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This paper investigates the new KP equation with variable coefficients of time ‘t’, broadly used to elucidate shallow water waves that arise in plasma physics, marine engineering, ocean physics, nonlinear sciences, and fluid dynamics. In 2020, Wazwaz [1] proposed two extensive KP equations with time-variable coefficients to obtain several soliton solutions and used Painlevé test to verify their integrability. In light of the research described above, we chose one of the integrated KP equations with time-variable coefficients to obtain multiple solitons, rogue waves, breather waves, lumps, and their interaction solutions relating to the suitable choice of time-dependent coefficients. For this KP equation, the multiple solitons and rogue waves up to fourth-order solutions, breather waves, and lump waves along with their interactions are achieved by employing Hirota's method. By taking advantage of Wolfram Mathematica, the time-dependent variable coefficient's effect on the newly established solutions can be observed through the three-dimensional wave profiles, corresponding contour plots. Some newly formed mathematical results and evolutionary dynamical behaviors of wave-wave interactions are shown in this work. The obtained results are often more advantageous for the analysis of shallow water waves in marine engineering, fluid dynamics, and dusty plasma, nonlinear sciences, and this approach has opened up a new way to explain the dynamical structures and properties of complex physical models. This study examines to be applicable in its influence on a wide-ranging class of nonlinear KP equations.

This research aims to investigate a generalized fifth-order nonlinear partial differential equation for the Sawada-Kotera (SK), Lax, and Caudrey-Dodd-Gibbon (CDG) equations to study the nonlinear wave phenomena in shallow water, ion-acoustic waves in plasma physics and other nonlinear sciences. The Painlev\'e analysis is used to determine the integrability of the equation, and the simplified Hirota technique is applied to construct multiple soliton solutions with an investigation of the dispersion relation and phase shift of the equation. We utilize a linear combination approach to construct a system of equations to obtain a general logarithmic transformation for dependent variable. We generate one-soliton, two-soliton, and three-soliton wave solutions using the simplified Hirota method and showcase the dynamics of these solutions graphically through interaction between one, two, and three solitons. We investigate the impact of the system's parameters on the solitons and periodic waves. The SK, Lax, and CDG equations have a wide range of applications in nonlinear dynamics, plasma physics, oceanography, soliton theory, fluid dynamics, and other sciences.

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In this work, we formulate a new generalized nonlinear KdV-type equation of fifth-order using the recursion operator. This equation generalizes the Sawada-Kotera equation and the Lax equation that study the vibrations in mechanical engineering, nonlinear waves in shallow water, and other sciences. To determine the integrability, we use Painlevé analysis and construct solutions for multiple solitons by employing the Hirota bilinear technique to the established equation. It produces a bilinear form for the driven equation and utilizes the Lagrange interpolation to create a dependent variable transformation. We construct the solutions for multiple solitons and show the graphics for these built solutions. The mathematical software program Mathematica employs symbolic computation to obtain the multiple solitons and various dynamical behaviors of the newly generated solutions. The Sawada-Kotera equation and Lax equation have various applications in mechanical engineering, plasma physics, nonlinear water waves, soliton theory, mathematical physics, and other nonlinear fields.