Dark-singular Soliton Solutions Research Articles

New soliton solutions for the space-time fractional modified third order Korteweg–de Vries equation

In present research work, the soliton solutions of the space-time fractional modified third order Korteweg–de Vries (KdV) equation are explored by Sardar-subequation method (SSM). The (KdV) equation has an established version based on the shallow water waves in the oceans, oceanography theory and the ion-acoustic waves in plasma. The fractional order partial differential equation (PDE) is transform to a non-linear ordinary differential equation (ODE) by using traveling wave transformation. It is noted that in this work the fractional differential equation is solved in the sense of conformable derivative. Then soliton solutions of reduced equation are established by SSM. The present technique provides bright, dark, singular, periodic singular, combined bright-dark and combined dark-singular soliton solutions.

Pure-cubic optical soliton perturbation with full nonlinearity by a new generalized approach

This work introduces a new generalized integration scheme to construct the pure-cubic optical solitons in a polarization-preserving fiber with Kerr law nonlinearity. First, Lie symmetry analysis is performed to find the similarity transformation. Then, the exact solutions of the governing equation are derived via a purposed new generalized integration scheme. In particular, we have derived the pure-cubic dark, singular and combo dark-singular soliton solutions. The existence criteria of such solutions are also provided.

Optical solitons of NLS-type differential equations by extended direct algebraic method

In this paper, we examine the optical soliton solutions of nonlinear partial differential equations belonging to the nonlinear Schrödinger (NLS) class which includes cubic focusing NLS equation and paraxial NLS equation in a Kerr media. We construct the optical soliton of these models using a new extended direct algebraic method (NEDAM). The resulting solutions carry a variety of new families including singular solutions of Types 1 and 2, dark, darkly bright and dark singular soliton solutions. Sufficient conditions are provided for these structures to exist. NEDAM is successfully applied to supposed nonlinear model and provides us some new and interesting solitary wave solutions. These solutions contain the trigonometric-, hyperbolic- and exponential-type functions. Moreover, to present the physical importance of the considered model, some 3D and 2D diagrams of obtained solutions are plotted by using Mathematica under the suitable choice of involving parameters values.

Complex and Real Optical Soliton Properties of the Paraxial Non-linear Schrödinger Equation in Kerr Media With M-Fractional

In this paper, we use the modified exponential function method in terms of 𝐾𝑓(𝑥) instead of 𝑒𝑓(𝑥) and the extended sinh-Gordon method to find some new family solution of the M-fractional paraxial nonlinear Schrodinger equation. The novel complex and real optical soliton solutions are plotted in 2-D, 3-D with a contour plot. Moreover, the dark exact solution, singular soliton solutions, kink-type soliton solution and periodic dark-singular soliton solutions for M-fractional paraxial nonlinear Schrodinger equation are constructed. We guarantee that all solutions are new and verified the main equation of the M-fractional paraxial wave equation. For existence, the constraint condition is also added.

Bright, dark and dark-singular soliton solutions of nonlinear Schrödinger's equation with spatio-temporal dispersion

ABSTRACTThis article presents an analytical study of the propagation of solitons through optical fibres. We consider a nonlinear Schrödinger equation with spatio-temporal dispersion and quadratic–cubic nonlinearity. Jacobi elliptic functions are used as an ansatz to extract optical dark and bright solitons as well as Jacobi elliptic solutions. The extended direct algebraic method gives dark and dark-singular soliton solutions. The constraint conditions which guarantee the existence of soliton solutions are listed.

Optical dark and dark-singular soliton solutions of (1+2)-dimensional chiral nonlinear Schrodinger’s equation

ABSTRACTIn this study, the dynamical analysis of optical dark and singular solitons is carried out for chiral (1+2)-dimensional nonlinear Schrödinger’s equation with the implementation of extended direct algebraic and extended trial equation method independently. The constraint conditions guarantee the perseverance of these soliton solutions. Along with optical dark and singular solitons, these integration techniques yield other wave solutions such as Jacobi elliptic function, rational function, and hyperbolic function solutions as outgrowth.

Solitons and other solutions to the nonlinear Bogoyavlenskii equations using the generalized Riccati equation mapping method

Recently, the authors of the article (Zahran and khater in Appl Math Model 40:1769–1775, 2016) have applied the modified extended tanh-function method to the nonlinear Bogoyavlenskii equations and found very few special solutions. In our article, we generalize this work by showing that the modified extended tanh-function method is just a special case of the generalized Riccati equation mapping method. Many families of exact solutions of the nonlinear Bogoyavlenskii equations have been found using the generalized Riccati equation mapping method. Bright–dark–singular soliton solutions and other solutions are obtained. Comparing our results with the well-known results are given.

Exact solutions and optical soliton solutions for the (2 + 1)-dimensional hyperbolic nonlinear Schrödinger equation

This paper studies the exact solutions with parameters and optical soliton solutions of the (2+1)-dimensional hyperbolic nonlinear Schrödinger equation which describes space–time evolutions of slowly varying envelopes. When these parameters are taken special values, the optical solitary wave solutions are derived from the exact solutions. There are some integration tools that are adopted to retrieve soliton solutions. They are the modified simple equation method, the exp-function method, the soliton ansatz method and other two Sub-ODE methods. Bright–dark-singular soliton solutions and some trigonometric function solutions are obtained along with their respective constraint conditions. We compare between the results yielding from these integration tools. A comparison between our results in this paper and the well-known results is also given.