Exploring novel soliton, wave solutions, and modulation instability analysis for the (3+1)-dimensional KP-SKR...

In this work, new soliton and wave solutions for the (3+1)-dimensional Kadomtsev-Petviashvili-Sawada-Kotera-Ramani equation (KP-SKRE) were derived using the improved generalized Riccati equation mapping method (IGREMM). The KP-SKRE plays an essential role in modeling the phenomena of wave nonlinearity. The wave nonlinearity appears in fluid dynamics, plasma physics, and other fields that involve interactions between dispersion and nonlinearity. The obtained solutions include bright soliton, combo bright-dark soliton, singular soliton in addition to singular periodic, hyperbolic, exponential, and rational solutions. The obtained solutions describe different behaviors of wave propagation through nonlinear media and provide a deeper understanding of the dynamics of nonlinear waves governed by the KP-SKRE. The novelty of this work includes four key original contributions to the study of the KP-SKRE. First, the novel application of IGREMM to the KP-SKRE. Second, the derivation of previously unreported exact solutions, including combo bright-dark soliton and singular soliton, along with hyperbolic, exponential, rational, and singular periodic solutions. Third, the Bäcklund transformation of IGREMM can be used to construct additional novel forms of solution. Fourth, the conduction of the modulation instability (MI) analysis by the linear stability technique. The MI analysis reveals the conditions under which the wave solutions become unstable. The direct substitution of the derived solutions in the original equation confirmed their validity. For visualization, 2D, 3D, and density plots of some selected exact solutions were presented. This visualization illustrates the solution wave profile with the choice of parameters' values that satisfy the constraints. Furthermore, graphical representations of the outcomes of MI analysis were provided to enhance the physical interpretation of the results. The findings of this study contribute to a better understanding of nonlinear wave dynamics. Potential applications of these findings can be found in oceanography, optical communications, and plasma physics where soliton and nonlinear wave solutions have significant importance.