solutions are graphically illustrated. Furthermore, we apply the sine-cosine method to find other traveling wave solutions for this equation. [ABSTRACT FROM AUTHOR]

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 (-\Omega (\xi)) $ 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. [ABSTRACT FROM AUTHOR]

In this study, the nonlinear (2+1)-dimensional generalized breaking soliton system has been studied by introducing an appropriate travelling wave transformation. The model under consideration is integrable with constant coefficients and demonstrates Riemann wave propagation characteristics, it principally transformed into a third-order nonlinear ODE. The reliable and robust method, namely the modified exponential function method, is applied to the nonlinear system for the first time. The main goal is to investigate and obtain some explicitly exact travelling waves, periodic waves, and soliton solutions. The obtained solutions are in the form of exponential functions, trigonometric hyperbolic functions, and combined structures of the trigonometric hyperbolic with logarithmic functions. Furthermore, the obtained solutions are new and significant in revealing the pertinent features of the physical phenomenon. The results have been expressed in several graphs, including two- and three-dimensional plots, for the best visual assessment of the physical significance and dynamic characteristics. The most potent and efficacious tools are the computer software packages we utilize to derive solutions and graphs. [ABSTRACT FROM AUTHOR]

The current study is dedicated for operating the Lie symmetry approach, to complex short pulse equation. The method reduces the complex short pulse equation to a system of ordinary differential equations with the help of suitable similarity transformations. Consequently, these systems of nonlinear ordinary differential equations under each subalgebras are solved for exact solutions. Further, with the help of similarity variable, similarity solutions and exact solutions of nonlinear ordinary differential equation, soliton solutions of the complex short pulse equation are obtained which are in form of hyperbolic functions and trigonometric functions. [ABSTRACT FROM AUTHOR]

In this paper, we study a (3+1)-dimensional generalized shallow water wave equation with variable coefficients, which describes the flow below a pressure surface in a fluid. We give the Kadomtsev–Petviashvili hierarchy reduction and construct the multi-soliton solutions and semi-rational solutions in terms of the Gramian. For the multi-soliton solutions, we conclude that: (1) v 1 (t) affects the directions for the two solitons to move; (2) there is the periodic interaction of the two solitons when v 2 (t) is a periodic function and (3) the magnitudes of the velocities for the two solitons increase as the amplitude of the periodic function v 2 (t) increases, where v 1 (t) represents the perturbed effect, v 2 (t) indicates the dispersion effect and t is an independent variable. For the first-order semi-rational solutions, we see that: (1) the fission with v 1 (t) < 0 and fusion with v 1 (t) > 0 appear; (2) there is the periodic interaction when v 2 (t) is a periodic function; (3) the magnitude of the velocity for the soliton increases as the amplitude of the periodic function v 2 (t) increases and (4) the lump becomes narrower as the amplitude of the periodic function v 4 (t) decreases, where v 4 (t) indicates the perturbed effect. [ABSTRACT FROM AUTHOR]

We present a paper on the different methods for finding solutions to the Caudrey–Dodd–Gibbon equation with variable coefficients, which has wide applications in quantum mechanics and nonlinear optics. We have studied the equation analytically by using the unified method and the modified Kudryashov method. With the aid of symbolic computation and several types of auxiliary equations, we have obtained soliton solutions and other solutions. Then, we assign different values to the parameters, generating two-dimensional and three-dimensional graphics of the solutions, and discuss the interaction of several groups of solion solutions to form periodic wave solutions and kink solitary wave solutions. [ABSTRACT FROM AUTHOR]

In this article, a new solution for the (3 + 1) dimensions Boiti–Leon–Manna–Pempinelli (BLMP) equation using the sine Gordon expansion method and the extended tanh function method is given. The methods were chosen very carefully due to the precision of their solutions. Also, some of the two- and three-dimensional figures and some of the contours plots of the obtained solutions were presented. Finally, a discussion of the results was given. [ABSTRACT FROM AUTHOR]

The current research manifests kink wave answers, mixed singular optical solitons, the mixed dark-bright lump answer, the mixed dark-bright periodic wave answer, and periodic wave answers to the conformable fractional ZK model, including power law nonlinearity by plugging the revised (G ′ / G) -expansion process. The constraint requirements for the occurrence of substantial solitons are provided. Under the selection of proper values of a, b, n, t, λ, μ and α, the 2D and 3D pictures to a few of the recorded answers are sketched. From our obtained solutions, we might decide that the investigated procedure is hugely muscular, sincere, and essential in rendering various new soliton solutions of distinct nonlinear conformable fractional evolution equations and accordingly, we shall bring it up in our future investigations. [ABSTRACT FROM AUTHOR]

