New abundant solitary wave structures for a variety of some nonlinear models of surface wave propagation with their geometric interpretations

Abstract

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.

Exploring the Analytical Solutions to the Economical Model via Three Different Methods

“…Many mathematical models have been developed in many areas of sciences in the form of nonlinear partial diferential equations (NLPDEs). Numerous techniques are made to gain exact solutions of NLPDEs such as generalized exponential rational function scheme (GERFS) [1][2][3][4], (m + 1/G)-expansion and Adomian decomposition schemes [5], new generalized expansion method [6], simplest equation and Kudryashov's new function techniques [7], modifed simple equation scheme [8], modifed Kudryashov simple equation technique [9], frst integral technique [10], Bäcklund transformation scheme [11], extended jacobi elliptic function expansion technique [12], and extended (G/G)-expansion and improved (G ′ /G)-expansion schemes [13].…”

Optical solitons of the complex Ginzburg-Landau equation having dual power nonlinear form using $\varphi^{6}$-model expansion approach

“…As a result, a number of sophisticated mathematical techniques have been developed to generate soliton solutions for a wide range of physical models such as the Kadomtsev-Petviashvili equation [13], the Benjamin-Ono equation [14], the disturbance effect in intracellular calcium dynamic on fibroblast cells [15], the Fisher equation [16], the nonlinear Schrödinger equation [17,18], the Sharma-Tasso-Olver equation [19], the Murnaghan model [20], the Kaup-Kupershmidt equation [21], Navier-Stokes equation [22], the Zakharov-Kuznetsov equation [23], the B-type Kadomtsev-Petviashvili-Boussinesq equation [24] and others [25][26][27]. Recent analytical methods for solving PDEs, such as the eMETEM method [28], the generalized exponential rational function method [29], the extended sinh-Gordon equation expansion method [30], the q-homotopy analysis transform technique [31], the new extended direct algebraic method [32], the direct method [33], the Kudryashov's new function method [34], the split-step Fourier transform [35], the new modified unified auxiliary equation method [36], the 1 G ′ -expansion method [37][38][39], the Jacobi elliptic functions [40].…”

Construction of traveling wave solutions of the (2 + 1)‐dimensional modified KdV‐KP equation

“…Nonlinear evolution equations, such as Lotka-Volterra type competition and predator-prey models, are useful tools to study the dynamics of biological and ecological invasion. 1,2 Several analytical and numerical techniques have been successfully developed by diverse groups of mathematicians and physicists to deal with NLEEs, such as Hirota's bilinear transformation method, 3 the tanh-function method, 4 the Exp-function method, 5,6 the modified simple equation method, [7][8][9] the Kudryashov's new function method, 10 the G 0 =G ð Þ-expansion method, 11 the Riccati equation rational expansion method, 12 the generalized Kudryashov method, 12 the Jacobi elliptic function method, 13 the complex method, [14][15][16][17][18] various numerical approaches, 1,2,[19][20][21][22] and the references therein.…”