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In this paper, we investigate and use the new modified exp ( − Ω ( ξ ) ) -expansion method, (MEM). We apply the new MEM to nonlinear long–short-wave interaction systems (NLSWIS). Among our findings are sets of solutions including, but not limited to, new hyperbolic, complex, and dark soliton solutions. Not only is MEM shown to be highly adaptable for partial differential equations with strong nonlinearities, but also, it turns out to be highly efficient, despite its ease. [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]

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]

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]

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]

Under investigation in this paper is the (2 + 1) -dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation. Based on bilinear method, the multiple rogue wave solutions and the novel multiple soliton solutions are constructed by giving some specific activation functions in the considered model. By means of symbolic computation, these analytical solutions and corresponding rogue waves are obtained, via Maple 18. By utilizing improved tan (Φ (ρ) ∕ 2) -expansion technique the series of novel exact solutions in terms of rational, periodic and hyperbolic functions for the fractional cases are derived. Also, the semi-inverse variational principle is offered to get the solitary solutions. We construct the exact lump and rogue wave solutions, by solving the under-determined nonlinear system of algebraic equations for the specified parameters. Via various three-dimensional plots, curve plots, density plots and contour plots, dynamical characteristics of these waves are represented. [ABSTRACT FROM AUTHOR]