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The propagation of waves of water on the surface is characterized by various mathematical models. In this work, we adopt the simplest equation method as well as the Kudryashov's new function method, to a variety of some nonlinear models of surface wave propagation to extract their abundant solitary wave structures. These nonlinear models include the Benjamin‐Bona‐Mahony (BBM) equation, the Ostrovsky (OS) equation, the Ostrovsky‐Benjamin‐Bona‐Mahony (OS‐BBM) equation, and the Boussinesq system of equations. Based on the solutions of two distinct auxiliary equations, various wave solutions are successfully obtained. In addition, the geometric interpretations of the obtained solutions are analyzed. To illuminate the physical nature of the obtained solutions of this paper, some graphical representations (numerical simulations) of the acquired solutions have been depicted in two‐dimensional density, two‐ and three‐dimensional graphs. Furthermore, a comparison between the obtained results of this paper and the solutions for the considered models obtained in the literature is also provided. It is shown that the used approaches are very effective and powerful strategies for a wider class of nonlinear partial differential equations in physical problems. [ABSTRACT FROM AUTHOR]

A nonlinear fractional model arising in water waves theory, namely the new conformable space–time fractional (4+1)-dimensional Fokas equation, is explored via some recently developed approaches, namely, the functional variable, the generalized Kudryashov, the Jacobi elliptic function expansion and the generalized Riccati equation mapping methods. As a result, several (abundant) distinct types of new exact solutions are constructed with the aid of symbolic computation via Mathematica 9 in terms of solitary wave, periodic wave, Jacobi elliptic function and other solutions. Furthermore, some graphical representations (numerical simulations) have been provided to illuminate the physical behavior of the resultant solutions that may be very important in real world problems and applications with fractional order. [ABSTRACT FROM AUTHOR]

This paper secures dark and singular resonant optical solitons that is studied with dual-power law nonlinearity and fractional temporal evolution. Khalil's conformable fractional derivative is put into perspective to retrieve these soliton solutions. The parameter restrictions for the existence of such solitons are also indicated. [ABSTRACT FROM AUTHOR]

In this paper, two distinct methods are applied to look for exact traveling wave solutions of the (4+1)-dimensional nonlinear Fokas equation, namely the modified simple equation method (MSEM) and the extended simplest equation method (ESEM). Some new exact traveling wave solutions involving some parameters are obtained. The solitary wave solutions can be extracted by assigning special values of these parameters. The obtained solutions show the simplicity and efficiency of the used approaches that can be applied for nonlinear equations as well as linear ones. [ABSTRACT FROM AUTHOR]

In this paper, the modified simple equation method (MSEM) is applied to construct exact solutions of the generalized ( 2 + 1 )-dimensional nonlinear evolution equations (NLEEs) involving parameters via the ( 2 + 1 )-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation, the ( 2 + 1 )-dimensional breaking soliton equation and the ( 2 + 1 )-dimensional Bogoyavlenskii’s breaking soliton equation. The solitary wave solutions are derived from the exact solutions by assigning special values of the parameters. [ABSTRACT FROM AUTHOR]

This paper studies the propagation of the short pulse optics model governed by the higher-order nonlinear Schrödinger equation (NLSE) with non-Kerr nonlinearity. Exact one-soliton solutions are derived for a generalized case of the NLSE with the aid of software symbolic computations. The modified Kudryashov simple equation method (MSEM) is employed for this purpose under some parametric constraints. The computational work shows the difference, effectiveness, reliability, and power of the considered scheme. This method can treat several complex higher-order NLSEs that arise in mathematical physics. Graphical illustrations of some obtained solitons are presented. [ABSTRACT FROM AUTHOR]