We present a Riemann–Hilbert approach for the modified short pulse equation on the line. This approach allows us to give a representation of the solution to the Cauchy problem, which can be efficiently used in studying its long-time behavior, also to describe the soliton solutions. [ABSTRACT FROM AUTHOR]

We study the derivative nonlinear Schrödinger (DNLS) equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but exclude spectral singularities). We show that the set of such initial data is open and dense in a weighted Sobolev space, and includes data of arbitrarily large L2-norm. We prove global well-posedness on this open and dense set. In a subsequent paper, we will use these results and a steepest descent analysis to prove the soliton resolution conjecture for the DNLS equation with the initial data considered here and asymptotic stability of N-soliton solutions. [ABSTRACT FROM AUTHOR]

Numerical solutions of the generalized variable-coefficient Korteweg-de Vries equation are obtained using a discrete singular convolution and a fourth order singly diagonally implicit Runge-Kutta method for space and time discretisation, respectively. The theoretical convergence of the proposed method is rigorously investigated. Test problems including propagation of single solitons and interaction of solitary waves are performed to verify the efficiency and accuracy of the method. The numerical results are checked against available analytical solutions and compared with the Sinc numerical method. We find that our approach is a very accurate, efficient and reliable method for solving nonlinear partial differential equations. [ABSTRACT FROM PUBLISHER]

In this paper, we study the standing waves for the relativistic quasilinear asymptotically Schrödinger equation This model has been proposed in the self-channelling of a high-power ultrashort laser in matter. [ABSTRACT FROM AUTHOR]

The modified unified algebraic (MUA) and auxiliary equation expansion (AEE) methods are applied to reach the different type soliton solutions of Kundu–Eckhaus (K–E) equation. It is illustrated that MUA and AEE are very effective methods to reach the various types of the solutions of nonlinear evolution equations. [ABSTRACT FROM PUBLISHER]

In this paper, we construct soliton solutions for a generalized variable-coefficient coupled Hirota–Maxwell–Bloch system, which can describe the ultrashort optical pulse propagation in a nonlinear, dispersive fiber doped with two-level resonant atoms. Under certain transformations and constraints, one- and two-soliton solutions are obtained via the Hirota method and symbolic computation, and soliton collisions are graphically presented and analyzed. One soliton is shown to maintain its amplitude and shape during the propagation. Soliton collision is elastic, while bright two-peak solitons and dark two-peak solitons are also observed. We discuss the influence of the coefficients for the group velocity, group-velocity dispersion (GVD), self-phase modulation, distribution of the dopant, and Stark shift on the soliton propagation and collision features, with those coefficients are set as some constants and functions, respectively. We find the group velocity and self-phase modulation can change the solitons’ amplitudes and widths, and the solitons become curved when the GVD and distribution of the dopant are chosen as some functions. When the Stark shift is chosen as a certain constant, the two peaks of bright two-peak solitons and dark two-peak solitons are not parallel. In addition, we observe the periodic collision of the two solitons. [ABSTRACT FROM AUTHOR]

Soliton interactions affect the pulse quality and capacity of communication systems. In this paper, parallel transmission of solitons without interactions are investigated to improve the pulse quality and capacity. Analytic two-soliton solutions for the variable coefficient nonlinear Schrödinger equation are obtained.The exponential profile of optical fibers is suggested to derive the parallel transmission of solitons, and influences of corresponding parameters are analyzed. [ABSTRACT FROM AUTHOR]

The first integral method was used to construct exact solutions of the Zoomeron and Klein-Gordon-Zakharov equations. The obtained results include new soliton and periodic solutions. The work confirms the significant features of the employed method and shows the variety of the obtained solutions. Throughout the paper, all the calculations are made with the aid of the Maple packet program. [ABSTRACT FROM AUTHOR]

We consider the inhomogeneous (1+1)-dimensional coupled nonlinear Schrodinger equations from the integrability point of view. This system has potential applications in inhomogeneous birefringent fiber media or in optical communication links with variable parameters. The exact analytical multisoliton solutions are obtained by employing the simple, straightforward Darboux transformation based on the obtained 3 × 3 Lax pair. Some main propagation and interaction properties of the one- and two-soliton solutions are discussed simultaneously. Also, the interaction between two neighboring combined solitary waves is numerically discussed. Excellent agreement with analytical results are found in the case of no collision. Moreover, the stability of these solutions are discussed in detail numerically with respect to a finite perturbation. [ABSTRACT FROM AUTHOR]

In this article, we discuss the analytic solution of the fully developed shock waves. The homotopy perturbation method is used to solve the shock wave equation, which describes the flow of gases. Unlike the various numerical techniques, which are usually valid for short period of time, the solution of the presented equation is analytic for 0 < t < ∞. The results presented converge very rapidly, indicating that the method is reliable and accurate. [ABSTRACT FROM AUTHOR]