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The (3+1)-dimensional hyperbolic nonlinear Gardner–Kadomtsov–Petviashvili (GKP) equation is a mathematical model that describes the propagation of nonlinear waves in a four-dimensional space-time. The GKP equation is used as a mathematical tool for studying the dynamics of nonlinear waves in a variety of physical systems including fluid dynamics, nonlinear optics, plasma physics. The GKP equation is a nonlinear partial differential equation (PDE) with mixed spatial and temporal derivatives. It exhibits various interesting phenomena, such as soliton solutions, wave interactions and wave turbulence depending on the values of the conditions. The improved F-expansion technique and generalized Kudryashov technique are successfully employed to evaluate the soliton solutions of governing model. The GKP equation supports various types of soliton solutions, such as bright solitons and dark solitons, depending on the specific parameters. The study of solitons in the GKP equation provides insights into the stability, propagation, and interaction of nonlinear waves in higher-dimensional spaces. The nonlinear terms in the GKP equation are responsible for wave–wave interactions. When multiple waves co-exist in a medium, they can interact with each other, leading to complex phenomena. The nature of wave interactions in the GKP equation depends on the specific values of the coefficients. The GKP equation has many applications in various areas of physics and mathematics. It has been used to model nonlinear waves in several fields and Bose–Einstein condensates. [ABSTRACT FROM AUTHOR]

In this study, we solve by involving conformable fractional derivative on time-fractional perturbed nonlinear Schrödinger model which describe the behaviour of soliton transmission on fibers optical system in physics specially in transmission of data over long distances with large bandwidth. There are two modern techniques are utilized for solving the suggested model namely the Extended hyperbolic function technique and the polynomial expansion technique. These techniques give unique, robust, and powerful solutions that are useful in many research areas. We attain different types of solutions that give unique behaviour of V shaped, singular solution, and periodic soliton solutions. These solutions are play a vital role in the soliton theory along with data transmission in optical fibers. Further we also discuss the stability of obtain solutions as well as the modulation instability of the governing NLSE. To grasp and better understanding of the solution behavior we include the 3D, contour and 2D graphics. [ABSTRACT FROM AUTHOR]

In this study, the (2+1)-dimensional generalized Hirota–Satsuma–Ito model is studied extensively to investigate wave dynamics of shallow water. A set of state of the art analytical approaches are being employed to acquire several fascinating wave structures. We have established a comparison between exact and numerical solutions to determine the absolute error. Moreover, the modulation instability as well as the stability of the solutions are adequately discussed. The 3D, contour and 2D plots are displayed for several interesting exact solutions to understand their behaviour. These strategies are demonstrated as more strong, effective, and efficient in developing various new wave structures which have numerous applications in several mathematical physics domains. [ABSTRACT FROM AUTHOR]