Analytical soliton solutions of the time fractional Kadomtsev-Petviashvili (KP) equation using sine – cosine and tanh-coth methods in the sense of conformable fractional derivative

The time-fractional Kadomtsev–Petviashvili (TF-KP) equation is one of the essential equations that describe the propagation of weakly nonlinear, weakly dispersive waves in media with transverse perturbations in mathematical physics. Physical Applications include Shallow Water Hydrodynamics, Nonlinear optics, and ion-acoustic plasma waves. The use of a fractional-order time derivative replaces the classical integer-order time derivative, allowing the description of memory effects, anomalous diffusion, and non-local dynamics, which are not captured by standard calculus. Encouraged by the analytical tractability and physically relevant properties of the conformable fractional derivative, this work focuses on the TF-KP equation, exploring exact soliton solutions within this framework. Two systematic analytical methods, namely the Sine–Cosine method and the Tanh–Coth method, are applied. By first transforming the governing equation into a nonlinear ordinary differential equation and then solving the nonlinear equation by each method through algebraic balancing and symbolic computation, we use the fractional travelling-wave transformation to solve a nonlinear ordinary differential equation. We use the fractional travelling-wave transformation to solve the nonlinear ordinary differential equation, which is first transformed into a nonlinear ordinary differential equation by the fractional travelling-wave transformation and then solved by each method through algebraic balancing and symbolic computation. A rich family of exact solutions is obtained, such as bright solitons, periodic wave solutions, kink-type travelling waves, V-shaped line solitons, and singular solutions. The effect of the fractional order parameter is systematically investigated using three-dimensional surface plots: As the fractional order parameter α decreases, the wave profiles become broader, smoother, and more stable, and these properties are due to memory effects, while letting α = 1 restores the classical KP wave profiles with sharp and high-amplitude features. Both approaches provide complementary and, in some instances, different families of solutions, and are useful and powerful in their own right. The present study adds to the vast collection of research on the analysis of the nonlinear fractional differential equations and is the foundation for further studies on variable-order or coupled fractional systems.