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This study reveals the dark, bright, combined dark–bright, singular optical solitons and other solutions to the conformable space–time fractional Fokas–Lenells equation. We use two integral schemes in reaching such solutions. The constraints conditions which guarantee the existence of valid solutions are stated. By choosing the suitable values of the parameters involved, we plot the 2-dimensional and 3-dimensional surfaces to some of the obtained solutions. [ABSTRACT FROM AUTHOR]

This study uses the extended sinh-Gordon equation expansion method in constructing various optical soliton solutions to the resonant nonlinear Schrödinger equation with both spatio-temporal and inter-modal dispersions. Resonant nonlinear Schrödinger equation expresses the propagation dynamics of optical solitons and Madelung fluids. Dark, bright, combined dark–bright and singular optical solitons are successfully constructed. Under the choice of suitable values of parameters, the two-dimensional, three-dimensional and the contour graphs to some of the acquired results are plotted. The reported results may be useful in explaining the physical meaning of the studied model. [ABSTRACT FROM AUTHOR]

This paper constructs dark, bright, combined dark-bright, singular, combined singular soliton optical and singular periodic solutions to the conformable space-time fractional second order nonlinear Schrödinger equation with group velocity dispersion coefficient and spatiotemporal dispersion by using the extended sinh-Gordon equation expansion method. The constraint conditions for the existence of valid soliton solutions are given. The reported results in this study may be helpful in explaining the physical meaning of the studied model. [ABSTRACT FROM AUTHOR]

In this article, we employ the sine-Gordon expansion method to construct a new solitary and optical wave structures to the (1 + 1)-dimensional combined KdV–mKdV equation. Sine-Gordon expansion method is developed based on the travelling wave transformation u ( x , y ) = U ( ξ ), ξ = μ ( x − ct ) and the sine-Gordon equation u xx − u tt = m 2 sin( u ). We find the travelling wave solutions to (1 + 1)-dimensional combined KdV–mKdV equation in the form of complex, hyperbolic, trigonometric and rational function. Our results show that all the travelling wave solutions we obtain are solitary and optical wave structures solutions. All the computations and graphics plot in this article are carried out with the help of Wolfram Mathematica 9. [ABSTRACT FROM AUTHOR]

In this study, we apply the modified exp ( - Ω ( ξ ) ) - expansion function method to the new coupled Konno-Oono equation. We obtain some new travelling wave solutions such as complex function, hyperbolic function and rational function solutions. We observe that all travelling wave solutions have been verified the new coupled Konno-Oono equation by using Wolfram Mathematica 9. Then, we plot the two- and three-dimensional surfaces for all travelling wave solutions obtained in this paper by the same computer program. [ABSTRACT FROM AUTHOR]

This study reveals the dark, bright, mixed dark-bright, singular and mixed singular optical solitons to the space-time fractional (1+1)-dimensional coupled nonlinear Schrödinger equation by using the extended sinh-Gordon equation expansion method. The constraint conditions for the existence of valid solitons are given. Under the choice of suitable values of the parameters and the fractional values of α and β , the 2- and 3-dimensional graphs to some of the reported solutions are plotted. [ABSTRACT FROM AUTHOR]

In this study, we consider the t a n ( F ( ξ ) 2 ) -expansion methodfor the traveling wave solutions of the potential KdV equation (pKdV)and the (3 + 1)-dimensional shallow water wave equation (SWWE). We find some traveling wave solutions such as trigonometric function, hyperbolic function, exponential function solutions. Then, we showthe two and three dimensional surfaces for some traveling wave solutions obtained in this study by the Mathematica. [ABSTRACT FROM AUTHOR]

In this paper, we seek for the new soliton solutions to the coupled nonlinear Maccari's system by using sine-Gordon expansion method. Sine-Gordon expansion method is based on sine-Gordon Eq. (2.1) and a travelling wave transformation. We obtain new complex hyperbolic functions solutions to this system with help of Wolfram Mathematica 9, we also plot 3-dimensional and 2-dimensional graphics to some obtained solution using the same program in Wolfram Mathematica 9. Finally, we submit a comprehensive conclusions. [ABSTRACT FROM AUTHOR]

This study is related to new soliton solutions compound KdV-Burgers equation, generalized KdV equation and (2+1) dimensional Broer–Kaup equations with variable coefficients. The generalized Kudryashov method (GKM) has been tackled for procuring exact solutions of these equations. By using this method, dark and bright soliton solutions of these equations have been found. Also, some graphical representations have been given to observe the structure of these solutions. [ABSTRACT FROM AUTHOR]

In this paper, we apply auto-Bäcklund transformation for Sharma-Tasso-Olver (STO) equation and fourth order equation of the Burgers hierarchy. Auto-Bäcklund transformation was developed as a direct and simple method to obtain solutions of nonlinear partial differential equations by Fan.We obtain solitary wave solutions of these equations.The numerical simulations are added to these obtained solutions [ABSTRACT FROM AUTHOR]

Obtaining new and important travelling wave solutions of wave propagation modelling of waves such as nonlinear optics models, propagation and transmission models of waves by using different methods plays an important role in maritime engineering, ocean, beach science and floating structures, and also for understanding new physical meanings of coastal structural properties. New complex travelling wave solutions obtained by using different newly improved methods explain new and general properties of nonlinear optic structures, wave propagations and motions, major structures of the impact of environmental factors on the beach like tsunami, the impact of the waves on the vessel, the power of the effects of the waves on wave distribution panels. These solutions structures may be trigonometric, complex function, hyperbolic function, exponential and rational function. In this study, we apply two effective methods to the nonlinear evolution equation used to describe the new versions of different mathematical models for wave motion and propagations. The first is improved Bernoulli sub-equation function method (IBSEFM), the latter is modified exp ( - Ω ( ξ ) ) -expansion function method (MEFM). We obtain some new travelling wave structures such as complex function, hyperbolic function and rational function, exponential function solutions. We observe that all travelling wave solutions have been verified the nonlinear partial differential equation by using Wolfram Mathematica 9. Then, we plot the two and three dimensional surfaces for all travelling wave structures obtained in this paper by the same computer program. [ABSTRACT FROM AUTHOR]

Modelling of wave motion and propagation characteristics of waves plays an important role in coastal, ocean and maritime engineering, and for most geographical areas, waves are the major source of environmental actions on beaches or on man-made fixed or floating structures. The travelling wave solutions are described by the hyperbolic functions, the trigonometric functions and the rational functions. In this paper, exact travelling wave solutions of Hirota equation and Hirota–Maccari system are successfully surveyed. Extended trial equation method and generalized Kudryashov method are applied to gain some exact travelling wave solutions of Hirota equation and Hirota–Maccari system. Firstly, we find some exact travelling wave solutions of Hirota equation and Hirota–Maccari system with the aid of extended trial equation method. Secondly, we obtain dark soliton solutions of Hirota equation and Hirota–Maccari system by means of generalized Kudryashov method. Finally, for certain parameter values, we plot two and three dimensional graphics of imaginary and real values of some solutions that we obtained by using these methods. [ABSTRACT FROM AUTHOR]