A study on solitary wave solutions for the Zoomeron equation supported by two-dimensional dynamics

Abstract

This study emphasizes the importance of understanding natural phenomena through various observations and relating them to scientific studies. Nonlinear partial differential equations serve as fundamental tools for modeling these phenomena, with a focus on nonlinear evolution equations when involving time. This paper investigates the dynamics of the Zoomeron equation, highlighting not only a result derived from the KdV and Schrödinger equations but also its contribution to the modeling of Boomeron and Trappon solitons. Different analytical techniques yield various solutions for the Zoomeron equation, each offering unique insights. The behavior of the solutions generated in kink and singular soliton forms in various dynamics using the G ′ / G 2 -expansion method is investigated and compared under restrictive conditions, and the distribution of energy density is explained with the assistance of the gradient function. It is aimed at gaining a physically unique perspective by revealing the connections between the velocities of solitons produced for this equation and the effects of gradient flow directions. In addition, stability analysis for some of the solutions generated in the kink form is investigated. The structure of the paper includes method introduction, application, stability property, results and discussion, resulting in a unique perspective on understanding the physical dynamics of the Zoomeron model.

Complexiton, complex multiple kink soliton and the rational wave solutions to the generalized (3 + 1)-dimensional kadomtsev-petviashvili equation

“…In order to better understand and explain the nonlinear phenomena, finding exact solutions to the NPDEs has become an important focus of scholars' attention and research. In the past half century, mathematicians and physicists have been dedicated to studying exact solutions to NPDEs, including the Jacobi elliptic function expansion approach [12], modified generalized exponential rational function method [13], direct algebraic approach [14], (G'/G 2 )-expansion method [15], variational approach [16], trial-equation technique [17,18], Bäcklund transformation approach [19,20], subequation approach [21,22], Darboux transformation technique [23,24], exp-function approach [25], modified Kudryashov method [26], extended F-Expansion approach [27], sinh-Gordon equation expansion method [28] and so on. Although mathematical physicists have developed a large number of methods, it has been found that, due to the diversity and complexity of NPDEs, there is currently no unified method to solve them, and often only the corresponding methods can be selected based on specific equations.…”

Multi-lump, resonant Y-shape soliton, complex multi kink solitons and the solitary wave solutions to the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation for incompressible fluid

“…As is well known, a series of important principles such as the linear superposition principle of solutions no longer hold for NPDEs, so there is no universal method for solving the NPDEs. Although there is no universal and effective method for obtaining the exact solutions to NPDEs, several approaches for constructing the exact solutions for the NPDEs have been put forward in different applicable situations such as the extended F-expansion technique [1][2][3][4], Darboux transformation approach [5][6][7], general integral technique [8], unified Riccati equation approach [9], Bäcklund transformation [10][11][12][13], exp-function method [14][15][16][17], Subequation technique [18][19][20], tanh function method [21,22], (G′/G)-expansion approach [23,24] and many others [25][26][27][28][29]. In this exploration, we aim to give a study to the (2+1)-dimensional BLMPE, which reads as:…”